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+# STRUCTURED ADVERSARIAL ATTACK: TOWARDS GENERAL IMPLEMENTATION AND BETTER INTERPRETABILITY
+
+Kaidi $\mathbf { X } \mathbf { u } ^ { 1 * }$ Sijia $\mathbf { L i u ^ { 2 * } }$ Pu Zhao1 Pin-Yu Chen2 Huan Zhang3 Quanfu Fan2   
+Deniz Erdogmus1 Yanzhi Wang1 Xue Lin1   
+1Northeastern University, USA   
+2MIT-IBM Watson AI Lab, IBM Research, USA   
+3University of California, Los Angeles, USA
+
+# ABSTRACT
+
+When generating adversarial examples to attack deep neural networks (DNNs), $\ell _ { p }$ norm of the added perturbation is usually used to measure the similarity between original image and adversarial example. However, such adversarial attacks perturbing the raw input spaces may fail to capture structural information hidden in the input. This work develops a more general attack model, i.e., the structured attack (StrAttack), which explores group sparsity in adversarial perturbations by sliding a mask through images aiming for extracting key spatial structures. An ADMM (alternating direction method of multipliers)-based framework is proposed that can split the original problem into a sequence of analytically solvable subproblems and can be generalized to implement other attacking methods. Strong group sparsity is achieved in adversarial perturbations even with the same level of $\ell _ { p }$ -norm distortion $( p \in \{ 1 , 2 , \infty \} )$ as the stateof-the-art attacks. We demonstrate the effectiveness of StrAttack by extensive experimental results on MNIST, CIFAR-10 and ImageNet. We also show that StrAttack provides better interpretability (i.e., better correspondence with discriminative image regions) through adversarial saliency map (Papernot et al., 2016b) and class activation map (Zhou et al., 2016). Our code is available at https://github.com/KaidiXu/StrAttack.
+
+# 1 INTRODUCTION
+
+Deep learning achieves exceptional successes in domains such as image recognition (He et al., 2016; Geifman & ElYaniv, 2017), natural language processing (Hinton et al., 2012; Harwath et al., 2016), medical diagnostics (Chen et al., 2016; Shi et al., 2018) and advanced control (Silver et al., 2016; Fu et al., 2017). Recent studies (Szegedy et al., 2013; Goodfellow et al., 2014; Nguyen et al., 2015; Kurakin et al., 2016; Carlini & Wagner, 2017) show that DNNs are vulnerable to adversarial attacks implemented by generating adversarial examples, i.e., adding well-designed perturbations to original legal inputs. Delicately crafted adversarial examples can mislead a DNN to recognize them as any target image label, while the perturbations appears unnoticeable to human eyes. Adversarial attacks against
+
+![](images/59a8c6dcdd6028fc60d9a97f6c39a64c4d378dfa203f33309f9fc9d76ca96306.jpg)  
+Figure 1: Group sparsity demonstrated in adversarial perturbations obtained by C&W attack and our StrAttack, where ‘ostrich’ is the original label, and ‘unicycle’ is the misclassified label. Here each group is a region of $1 3 \times 1 3 \times 3$ pixels and the strength of adversarial perturbations (through their $\ell _ { 2 }$ norm) at each group is represented by heatmap. C&W attack perturbs almost all groups, while StrAttack yields strong group sparsity, with more semantic structure: the perturbed image region matches the feature of the target object, namely, the frame of the unicycle.
+
+DNNs not only exist in theoretical models
+
+but also pose potential security threats to the real world (Kurakin et al., 2016; Evtimov et al., 2017; Papernot et al., 2017). Several explanations are proposed to illustrate why there exist adversarial examples to DNNs based on hypotheses such as model linearity and data manifold (Goodfellow et al., 2014; Gilmer et al., 2018). However, little is known to their origins, and convincing explanations remain to be explored.
+
+Besides achieving the goal of (targeted) mis-classification, an adversarial example should be as “similar” to the original legal input as possible to be stealthy. Currently, the similarity is measured by the $\ell _ { p }$ norm $( p = 0 , 1 , 2 , \infty )$ of the added perturbation (Szegedy et al., 2013; Carlini & Wagner, 2017; Chen et al., 2017b;a), i.e., $\ell _ { p }$ norm is being minimized when generating adversarial example. However, measuring the similarity between the original image and its adversarial example by $\ell _ { p }$ norm is neither necessary nor sufficient (Sharif et al., 2018). Besides, no single measure can be perfect for human perceptual similarity (Carlini & Wagner, 2017) and such adversarial attacks may fail to capture key information hidden in the input such as spatial structure or distribution. Spurred by that, this work implements a new attack model i.e., structured attack (StrAttack) that imposes group sparsity on adversarial perturbations by extracting structures from the inputs. As shown in Fig. 1, we find that StrAttack identifies minimally sufficient regions that make attacks successful, but without incurring extra pixel-level perturbation power. The major contributions are summarized as below.
+
+• (Structure-driven attack) This work is the first attempt towards exploring group-wise sparse structures when implementing adversarial attacks, but without losing $\ell _ { p }$ distortion performance when compared to state-of-the-art attacking methods. (Generality) We show that the proposed attack model covers many norm-ball based attacks such as C&W (Carlini & Wagner, 2017) and EAD (Chen et al., 2017a). (Efficient implementation) We develop an efficient algorithm to generate structured adversarial perturbations by leveraging the alternating direction method of multipliers (ADMM). We show that ADMM splits the original complex problem into subproblems, each of which can be solved analytically. Besides, we show that ADMM can further be used to refine an arbitrary adversarial attack under the fixed sparse structure. (Interpretability) The generated adversarial perturbations demonstrate clear correlations and interpretations between original and target images. With the aid of adversarial saliency map (Papernot et al., 2016b) and class activation map (Zhou et al., 2016), we show that the obtained group-sparse adversarial patterns better shed light on the mechanisms of adversarial perturbations to fool DNNs.
+
+Related work Many works studied norm-ball constrained adversarial attacks. For example, FGM (Goodfellow et al., 2014) and IFGSM (Kurakin et al., 2017) attack methods were proposed to maximize the classification error subject to $\ell _ { \infty }$ -norm based distortion constraints. Moreover, L-BFGS (Szegedy et al., 2013) and C&W (Carlini & Wagner, 2017) attacks found an adversarial example by minimizing its $\ell _ { 2 }$ -norm distortion. By contrast, JSMA (Papernot et al., 2016b) and one-pixel (Su et al., 2017) attacks attempted to generate adversarial examples by perturbing the minimum number of pixels, namely, minimizing the $\ell _ { 0 }$ norm of adversarial perturbations. Different from the above norm-ball constrained attacks, some works (Karmon et al., 2018; Brown et al., 2017) crafted adversarial examples by adding noise patches. However, the resulting adversarial perturbations are no longer imperceptible to humans. Here we argue that imperceptibility could be important since it helps us to understand how/why DNNs are vulnerable to adversarial attacks while perturbing natural examples just by indistinguished adversarial noise.
+
+In the aforementioned norm-ball constrained adversarial attacks, two extremely opposite principles have been applied: C&W attack (or $\ell _ { \infty }$ attacks) seeks the minimum image-level distortion but allows to modify all pixels; one-pixel attack only perturbs a few pixels but suffers a high pixel-level distortion. Both attacking principles might lead to a high noise visibility due to perturbing too many pixels or perturbing a few pixels too much. In this work, we wonder if there exists a more effective attack that can be as successful as existing attacks but achieves a tradeoff between the perturbation power and the number of perturbed pixels. We will show that the proposed StrAttack is able to identify sparse perturbed regions that make attacks successful, but without incurring extra pixel-level perturbations. It is also worth mentioning that one-pixel attack has much lower attack success rate on ImageNet than C&W attack and StrAttack.
+
+In addition to adversarial attacks, many defense works have been proposed. Examples include defensive distillation (Papernot et al., 2016c) that distills the original DNN and introduces temperature into the softmax layer, random mask (Anonymous, 2019) that modifies the DNN structures by randomly removing certain neurons before training, adversarial training through enlarging the training dataset with adversarial examples, and robust adversarial training (Madry et al., 2017; Sinha et al., 2018) through the min-max optimization. It is commonly known that the robust adversarial training method ensures the strongest defense performance against adversarial attacks on MNIST and CIFAR-10. In this work, we will evaluate the effectiveness of StrAttack to three defense methods, a) defensive distillation (Papernot et al., 2016c), b) adversarial training via data augmentation (Tramèr et al., 2018) and c) robust adversarial training (Madry et al., 2017).
+
+Although the adversarial attack and defense have attracted an increasing amount of attention, the visual explanation on adversarial perturbations is less explored since the distortion power is minimized and the resulting adversarial effects become imperceptible to humans. The work (Dong et al., 2017) attempted to understand how the internal representations of DNNs are affected by adversarial examples. However, only an ensemble-based attack was considered, which fails to distinguish the effectiveness of different norm-ball constrained adversarial attacks. Unlike (Dong et al., 2017), we employ the interpretability tools, adversarial saliency map (ASM) (Papernot et al., 2016b) and class activation map (CAM) (Zhou et al., 2016) to measure the effectiveness of different attacks in terms of their interpretability. Here ASM provides sensitivity analysis for pixel-level perturbation’s impact on label classification, and CAM localizes class-specific image discriminative regions (Xiao et al., 2018). We will show that the sparse adversarial pattern obtained by StrAttack offers a great interpretability through ASM and CAM compared with other norm-ball constrained attacks.
+
+# 2 STRUCTURED ATTACK: EXPLORE GROUP STRUCTURES FROM IMAGES
+
+In the section, we introduce the concept of StrAttack, motivated by the question: ‘what possible structures could adversarial perturbations have to fool DNNs?’ Our idea is to divide an image into sub-groups of pixels and then penalize the corresponding group-wise sparsity. The resulting sparse groups encode minimally sufficient adversarial effects on local structures of natural images.
+
+Let $\pmb { \Delta } \in \mathbb { R } ^ { W \times H \times C }$ be an adversarial perturbation added to an original image $\mathbf { X } _ { 0 }$ , where $W \times H$ gives the spatial region, and $C$ is the depth, e.g., $C = 3$ for RGB images. To characterize the local structures of $\pmb { \Delta }$ , we introduce a sliding mask $\mathcal { M }$ with stride $S$ and size $r \times r \times C$ . When $S = 1$ , the mask moves one pixel at a time; When $S = 2$ , the mask jumps 2 pixels at a time while sliding. By adjusting the stride $S$ and the mask size $r$ , different group splitting schemes can be obtained. If $S \ < \ r$ , the resulting groups will contain overlapping pixels. By contrast, groups will become non-overlapped when $S = r$ .
+
+A sliding mask $\mathcal { M }$ finally divides $\pmb { \Delta }$ into a set of groups $\{ \Delta _ { { \mathcal G } _ { p , q } } \}$ for $p \in [ P ]$ and $q \in [ Q ]$ , where $P = ( W - r ) / S + 1$ , $Q = ( H - r ) / S + 1$ , and $[ n ]$ denotes the integer set $\{ 1 , 2 , \ldots , n \}$ . Given the groups $\{ \Delta _ { { \mathcal G } _ { p , q } } \}$ , the group sparsity can be characterized through the following sparsity-inducing function (Yuan $\&$ Lin, 2006; Bach et al., 2012; Liu et al., 2015), motivated by the problem of group Lasso (Yuan & Lin, 2006):
+
+$$
+\begin{array} { r } { g ( \Delta ) = \sum _ { p = 1 } ^ { P } \sum _ { q = 1 } ^ { Q } \| \Delta { \mathcal { G } } _ { p , q } \| _ { 2 } , } \end{array}
+$$
+
+where $\Delta _ { { \mathcal G } _ { p , q } }$ denotes the set of pixels of $\pmb { \Delta }$ indexed by $\mathcal { G } _ { p , q }$ , and $\| \cdot \| _ { 2 }$ is the $\ell _ { 2 }$ norm. We refer readers to Fig. A1 for an illustrative example of our concepts on groups and group sparsity.
+
+# 3 STRUCTURED ADVERSARIAL ATTACK WITH ADMM
+
+In this section, we start by proposing a general framework to generate prediction-evasive adversarial examples, where the adversary relies only on gradients of the loss function with respect to inputs of DNNs. Our model takes into account both commonly-used adversarial distortion metrics and the proposed group-sparsity regularization that encodes spatial structures in attacks. We show that the process of generating structured adversarial examples leads to an optimization problem that is difficult to solve using the existing optimizers Adam (for C&W attack) and FISTA (for EAD attack) (Carlini & Wagner, 2017; Chen et al., 2017a). To circumvent this challenge, we develop an efficient optimization method via alternating direction method of multipliers (ADMM).
+
+Given an original image $\mathbf { x } _ { 0 } \in \mathbb { R } ^ { n }$ , we aim to design the optimal adversarial perturbation $\pmb { \delta } \in \mathbb { R } ^ { n }$ so that the adversarial example $( \mathbf { x } _ { 0 } + \pmb { \delta } )$ misleads DNNs trained on natural images. Throughout this paper, we use vector representations of the adversarial perturbation $\pmb { \Delta }$ and the original image $\mathbf { X } _ { 0 }$ without loss of generality. A well designed perturbation $\pmb { \delta }$ can be obtained by solving optimization problems of the following form,
+
+$$
+\begin{array} { r l } { \underset { \delta } { \mathrm { m i n i m i z e } } } & { \ f ( \mathbf { x } _ { 0 } + \pmb { \delta } , t ) + \gamma D ( \pmb { \delta } ) + \tau g ( \pmb { \delta } ) } \\ { \mathrm { s u b j e c t ~ t o } } & { \ ( \mathbf { x } _ { 0 } + \pmb { \delta } ) \in [ 0 , 1 ] ^ { n } , \ \lVert \pmb { \delta } \rVert _ { \infty } \leq \epsilon , } \end{array}
+$$
+
+where $f ( \mathbf { x } , t )$ denotes the loss function for crafting adversarial example given a target class $t$ , $D ( \delta )$ is a distortion function that controls the perceptual similarity between a natural image and a perturbed image, $\begin{array} { r } { g ( \delta ) = \sum _ { p = 1 } ^ { P } \sum _ { q = 1 } ^ { Q } \| \delta _ { { \mathcal G } _ { p , q } } \| _ { 2 } } \end{array}$ is given by (1), and $\| \cdot \| _ { p }$ signifies the $\ell _ { p }$ norm. In problem (2), the ‘hard’ constraints ensure the validness of created adversarial examples with $\epsilon$ - tolerant perturbed pixel values. And the non-negative regularization parameters $\gamma$ and $\tau$ place our emphasis on the distortion of an adversarial example (to an original image) and group sparsity of adversarial perturbation. Tuning the regularization parameters will be discussed in Appendix F.
+
+Problem (2) gives a quite general formulation for design of adversarial examples. If we remove the group-sparsity regularizer $g ( \delta )$ and the $\ell _ { \infty }$ constraint, problem (2) becomes the same as the C&W attack (Carlini & Wagner, 2017). More specifically, if we further set the distortion function $D ( \delta )$ to the form of $\ell _ { 0 }$ , $\ell _ { 2 }$ or $\ell _ { \infty }$ norm, then we obtain C&W $\ell _ { 0 }$ , $\ell _ { 2 }$ or $\ell _ { \infty }$ attack. If $D ( \delta )$ is specified by the elastic-net regularizer, then problem (2) becomes the formulation of EAD attack (Chen et al., 2017a).
+
+In this paper, we specify the loss function of problem (2) as below, which yields the best known performance of adversaries (Carlini & Wagner, 2017),
+
+$$
+f ( \mathbf { x } _ { 0 } + \pmb { \delta } , t ) = c \cdot \operatorname* { m a x } \{ \operatorname* { m a x } _ { j \neq t } Z ( \mathbf { x } _ { 0 } + \pmb { \delta } ) _ { j } - Z ( \mathbf { x } _ { 0 } + \pmb { \delta } ) _ { t } , - \kappa \} ,
+$$
+
+where $Z ( \mathbf { x } ) _ { j }$ is the $j$ th element of logits $Z ( \mathbf { x } )$ , representing the output before the last softmax layer in DNNs, and $\kappa$ is a confidence parameter that is usually set to zero if the attack transferability is not much cared. We choose $D ( \delta ) \overset { \cdot } { = } \lVert \delta \rVert _ { 2 } ^ { 2 }$ for a fair comparison with the $\mathrm { C } \& \mathbf { W } \ \ell _ { 2 }$ adversarial attack. In this section, we assume that $\{ \mathcal { G } _ { p , q } \}$ are non-overlapping groups, i.e., $\mathcal { G } _ { p , q } \cap \mathcal { G } _ { p ^ { \prime } , q ^ { \prime } } = \emptyset$ for $q \neq q ^ { \prime }$ or $p \neq p ^ { \prime }$ . The overlapping case will be studied in the next section.
+
+The presence of multiple non-smooth regularizers and ‘hard’ constraints make the existing optimizers Adam and FISTA (Carlini & Wagner, 2017; Chen et al., $2 0 1 7 \mathrm { a }$ ; Kingma & Ba, 2015; Beck & Teboulle, 2009) inefficient for solving problem (2). First, the subgradient of the objective function of problem (2) is difficult to obtain especially when $\{ \mathcal { G } _ { p , q } \}$ are overlapping groups. Second, it is impossible to compute the proximal operations required for FISTA with respect to all non-smooth regularizers and ‘hard’ constraints. Different from the existing work, we show that ADMM, a firstorder operator splitting method, helps us to split the original complex problem (2) into a sequence of subproblems, each of which can be solved analytically.
+
+We reformulate problem (2) in a way that lends itself to the application of ADMM,
+
+$$
+\begin{array} { r l } { \underset { \delta , { \mathbf z } , { \mathbf w } , { \mathbf y } } { \mathrm { m i n i m i z e } } } & { \ f ( \mathbf z + \mathbf x _ { 0 } ) + \gamma D ( \delta ) + \tau \sum _ { i = 1 } ^ { P Q } \| \mathbf y _ { \mathcal D _ { i } } \| _ { 2 } + h ( \mathbf w ) } \\ { \mathrm { s u b j e c t ~ t o } } & { \ \mathbf z = \delta , \ \mathbf z = \mathbf y , \ \mathbf z = \mathbf w , } \end{array}
+$$
+
+where $\mathbf { z } , \mathbf { y }$ and w are newly introduced variables, for ease of notation let $\mathcal { D } _ { ( q - 1 ) P + p } = \mathcal { G } _ { p , q }$ , and $h ( \mathbf { w } )$ is an indicator function with respect to the constraints of problem (2),
+
+$$
+h ( \mathbf { w } ) = { \left\{ \begin{array} { l l } { 0 } & { { \mathrm { ~ i f ~ } } ( \mathbf { x } _ { 0 } + \mathbf { w } ) \in [ 0 , 1 ] ^ { n } , \ \| \mathbf { w } \| _ { \infty } \leq \epsilon , } \\ { \infty } & { { \mathrm { ~ o t h e r w i s e . } } } \end{array} \right. }
+$$
+
+ADMM is performed by minimizing the augmented Lagrangian of problem (4),
+
+$$
+\begin{array} { r l r } & { } & { L ( { \bf z } , \delta , { \bf y } , { \bf w } , { \bf u } , { \bf v } , { \bf s } ) = f ( { \bf z } + { \bf x } _ { 0 } ) + \gamma D ( \delta ) + \tau \sum _ { i = 1 } ^ { P Q } \| { \bf y } { \boldsymbol { \pi } } _ { i } \| _ { 2 } + h ( { \bf w } ) + { \bf u } ^ { T } ( \delta - { \bf z } ) } \\ & { } & { + { \bf v } ^ { T } ( { \bf y } - { \bf z } ) + { \bf s } ^ { T } ( { \bf w } - { \bf z } ) + \frac { \rho } { 2 } \| \delta - { \bf z } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| { \bf y } - { \bf z } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| { \bf w } - { \bf z } \| _ { 2 } ^ { 2 } , \quad } \end{array}
+$$
+
+where u, v and s are Lagrangian multipliers, and $\rho > 0$ is a given penalty parameter. ADMM splits all of optimization variables into two blocks and adopts the following iterative scheme,
+
+$$
+\begin{array} { r l } & { \{ \delta ^ { k + 1 } , \mathbf w ^ { k + 1 } , \mathbf y ^ { k + 1 } \} = \underset { \delta , \mathbf w , \mathbf y } { \operatorname* { a r g m i n } } L ( \delta , \mathbf z ^ { k } , \mathbf w , \mathbf y , \mathbf u ^ { k } , \mathbf v ^ { k } , \mathbf s ^ { k } ) , } \\ & { } \\ & { \mathbf z ^ { k + 1 } = \underset { \mathbf z } { \operatorname* { a r g m i n } } L ( \delta ^ { k + 1 } , \mathbf z , \mathbf w ^ { k + 1 } , \mathbf y ^ { k + 1 } , \mathbf u ^ { k } , \mathbf v ^ { k } , \mathbf s ^ { k } ) , } \\ & { \left\{ \begin{array} { l l } { \mathbf u ^ { k + 1 } = \mathbf u ^ { k } + \rho ( \delta ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , } \\ { \mathbf v ^ { k + 1 } = \mathbf v ^ { k } + \rho ( \mathbf y ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , } \\ { \mathbf s ^ { k + 1 } = \mathbf s ^ { k } + \rho ( \mathbf w ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , } \end{array} \right. } \end{array}
+$$
+
+where $k$ is the iteration index, steps (7)-(8) are used for updating primal variables, and the last step (9) is known as the dual update step. We emphasize that the crucial property of the proposed ADMM approach is that, as we demonstrate in Proposition 1, the solution to problem (7) can be found in parallel and exactly.
+
+Proposition 1 When $D ( \delta ) = \| \delta \| _ { 2 } ^ { 2 }$ , the solution to problem (7) is given by
+
+$$
+\begin{array} { r l } & { \delta ^ { k + 1 } = \frac { \rho } { \rho + 2 \gamma } \mathbf { a } , } \\ & { [ \mathbf { w } ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { \operatorname* { m i n } \{ 1 - [ \mathbf { x } _ { 0 } ] _ { i } , \epsilon \} } & { b _ { i } > \operatorname* { m i n } \{ 1 - [ \mathbf { x } _ { 0 } ] _ { i } , \epsilon \} } \\ { \operatorname* { m a x } \{ - [ \mathbf { x } _ { 0 } ] _ { i } , - \epsilon \} } & { b _ { i } < \operatorname* { m a x } \{ - [ \mathbf { x } _ { 0 } ] _ { i } , - \epsilon \} \quad f o r \ i \in [ n ] , } \\ { b _ { i } } & { o t h e r w i s e , } \end{array} \right. } \\ & { [ \mathbf { y } ^ { k + 1 } ] _ { \mathcal { D } _ { i } } = \Big ( 1 - \frac { \tau } { \rho \| [ \mathbf { c } ] _ { \mathcal { D } _ { i } } \| _ { 2 } } \Big ) _ { + } [ \mathbf { c } ] _ { \mathcal { D } _ { i } } , \ i \in [ P Q ] , } \end{array}
+$$
+
+where $\mathbf { a } : = \mathbf { z } ^ { k } - \mathbf { u } ^ { k } / \rho , \mathbf { b } : = \mathbf { z } ^ { k } - \mathbf { s } ^ { k } / \rho , \mathbf { c } : = \mathbf { z } ^ { k } - \mathbf { v } ^ { k } / \rho , ( x ) _ { + } = x$ if $x \geq 0$ and 0 otherwise, $[ \mathbf { x } ] _ { i }$ denotes the ith element of $\mathbf { x }$ , and $[ \mathbf { x } ] _ { \mathcal { D } _ { i } }$ denotes the sub-vector of $\mathbf { x }$ indexed by $\mathcal { D } _ { i }$ .
+
+Proof: See Appendix B.
+
+It is clear from Proposition 1 that introducing auxiliary variables does not increase the computational complexity of ADMM since (10)-(12) can be solved in parallel. Moreover, if another distortion metric (different from $D ( \delta ) = \lvert \lvert \delta \rvert \rvert _ { 2 } ^ { 2 } )$ is used, then ADMM only changes at the $\delta$ -step (10).
+
+We next focus on the $\mathbf { z }$ -minimization step (8), which can be equivalently transformed into
+
+$$
+\operatorname* { m i n i m i z e } _ { \mathbf { z } } \quad f ( \mathbf { x } _ { 0 } + \mathbf { z } ) + \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { b } ^ { \prime } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { c } ^ { \prime } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\mathbf { a } ^ { \prime } : = \delta ^ { k + 1 } + \mathbf { u } ^ { k } / \rho$ , $\mathbf b ^ { \prime } : = \mathbf w ^ { k + 1 } + \mathbf s ^ { k } / \rho$ , and $\mathbf { c } ^ { \prime } : = \mathbf { y } ^ { k + 1 } + \mathbf { v } ^ { k } / \rho$ . We recall that attacks studied in this paper belongs to ‘first-order’ adversaries (Madry et al., 2017), which only have access to gradients of the loss function $f$ . Spurred by that, we solve problem (13) via a linearization technique that is commonly used in stochastic/online ADMM (Ouyang et al., 2013; Suzuki, 2013; Liu et al., 2018) or linearized ADMM (Boyd et al., 2011; Liu et al., 2017). Specifically, we replace the function $f$ with its first-order Taylor expansion at the point $\mathbf { z } ^ { k }$ by adding a Bregman divergence term $( \eta _ { k } / 2 ) \lvert | \mathbf { z } - \mathbf { z } ^ { k } \rvert | _ { 2 } ^ { 2 }$ . As a result, problem (13) becomes
+
+$$
+\begin{array} { r l } { \underset { \mathbf { z } } { \mathrm { m i n i m i z e } } } & { ( \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) ) ^ { T } ( \mathbf { z } - \mathbf { z } ^ { k } ) + \displaystyle \frac { \eta _ { k } } { 2 } \| \mathbf { z } - \mathbf { z } ^ { k } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } } \\ & { + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { b } ^ { \prime } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { c } ^ { \prime } \| _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+where $1 / \eta _ { k } > 0$ is a given decaying parameter, e.g., $\eta _ { k } = \alpha \sqrt { k }$ for some $\alpha > 0$ , and the Bregman divergence term stabilizes the convergence of $\mathbf { z }$ -minimization step. It is clear that problem (14) yields a quadratic program with the closed-form solution
+
+$$
+\begin{array} { r } { \mathbf { z } ^ { k + 1 } = \left( 1 / \left( \eta _ { k } + 3 \rho \right) \right) \left( \eta _ { k } \mathbf { z } ^ { k } + \rho \mathbf { a } + \rho \mathbf { b } + \rho \mathbf { c } - \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) \right) . } \end{array}
+$$
+
+In summary, the proposed ADMM algorithm alternatively updates (7)-(9), which yield closed-form solutions given by (10)-(12) and (15). The convergence of linearized ADMM for nonconvex optimization was recently proved by (Liu et al., 2017), and thus provides theoretical validity of our approach. Compared to the existing solver for generation of adversarial examples (Carlini & Wagner, 2017; Papernot et al., 2016b), our algorithm offers two main benefits, efficiency and generality. That is, the computations for every update step are efficiently carried out, and our approach can be applicable to a wide class of attack formulations.
+
+# 4 OVERLAPPING GROUP AND REFINED STRATTACK
+
+In this section, we generalize our proposed ADMM solution framework to the case of generating adversarial perturbations with overlapping group structures. We then turn to an attack refining model under fixed sparse structures. We will show that both extensions can be unified under the ADMM framework. In particular, the refined approach will allow us to gain deeper insights on the structural effects on adversarial perturbations.
+
+# 4.1 OVERLAPPING GROUP STRUCTURE
+
+We recall that groups $\{ \mathcal { D } _ { i } \}$ (also denoted by $\{ \mathcal { G } _ { p , q } \} )$ studied in Sec. 3 could be overlapped with each other; see an example in Fig. A1. Therefore, $\{ \mathcal { D } _ { i } \}$ is in general a cover rather than a partition of $[ n ]$ . To address the challenge in coupled group variables, we introduce multiple copies of the variable y in problem (4), and achieve the following modification
+
+$$
+\begin{array} { r l } { \underset { \delta , { \mathbf { z } } , { \mathbf { w } } , \{ { \mathbf { y } } _ { i } \} } { \mathrm { m i n i m i z e } } } & { \ f ( { \mathbf { z } } + { \mathbf { x } } _ { 0 } ) + \gamma D ( \delta ) + \tau \sum _ { i = 1 } ^ { P Q } \| \mathbf { y } _ { i , \mathcal { D } _ { i } } \| _ { 2 } + h ( \mathbf { w } ) } \\ { \mathrm { s u b j e c t ~ t o } } & { \ \mathbf { z } = \delta , \ \mathbf { z } = \mathbf { w } , \ \mathbf { z } = \mathbf { y } _ { i } , \quad i \in [ P Q ] , } \end{array}
+$$
+
+where compared to problem (4), there exist $P Q$ variables $\mathbf { y } _ { i } \in \mathbb { R } ^ { n }$ for $i \in [ P Q ]$ , and ${ \bf y } _ { i , \mathcal { D } _ { i } }$ denotes the subvector of $\mathbf { y } _ { i }$ with indices given by $\mathcal { D } _ { i }$ . It is clear from (16) that groups $\{ \mathcal { D } _ { i } \}$ become nonoverlapped since each of them lies in a different copy $\mathbf { y } _ { i }$ . The ADMM algorithm for solving problem (16) maintains a similar procedure as (7)-(9) except $\mathbf { y }$ -step (12) and $\mathbf { z }$ -step (15); see Proposition 2.
+
+Proposition 2 Given the same condition of Proposition $I$ , the ADMM solution to problem (16) involves the $\delta$ -step same as $( I O )$ , the w-step same as $( l I )$ , and two modified y- and $\mathbf { z }$ -steps,
+
+$$
+\left\{ \begin{array} { l l } { \left[ \mathbf { y } _ { i } ^ { k + 1 } \right] _ { \mathcal { D } _ { i } } = \left( 1 - \frac { \tau } { \rho \| [ \mathbf { c } _ { i } ] _ { \mathcal { D } _ { i } } \| _ { 2 } } \right) _ { + } [ \mathbf { c } _ { i } ] _ { \mathcal { D } _ { i } } } \\ { \left[ \mathbf { y } _ { i } ^ { k + 1 } \right] _ { [ n ] / \mathcal { D } _ { i } } = [ \mathbf { c } _ { i } ] _ { [ n ] / \mathcal { D } _ { i } } } \\ { \mathbf { z } ^ { k + 1 } = \left( 1 / \left( \eta _ { k } + 2 \rho + P Q \rho \right) \right) \left( \eta _ { k } \mathbf { z } ^ { k } + \rho \mathbf { a } ^ { \prime } + \rho \mathbf { b } ^ { \prime } + \rho \sum _ { i = 1 } ^ { P Q } \mathbf { c } _ { i } ^ { \prime } - \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) \right) , } \end{array} \right.
+$$
+
+where $\mathbf { c } _ { i } : = \mathbf { z } ^ { k } - \mathbf { v } _ { i } ^ { k } / \rho , \mathbf { v }$ $\mathbf { v } _ { i }$ is the Lagrangian multiplier associated with equality constraint $\mathbf { y } _ { i } = \mathbf { z }$ , similar to (9) we obtain $\mathbf v _ { i } ^ { k + 1 } = \mathbf v _ { i } ^ { k } + \rho ( \mathbf y ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , [ \tau$ $[ n ] / \mathcal { D } _ { i }$ denotes the difference of sets $[ n ]$ and $\mathcal { D } _ { i } , \mathbf { a } ^ { \prime }$ and $\mathbf { b } ^ { \prime }$ have been defined in (13), and $\mathbf c _ { i } ^ { \prime } = \mathbf y _ { i } ^ { k + 1 } + \mathbf v _ { i } ^ { k } / \rho$ .
+
+Proof: See Appendix C.
+
+We note that updating $P Q$ variables $\left\{ \mathbf { y } _ { i } \right\}$ is decomposed as shown in (17). However, the side effect is the need of $P Q$ times more storage space than the $\mathbf { y }$ -step (12) when groups are non-overlapped.
+
+# 4.2 REFINED STRATTACK UNDER FIXED SPARSE PATTERN
+
+The approaches proposed in Sec. 3 and Sec. 4.1 help us to identify structured sparse patterns in adversarial perturbations. This section presents a method to refine structured attacks under fixed group sparse patterns. Let $\delta ^ { * }$ denote the solution to problem (2) solved by the proposed ADMM method. We define a $\sigma$ -sparse perturbation $\delta$ via $\delta ^ { * }$ ,
+
+$$
+\delta _ { i } = 0 \mathrm { i f } \delta _ { i } ^ { * } \leq \sigma , \mathrm { f o r a n y } i \in [ n ] ,
+$$
+
+where a hard thresholding operator is applied to $\delta ^ { * }$ with tolerance $\sigma$ . Our refined model imposes the fixed $\sigma$ -sparse structure (19) into problem (2). This leads to
+
+$$
+\begin{array} { r l } { \underset { \delta } { \mathrm { m i n i m i z e } } } & { f ( \mathbf { x } _ { 0 } + \pmb { \delta } ) + \gamma D ( \pmb { \delta } ) } \\ { \mathrm { s u b j e c t ~ t o } } & { ( \mathbf { x } _ { 0 } + \pmb { \delta } ) \in [ 0 , 1 ] ^ { n } , \| \pmb { \delta } \| _ { \infty } \leq \epsilon } \\ & { \delta _ { i } = 0 , \mathrm { i f ~ } i \in S _ { \sigma } , } \end{array}
+$$
+
+where $\scriptstyle { \mathcal { S } } _ { \sigma }$ is defined by (19), i.e., $S _ { \sigma } : = \left\{ j \vert \delta _ { j } ^ { * } \leq \sigma \right.$ , $j \in [ n ] \}$ . Compared to problem (2), the groupsparse penalty function is eliminated as it has been known as $a$ priori. With the priori knowledge of group sparsity, problem (20) is formulated to optimize and refine the non-zero groups, thus achieving better performance on highlighting and exploring the perturbation structure. Problem (20) can be solved using ADMM, and its solution is presented in Proposition 3.
+
+Proposition 3 The ADMM solution to problem (20) is given by
+
+$$
+\begin{array} { r } { [ \delta ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { 0 } & { i \in \mathcal { S } _ { \sigma } } \\ { \operatorname* { m i n } \{ 1 - [ { \mathbf { x } } _ { 0 } ] _ { i } , \epsilon \} } & { \frac { \rho } { 2 \gamma + \rho } a _ { i } > \operatorname* { m i n } \{ 1 - [ { \mathbf { x } } _ { 0 } ] _ { i } , \epsilon \} , i \notin \mathcal { S } _ { \sigma } } \\ { \operatorname* { m a x } \{ - [ { \mathbf { x } } _ { 0 } ] _ { i } , - \epsilon \} } & { \frac { \rho } { 2 \gamma + \rho } a _ { i } < \operatorname* { m a x } \{ - [ { \mathbf { x } } _ { 0 } ] _ { i } , - \epsilon \} , i \notin \mathcal { S } _ { \sigma } } \\ { \frac { \rho } { 2 \gamma + \rho } a _ { i } } & { o t h e r w i s e , } \end{array} \right. } \\ { [ \mathbf { z } ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { 0 } & { i \in \mathcal { S } _ { \sigma } } \\ { 1 / ( \eta _ { k } + \rho ) \left[ \eta _ { k } [ \mathbf { z } ^ { k } ] _ { i } + \rho [ \mathbf { a } ^ { \prime } ] _ { i } - [ \nabla f ( \mathbf { z } ^ { k } + { \mathbf { x } } _ { 0 } ) ] _ { i } \right] } & { i \notin \mathcal { S } _ { \sigma } , } \end{array} \right. } \end{array}
+$$
+
+for $i \in \lceil n \rceil$ , where $\mathbf { z } = \delta$ is the introduced auxiliary variable similar to (4), $\mathbf { a } : = \delta ^ { k + 1 } - \mathbf { u } ^ { k } / \rho ,$ $\bar { \mathbf { a } } ^ { \prime } : = \delta ^ { k + 1 ^ { \prime } } + \mathbf { u } ^ { k } / \rho$ , $\mathbf { u } ^ { k + 1 } = \mathbf { u } ^ { k } + \rho ( \delta ^ { k + 1 } - \mathbf { z } ^ { k + 1 } ) ,$ , and $\rho$ and $\eta _ { k }$ have been defined in (6) and $( I 4 )$ . The ADMM iterations can be initialized by $\delta ^ { * }$ , the known solution to problem (2).
+
+Proof: See Appendix D.
+
+# 5 EMPIRICAL PERFORMANCE OF STRATTACK
+
+We evaluate the performance of the proposed StrAttack on three image classification datasets, MNIST (Lecun et al., 1998), CIFAR-10 (Krizhevsky & Hinton, 2009) and ImageNet (Deng et al., 2009). To make fair comparison with the C&W $\ell _ { 2 }$ attack (Carlini & Wagner, 2017), we use $\ell _ { 2 }$ norm as the distortion function $D ( \delta ) = \| \delta \| _ { 2 } ^ { 2 }$ . And we also compare with FGM (Goodfellow et al., 2014) and IFGSM $\ell _ { 2 }$ attacks (Kurakin et al., 2017) as a reference. We evaluate attack success rate (ASR)1 as well as $\ell _ { p }$ distortion metrics for $p \in \{ 0 , 1 , 2 , \infty \}$ . The detailed experiment setup is presented in Appendix F. Our code is available at https://github.com/KaidiXu/StrAttack.
+
+For each attack method on MNIST or CIFAR-10, we choose 1000 original images from the test dataset as source and each image has 9 target labels. So a total of 9000 adversarial examples are generated for each attack method. On ImageNet, each attack method tries to craft 900 adverdarial examples with 100 random images from the test dataset and 9 random target labels for each image.
+
+Fig. 2 compares adversarial examples generated by StrAttack and C&W attack on each dataset. We observe that the perturbation of the C&W attack has poor group sparsity, i.e., many non-zeros groups with small magnitudes. However, the ASR of the C&W attack is quite sensitive to these small perturbations. As applying a threshold to have the same $\ell _ { 0 }$ norm as our attack, we find that only $6 . 7 \%$ of adversarial examples generated from C&W attack remain valid. By contrast, StrAttack is able to highlight the most important group structures (local regions) of adversarial perturbations without attacking other pixels. For example, StrAttack misclassifies a natural image (4 in MNIST) as an incorrect label 3. That is because the pixels that appears in the structure of 3 are more significantly perturbed by our attack; see the top right plots of Fig. 2. Furthermore, the ‘goose-sorrel’ example shows that misclassification occurs when we just perturb a small number of non-sparse group regions on goose’s head, which is more consistent with human perception. We refer readers to Appendix G for more results.
+
+By quatitatively analysis, we report $\ell _ { p }$ norms and ASR in Table 1 for $p \in \{ 0 , 1 , 2 , \infty \}$ . We show that StrAttack perturbs much fewer pixels (smaller $\ell _ { 0 }$ norm), but it is comparable to or even better than other attacks in terms of $\ell _ { 1 } , \ell _ { 2 }$ , and $\ell _ { \infty }$ norms. Specifically, the FGM attack yields the worst performance in both ASR and $\ell _ { p }$ distortion. On MNIST and CIFAR-10, StrAttack outperforms other attacks in $\ell _ { 0 }$ , $\ell _ { 1 }$ and $\ell _ { \infty }$ distortion. On ImageNet, StrAttack outperforms C&W attack in $\ell _ { 0 }$ and $\ell _ { 1 }$ distortion. Since the C&W attacking loss directly penalizes the $\ell _ { 2 }$ norm, it often causes smaller $\ell _ { 2 }$ distortion than StrAttack. We also observe that the overlapping case leads to the adversarial perturbation of less sparsity (in terms of $\ell _ { 0 }$ norm) compared to the non-overlapping case. This is not surprising, since the sparsity of the overlapping region is controlled by at least two groups. However, compared to C&W attack, the use of overlapping groups in StrAttack still yields sparser perturbations. Unless specified otherwise, we focus on the case of non-overlapping groups to generate the most sparse adversarial perturbations. We highlight that although a so-called one-pixel attack (Su et al., 2017) also yields very small $\ell _ { 0 }$ norm, it is at the cost of very large $\ell _ { \infty }$ distortion. Unlike one-pixel attack, StrAttack achieves the sparsity without losing the performance of $\ell _ { \infty }$ , $\ell _ { 1 }$ and $\ell _ { 2 }$ distortion.
+
+Furthermore, we compare the performance of StrAttack with the C&W $\ell _ { \infty }$ attack and IFGSM while attacking the robust model (Madry et al., 2017) on MNIST. We remark that all the considered attack methods are performed under the same $\ell _ { \infty }$ -norm based distortion constraint with an upper bound $\epsilon \in \{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 \}$ . Here we obtain a (refined) StrAttack subject to $\| \pmb { \delta } \| _ { \infty } \le \epsilon$ by solving problem (20) at $\gamma = 0$ . In Table 2, we demonstrate the ASR and the number of perturbed pixels for various attacks over 5000 (untargeted) adversarial examples. The ASR define as the proportion of the final perturbation results less than given $\epsilon \in \{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 \}$ bound over number of test images. Here an successful attack is defined by an attack that can fool DNNs and meets the $\ell _ { \infty }$ distortion constraint. As we can see, StrAttack can achieve the similar ASR compared to other attack methods, however, it perturbs a much less number of pixels. Next, we evaluate the performance of StrAttack against two defense mechanisms: defensive distillation (Papernot et al., 2016c) and adversarial training (Tramèr et al., 2018). We observe that StrAttack is able to break the two defense methods with $100 \%$ ASR. More details are provided in Appendix H.
+
+![](images/66bf25676453381db5aa7be4fa446f75d5345cd43a6cff506d80e98f6dcc21da.jpg)  
+Figure 2: C&W attack vs StrAttack. Here each grid cell represents a $2 \times 2 , 2 \times 2$ , and $1 3 \times 1 3$ small region in MNIST, CIFAR-10 and ImageNet, respectively. The group sparsity of perturbation is represented by heatmap. The colors on heatmap represent average absolute value of distortion scale to [0, 255]. The left two columns correspond to results of using C&W attack. The right two columns show results of StrAttack.
+
+Table 1: Adversarial attack success rate (ASR) and $\ell _ { p }$ distortion values for various attacks.   
+
+<table><tr><td rowspan="2">Data Set</td><td rowspan="2">Attack Method</td><td colspan="5">BestCase</td><td colspan="5">Average Case</td><td colspan="5">Worst Case</td></tr><tr><td>ASR</td><td>lo</td><td>l1</td><td>l2</td><td>lo</td><td>ASR</td><td>l</td><td>l1</td><td>l2</td><td>lo</td><td>ASR</td><td>lo</td><td>l1</td><td>l2</td><td>lo</td></tr><tr><td rowspan="5">MNIST</td><td>FGM</td><td>99.3</td><td>456.5 549.5</td><td>28.2</td><td>2.32</td><td>0.57</td><td>35.8</td><td>466</td><td>39.4</td><td>3.17</td><td>0.717</td><td>0</td><td>N.A</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>IFGSM</td><td>100</td><td></td><td>18.3</td><td>1.57</td><td>0.4</td><td>100</td><td>588</td><td>30.9</td><td>2.41</td><td>0.566</td><td>99.8</td><td>640.4</td><td>50.98</td><td>3.742</td><td>0.784</td></tr><tr><td>C&amp;W</td><td>100</td><td>479.8</td><td>13.3</td><td>1.35</td><td>0.397</td><td>100</td><td>493.4</td><td>21.3</td><td>1.9</td><td>0.528</td><td>99.7</td><td>524.3</td><td>29.9</td><td>2.45</td><td>0.664</td></tr><tr><td>StrAttack</td><td>100</td><td>73.2</td><td>10.9</td><td>1.51</td><td>0.384</td><td>100</td><td>119.4</td><td>18.05</td><td>2.16</td><td>0.47</td><td>100</td><td>182.0</td><td>26.9</td><td>2.81</td><td>0.5</td></tr><tr><td>+overlap</td><td>100</td><td>84.4</td><td>9.2</td><td>1.32</td><td>0.401</td><td>100</td><td>157.4</td><td>16.2</td><td>1.95</td><td>0.508</td><td>100</td><td>260.9</td><td>22.9</td><td>2.501</td><td>0.653</td></tr><tr><td rowspan="5">CIFAR-10</td><td>FGM</td><td>98.5</td><td>3049</td><td>12.9</td><td>0.389</td><td>0.046</td><td>44.1</td><td>3048</td><td>34.2</td><td>0.989</td><td>0.113</td><td>0.2</td><td>3071</td><td>61.3</td><td>1.76</td><td>0.194</td></tr><tr><td>IFGSM</td><td>100</td><td>3051</td><td>6.22</td><td>0.182</td><td>0.02</td><td>100</td><td>3051</td><td>13.7</td><td>0.391</td><td>0.0433</td><td>100</td><td>3060</td><td>22.9</td><td>0.655</td><td>0.075</td></tr><tr><td>C&amp;W</td><td>100</td><td>2954</td><td>6.03</td><td>0.178</td><td>0.019</td><td>100</td><td>2956</td><td>12.1</td><td>0.347</td><td>0.0364</td><td>99.9</td><td>3070</td><td>16.8</td><td>0.481</td><td>0.0536</td></tr><tr><td>StrAttack</td><td>100</td><td>264</td><td>3.33</td><td>0.204</td><td>0.031</td><td>100</td><td>487</td><td>7.13</td><td>0.353</td><td>0.050</td><td>100</td><td>772</td><td>12.5</td><td>0.563</td><td>0.075</td></tr><tr><td>+overlap</td><td>100</td><td>295</td><td>3.35</td><td>0.169</td><td>0.029</td><td>100</td><td>562</td><td>7.05</td><td>0.328</td><td>0.047</td><td>100</td><td>920</td><td>12.9</td><td>0.502</td><td>0.063</td></tr><tr><td rowspan="4">ImageNet</td><td>FGM</td><td>12</td><td>264917</td><td>152</td><td>0.477</td><td>0.0157</td><td>2</td><td>263585</td><td>51.3</td><td>0.18</td><td>0.00614</td><td>0</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>IFGSM</td><td>100</td><td>267079</td><td>299.32</td><td>0.9086</td><td>0.02964</td><td>100</td><td>267293</td><td>723</td><td>2.2</td><td>0.0792</td><td>98</td><td>267581</td><td>1378</td><td>4.22</td><td>0.158</td></tr><tr><td>C&amp;W</td><td>100</td><td>267916</td><td>127</td><td>0.471</td><td>0.016</td><td>100</td><td>263140</td><td>198</td><td>0.679</td><td>0.03</td><td>100</td><td>265212</td><td>268</td><td>0.852</td><td>0.041</td></tr><tr><td>StrAttack</td><td>100</td><td>14462</td><td>55.2</td><td>0.719</td><td>0.058</td><td>100</td><td>52328</td><td>152</td><td>1.06</td><td>0.075</td><td>100</td><td>80722</td><td>197</td><td>1.35</td><td>0.122</td></tr></table>
+
+\* Please refer to Appendix F for the definition of best case, best case and worst case. \*\* N.A. means not available in the case of zero ASR, +overlap means structured attack with overlapping groups.
+
+Lastly, we evaluate the transferability of StrAttack from Inception V3 (Szegedy et al., 2016) to other network models including Inception V2, Inception V4 (Szegedy et al., 2017), ResNet 50, ResNet 152 (He et al., 2016), DenseNet 121 and DenseNet 161 (Huang et al., 2017). For comparison, we also present the transferbility of IFGSM and C&W. This experiment is performed under 1000 (target) adversarial examples on ImageNet2. It can be seen from in Table 3 that StrAttack yields the largest attack success rate while transferring to almost every network model.
+
+Table 2: Attack success rate (ASR) and $\ell _ { 0 }$ norm of adversarial perturbations for various attacks against robust adversarial training based defense on MNIST.   
+
+<table><tr><td></td><td>ASR at ε = 0.1</td><td>ASR at e = 0.2</td><td>ASR at e = 0.3</td><td>ASR at ε = 0.4</td><td>l</td></tr><tr><td>IFGSM</td><td>0.01</td><td>0.02</td><td>0.09</td><td>0.94</td><td>654</td></tr><tr><td>C&amp;W loattack</td><td>0.01</td><td>0.02</td><td>0.10</td><td>0.96</td><td>723</td></tr><tr><td>StrAttack</td><td>0.01</td><td>0.02</td><td>0.10</td><td>0.99</td><td>279</td></tr></table>
+
+Table 3: Comparison of transferability of different attacks over 6 ImageNet models.   
+
+<table><tr><td></td><td>Incept V2</td><td>Incept V4</td><td>ResNet50</td><td>ResNet152</td><td>DenseNet121</td><td>DenseNet161</td></tr><tr><td>IFGSM</td><td>0.27</td><td>0.22</td><td>0.27</td><td>0.19</td><td>0.16</td><td>0.19</td></tr><tr><td>C&amp;W</td><td>0.25</td><td>0.24</td><td>0.23</td><td>0.23</td><td>0.15</td><td>0.15</td></tr><tr><td>StrAttack</td><td>0.28</td><td>0.27</td><td>0.25</td><td>0.25</td><td>0.26</td><td>0.25</td></tr></table>
+
+# 6 STRATTACK OFFERS BETTER INTERPRETABILITY
+
+In this section, we evaluate the effects of structured adversarial perturbations on image classification through adversarial saliency map (ASM) (Papernot et al., 2016b) and class activation map (CAM) (Zhou et al., 2016). Here we recall that ASM measures the impact of pixel-level perturbations on label classification, and CAM localizes class-specific image discriminative regions that we use to visually explain adversarial perturbations (Xiao et al., 2018). We will show that compared to C&W attack, StrAttack meets better interpretability in terms of (a) a higher ASM score and (b) a tighter connection with CAM, where the metric (a) implies interpretability at a micro-level, namely, perturbing pixels with largest impact on image classification, and the metric (b) demonstrates interpretability at a macro-level, namely, perturbations can be mapped to the most discriminative image regions localized by CAM.
+
+Given an input image $\mathbf { x } _ { \mathrm { 0 } }$ and a target class $t$ , let $\mathrm { A S M } ( \mathbf { x } _ { 0 } , t ) \in \mathbb { R } ^ { d }$ denote ASM scores for every pixel of $\mathbf { x } _ { \mathrm { 0 } }$ corresponding to $t$ . We elaborate on the mathematical definition of ASM in Appendix E. Generally speaking, the ith element of $\mathrm { A S M } ( \mathbf { x } _ { 0 } , t )$ , denoted by $\mathrm { A S M } ( \mathbf { x } _ { 0 } , t ) [ i ]$ , measures how much the classification score with respect to the target label $t$ will increase and that with respect to the original label $t _ { 0 }$ will decrease if a perturbation is added to the pixel $i$ . With the aid of ASM, we then define a Boolean map $\mathbf { B } _ { \mathrm { A S M } } \in \mathbb { R } ^ { \bar { d } }$ to encode the regions of $\mathbf { x } _ { \mathrm { 0 } }$ most sensitive to targeted adversarial attacks, where $\begin{array} { r } { \mathbf { B } _ { \mathrm { A S \bar { M } } } ( i ) = 1 } \end{array}$ if $\mathrm { A S M } ( \mathbf { x } _ { 0 } , t ) > \nu$ , and 0 otherwise. Here $\nu$ is a given threshold to highlight the most sensitive pixels. we then define the interpretability score (IS) via ASM,
+
+$$
+\mathrm { I S } ( \delta ) = \| \mathbf { B } _ { \mathrm { A S M } } \circ \pmb { \delta } \| _ { 2 } / \| \pmb { \delta } \| _ { 2 } ,
+$$
+
+where $\circ$ is the element-wise product. The rationale behind (23) is that $\mathrm { I S } ( \delta )  1$ if the sensitive region identified by ASM perfectly predicts the locations of adversarial perturbations. By contrast, if $\mathrm { \bar { I S } } ( \delta ) \to 0$ , then adversarial perturbations cannot be interpreted by ASM. In Fig. 3(a), we compare IS of our proposed attack with C&W attack versus the threshold $\nu$ , valued by different percentiles of ASM scores. We obsreve that our attack outperforms C&W attack in terms of IS, since the former is able to extract important local structures of images by penalizing the group sparsity of adversarial perturbations. It seems that our improvement is not significant. However, StrAttack just perturbs very few pixels to obtain this benefit, leading to perturbations with more semantic structure; see Fig. 3(b) for an illustrative example.
+
+Besides ASM, we show that the effect of adversarial perturbations can be visually explained through the class-specific discriminative image regions localized by CAM (Zhou et al., 2016). In Fig. 3(c), we illustrate CAM and demonstrate the differences between our attack and C&W in terms of their connections to the most discriminative regions of $\mathbf { x } _ { \mathrm { 0 } }$ with label $t _ { 0 }$ . We observe that the mechanism of StrAttack can be better interpreted from CAM: only a few adversarial perturbations are needed to suppress the feature of the original image with the true label. By replacing ASM with CAM, we can similarly compute IS in (23) averaged over 500 examples on ImageNet, yielding 0.65 for C&W attack and 0.77 for our attack. More examples of ASM and CAM can be viewed in Appendix E.
+
+To better interpret the mechanism of adversarial examples, we study adversarial attacks on some complex images, where the objects of the original and target labels exist simultaneously as shown in Fig. 4. It can be visualized from CAM that both C&W attack and StrAttack yields similar adversarial effects on natural images: Adversarial perturbations are used to suppress the most discriminative region with respect to the true label, and simultaneously promotes the discriminative region of the target label. The former principle is implied by the location of perturbed regions and $C ( \mathbf { x } _ { 0 } , t _ { 0 } )$ in Fig. 4, and the latter can be seen from $C ( \mathbf { x } _ { \mathrm { C W } } , t )$ or $\boldsymbol { C } ( \mathbf { x } _ { \mathrm { S t r } } , t )$ against $C ( \mathbf { x } _ { 0 } , t )$ . However, compared to C&W attack, StrAttack perturbs much less but ‘right’ pixels which have better correspondence with class-specific discriminative image regions localized by CAM.
+
+![](images/74cce4b5e2706017106661f57d73c9fc31a04ebded9f88c9210469c990e473de.jpg)  
+Figure 3: Interpretabilicy comparison of StrAttack and C&W attack. (a) ASM-based IS vs $\nu$ , given from the 30th percentile to the 90th percentile of ASM scores. (b) Overlay ASM and $\mathbf { B } _ { \mathrm { A S M } } \circ \delta$ on top of image with the true label ‘Tibetan Mastiff’ and the target label ‘streetcar’. From left to right: original image, ASM (darker color represents larger value of ASM score), $\mathbf { B } _ { \mathrm { A S M } } \circ \delta$ under StrAttack, and $\mathbf { B } _ { \mathrm { A S M } } \circ \delta$ under C&W attack. Here $\nu$ in $\mathbf { B } _ { \mathrm { A S M } }$ is set by the 90th percentile of ASM scores. (c) From left to right: original image with true label ‘stove’, CAM of ‘stove’, and perturbations with target label ‘water ouzel’ under StrAttack and C&W.
+
+# 7 CONCLUSION
+
+This work explores group-wise sparse structures when implementing adversarial attacks. Different from previous works that use $\ell _ { p }$ norm to measure the similarity between an original image and an adversarial example, this work incorporates group-sparsity regularization into the problem formulation of generating adversarial examples and achieves strong group sparsity in the obtained adversarial perturbations. Leveraging ADMM, we develop an efficient implementation to generate structured adversarial perturbations, which can be further used to refine an arbitrary adversarial attack under fixed group sparse structures. The proposed ADMM framewrok is general enough for implementing many state-of-the-art attacks. We perform extensive experiments using MNIST, CIFAR-10 and ImageNet datasets, showing that our structured adversarial attack (StrAttack) is much stronger than the existing attacks and its better interpretability from group sparse structures aids in uncovering the origins of adversarial examples.
+
+# ACKNOWLEDGEMENT
+
+This work is supported by Air Force Research Laboratory FA8750-18-2-0058, and U.S. Office of Naval Research. Sijia Liu, Pin-Yu Chen, Huan Zhang and Quanfu Fan were supported by the MITIBM Watson Ai Lab, IBM Research.
+
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+
+# APPENDIX
+
+# A ILLUSTRATIVE EXAMPLE OF GROUP SPARSITY
+
+![](images/495f2f1b7630c48a3fc9e71f27d4422891eb7fa55b1b6f64c1ef5711a31e6ed4.jpg)  
+Figure A1: An example of $4 \times 4$ perturbation matrix under sliding masks with different strides. The values of matrix elements are represented by color’s intensity (white stands for 0). Left: Non-overlapping groups with $r = 2$ and $S = 2$ . Right: Overlapping groups with $r = 2$ and $S = 1$ . In both cases, two groups $\mathcal { G } _ { 1 , 1 }$ and $\mathcal { G } _ { 1 , 2 }$ are highlighted, where $\mathcal { G } _ { 1 , 1 }$ is non-sparse, and $\mathcal { G } _ { 1 , 2 }$ is sparse.
+
+# B PROOF OF PROPOSITION 1
+
+We recall that the augmented Lagrangian function $L ( \delta , \mathbf { z } , \mathbf { w } , \mathbf { y } , \mathbf { u } , \mathbf { v } , \mathbf { s } )$ is given by
+
+$$
+\begin{array} { r l r } & { } & { L ( { \bf z } , \delta , { \bf y } , { \bf w } , { \bf u } , { \bf v } , { \bf s } ) = f ( { \bf z } + { \bf x } _ { 0 } ) + \gamma D ( \delta ) + \tau \sum _ { i = 1 } ^ { P Q } \| { \bf y } _ { \mathcal { D } _ { i } } \| _ { 2 } + h ( { \bf w } ) + { \bf u } ^ { T } ( \delta - { \bf z } ) } \\ & { } & { + { \bf v } ^ { T } ( { \bf y } - { \bf z } ) + { \bf s } ^ { T } ( { \bf w } - { \bf z } ) + \displaystyle \frac { \rho } { 2 } \| \delta - { \bf z } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| { \bf y } - { \bf z } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| { \bf w } - { \bf z } \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Problem (7), to minimize $L ( \delta , \mathbf { z } ^ { k } , \mathbf { w } , \mathbf { y } , \mathbf { u } ^ { k } , \mathbf { v } ^ { k } , \mathbf { s } ^ { k } )$ , can be decomposed into three sub-problems:
+
+$$
+\operatorname* { m i n i m i z e } _ { \delta } \gamma D ( \pmb { \delta } ) + \frac { \rho } { 2 } \| \pmb { \delta } - \mathbf { a } \| _ { 2 } ^ { 2 } ,
+$$
+
+$$
+\operatorname* { m i n i m i z e } _ { \mathbf { w } } { h ( \mathbf { w } ) } + \frac { \rho } { 2 } \| \mathbf { w } - \mathbf { b } \| _ { 2 } ^ { 2 } ,
+$$
+
+$$
+\underset { \mathbf { y } } { \mathrm { m i n i m i z e } } \ \tau \sum _ { i = 1 } ^ { P Q } \| \mathbf { y } _ { \mathcal { D } _ { i } } \| _ { 2 } + \frac { \rho } { 2 } \| \mathbf { y } - \mathbf { c } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\mathbf { a } : = \mathbf { z } ^ { k } - \mathbf { u } ^ { k } / \rho , \mathbf { b } : = \mathbf { z } ^ { k } - \mathbf { s } ^ { k } / \rho$ , and $\mathbf { c } : = \mathbf { z } ^ { k } - \mathbf { v } ^ { k } / \rho .$ .
+
+$\delta$ -step Suppose $D ( \delta ) = \| \delta \| _ { 2 } ^ { 2 }$ , then the solution to problem (25) is easily acquired as below
+
+$$
+\delta ^ { k + 1 } = \frac { \rho } { \rho + 2 \gamma } \mathbf { a }
+$$
+
+w-step Based on the definition of $h ( \mathbf { w } )$ , problem (26) becomes
+
+$$
+\begin{array} { r l } { \underset { \mathbf { w } } { \mathrm { m i n i m i z e } } } & { \| \mathbf { w } - \mathbf { b } \| _ { 2 } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { ( \mathbf { x } _ { 0 } + \mathbf { w } ) \in [ 0 , 1 ] ^ { n } , \ \| \mathbf { w } \| _ { \infty } \leq \epsilon . } \end{array}
+$$
+
+Problem (29) is equivalent to
+
+$$
+\begin{array} { r l } { \underset { w _ { i } } { \mathrm { m i n i m i z e } } } & { ( w _ { i } - a _ { i } ) _ { 2 } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { - [ \mathbf { x } _ { 0 } ] _ { i } \leq w _ { i } \leq 1 - [ \mathbf { x } _ { 0 } ] _ { i } , | w _ { i } | \leq \epsilon } \end{array}
+$$
+
+for $i \in [ n ]$ , where $x _ { i }$ or $[ \mathbf { x } ] _ { i }$ represents the $i$ th element of $\mathbf { x }$ , and $1 - [ { \bf x } _ { 0 } ] _ { i } > 0$ since $[ \mathbf { x } _ { 0 } ] _ { i } \in [ 0 , 1 ]$ . Problem (30) then yields the solution
+
+$$
+[ \mathbf { w } ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { \operatorname* { m i n } \{ 1 - [ \mathbf { x } _ { 0 } ] _ { i } , \epsilon \} } & { a _ { i } > \operatorname* { m i n } \{ 1 - [ \mathbf { x } _ { 0 } ] _ { i } , \epsilon \} } \\ { \operatorname* { m a x } \{ - [ \mathbf { x } _ { 0 } ] _ { i } , - \epsilon \} } & { a _ { i } < \operatorname* { m a x } \{ - [ \mathbf { x } _ { 0 } ] _ { i } , - \epsilon \} } \\ { a _ { i } } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+y-step Problem (27) becomes
+
+$$
+\underset { \mathbf { y } } { \mathrm { m i n i m i z e } } \sum _ { i = 1 } ^ { P Q } \| \mathbf { y } _ { \mathcal { D } _ { i } } \| _ { 2 } + \frac { \rho } { 2 \tau } \| \mathbf { y } - \mathbf { c } \| _ { 2 } ^ { 2 } ,
+$$
+
+The solution is given by the proximal operator associated with the $\ell _ { 2 }$ norm with parameter $\tau / \rho$ (Parikh et al., 2014)
+
+$$
+[ { \bf y } ^ { k + 1 } ] _ { \mathcal { D } _ { i } } = \left( 1 - \frac { \tau } { \rho \| [ { \bf c } ] _ { \mathcal { D } _ { i } } \| _ { 2 } } \right) _ { + } [ { \bf c } ] _ { \mathcal { D } _ { i } } , \ i \in [ P Q ] ,
+$$
+
+where recall that $\cup _ { i \in [ P Q ] } { \mathcal { D } } _ { i } = [ n ]$ , and $\mathcal { D } _ { i } \cap \mathcal { D } _ { j } = \emptyset$ if $i \neq j$ .
+
+# C PROOF OF PROPOSITION 2
+
+The augmented Lagrangian of problem (16) is given by
+
+$$
+\begin{array} { l } { { \displaystyle \langle { \bf z } , \delta , { \bf w } , \{ \bf y } _ { i } \rangle , { \bf u } , { \bf v } _ { i } , { \bf s } \rangle = f ( { \bf z } + { \bf x } _ { 0 } ) + \gamma D ( \delta ) + h ( { \bf w } ) + \tau \sum _ { i = 1 } ^ { P Q } \| { \bf y } _ { i } , { \mathcal D } _ { i } \| _ { 2 } + { \bf u } ^ { T } ( \delta - { \bf z } ) + { \bf s } ^ { T } ( { \bf w } - { \bf z } ) }  \\ { { \displaystyle \qquad + \sum _ { i = 1 } ^ { P Q } { \bf v } _ { i } ^ { T } ( { \bf y } _ { i } - { \bf z } ) + \frac { \rho } { 2 } \| \delta - { \bf z } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| { \bf w } - { \bf z } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \sum _ { i = 1 } ^ { P Q } \| { \bf y } _ { i } - { \bf z } \| _ { 2 } ^ { 2 } } , } \end{array}
+$$
+
+where u, $\mathbf { v } _ { i }$ and s are the Lagrangian multipliers.
+
+ADMM decomposes the optimization variables into two blocks and adopts the following iterative scheme,
+
+$$
+\begin{array} { r l } & { \{ \delta ^ { k + 1 } , \mathbf w ^ { k + 1 } , \mathbf y _ { i } ^ { k + 1 } \} = \underset { \delta , \mathbf w , \{ \mathbf y _ { i } \} } { \arg \operatorname* { m i n } } L ( \mathbf z ^ { k } , \delta , \mathbf w , \mathbf y _ { i } , \mathbf u ^ { k } , \mathbf v _ { i } ^ { k } , \mathbf s ^ { k } ) , } \\ & { } \\ & { \mathbf z ^ { k + 1 } = \underset { \mathbf z } { \arg \operatorname* { m i n } } L ( \mathbf z , \delta ^ { k + 1 } , \mathbf w ^ { k + 1 } , \mathbf y _ { i } ^ { k + 1 } , \mathbf u ^ { k } , \mathbf v _ { i } ^ { k } , \mathbf s ^ { k } ) , } \\ & { \left\{ \begin{array} { l l } { \mathbf u ^ { k + 1 } = \mathbf u ^ { k } + \rho ( \delta ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , } \\ { \mathbf v _ { i } ^ { k + 1 } = \mathbf v _ { i } ^ { k } + \rho ( \mathbf y _ { i } ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , \mathrm { ~ f o r ~ } i \in [ P Q ] , } \\ { \mathbf s ^ { k + 1 } = \mathbf s ^ { k } + \rho ( \mathbf w ^ { k + 1 } - \mathbf z ^ { k + 1 } ) , } \end{array} \right. } \end{array}
+$$
+
+where $k$ is the iteration index. Problem (35) can be split into three subproblems as shown below,
+
+$$
+\operatorname* { m i n i m i z e } _ { \delta } \gamma D ( \pmb { \delta } ) + \frac { \rho } { 2 } \| \pmb { \delta } - \mathbf { a } \| _ { 2 } ^ { 2 } ,
+$$
+
+$$
+\operatorname* { m i n i m i z e } _ { \mathbf { w } } ~ h ( \mathbf { w } ) + \frac { \rho } { 2 } \| \mathbf { w } - \mathbf { b } \| _ { 2 } ^ { 2 } ,
+$$
+
+$$
+\operatorname* { m i n i m i z e } _ { \mathbf { y } _ { i } } { \tau } | | \mathbf { y } _ { i , \mathcal { D } _ { i } } | | _ { 2 } + \frac { \rho } { 2 } | | \mathbf { y } _ { i } - \mathbf { c } _ { i } | | _ { 2 } ^ { 2 } , \mathrm { f o r } i \in [ P Q ] .
+$$
+
+where $\mathbf { a } = \mathbf { z } ^ { k } - \mathbf { u } ^ { k } / \rho$ , $\mathbf { b } = \mathbf { z } ^ { k } - \mathbf { s } ^ { k } / \rho$ and $\mathbf { c } _ { i } = \mathbf { z } ^ { k } - \mathbf { v } _ { i } ^ { k } / \rho$ . Each problem has a closed form solution. Note that the solutions to problem (38) and problem (39) are given (28) and (31).
+
+$\mathbf { y } _ { i }$ -step Problem (40) can be rewritten as
+
+$$
+\displaystyle \operatorname* { m i n i m i z e } _ { \mathbf { y } _ { i } } \ : \tau \| \mathbf { y } _ { i , \mathcal { D } _ { i } } \| _ { 2 } + \frac { \rho } { 2 } \| \mathbf { y } _ { i , \mathcal { D } _ { i } } - [ \mathbf { c } _ { i } ] _ { \mathcal { D } _ { i } } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| \mathbf { y } _ { i , [ n ] / \mathcal { D } _ { i } } - [ \mathbf { c } _ { i } ] _ { [ n ] / \mathcal { D } _ { i } } \| _ { 2 } ^ { 2 } , \mathrm { ~ f o r ~ } i \in [ P Q ] ,
+$$
+
+which can be decomposed into
+
+$$
+\begin{array} { r l } & { \underset { { \bf y } _ { i , \mathcal { D } _ { i } } } { \mathrm { m i n i m i z e ~ } } \tau \| { \bf y } _ { i , \mathcal { D } _ { i } } \| _ { 2 } + \frac { \rho } { 2 } \| { \bf y } _ { i , \mathcal { D } _ { i } } - [ { \bf c } _ { i } ] _ { \mathcal { D } _ { i } } \| _ { 2 } ^ { 2 } , \mathrm { ~ f o r ~ } i \in [ P Q ] , } \end{array}
+$$
+
+and
+
+$$
+\mathop { \operatorname* { m i n i m i z e } } _ { \mathbf { y } _ { i , [ n ] / \mathcal { D } _ { i } } } ~ \| \mathbf { y } _ { i , [ n ] / \mathcal { D } _ { i } } - [ \mathbf { c } _ { i } ] _ { [ n ] / \mathcal { D } _ { i } } \| _ { 2 } ^ { 2 } , \mathrm { f o r } i \in [ P Q ] .
+$$
+
+The solution to problem (42) can be obtained through the block soft thresholding operator (Parikh et al., 2014),
+
+$$
+\big [ \mathbf { y } _ { i } ^ { k + 1 } \big ] _ { \mathcal { D } _ { i } } = \left( 1 - \frac { \tau } { \rho \| \big [ \mathbf { c } _ { i } \big ] _ { \mathcal { D } _ { i } } \| _ { 2 } } \right) _ { + } [ \mathbf { c } _ { i } ] _ { \mathcal { D } _ { i } } , \mathrm { f o r } i \in [ P Q ] ,
+$$
+
+The solution to problem (43) is given by,
+
+$$
+\begin{array} { r } { \left[ \mathbf { y } _ { i } ^ { k + 1 } \right] _ { [ n ] / \mathcal { D } _ { i } } = [ \mathbf { c } _ { i } ] _ { [ n ] / \mathcal { D } _ { i } } , \mathrm { f o r } i \in [ P Q ] . } \end{array}
+$$
+
+z-step Problem (36) can be simplified to
+
+$$
+\operatorname* { m i n i m i z e } _ { \mathbf { z } } \quad f ( \mathbf { x } _ { 0 } + \mathbf { z } ) + \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { b } ^ { \prime } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \sum _ { i = 1 } ^ { P Q } \| \mathbf { z } - \mathbf { c } _ { i } ^ { \prime } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\mathbf { a } ^ { \prime } : = \delta ^ { k + 1 } + \mathbf { u } ^ { k } / \rho$ , $\mathbf b ^ { \prime } : = \mathbf w ^ { k + 1 } + \mathbf s ^ { k } / \rho$ , and $\mathbf c _ { i } ^ { \prime } : = \mathbf y _ { i } ^ { k + 1 } + \mathbf v _ { i } ^ { k } / \rho$ . We solve problem (46) using the linearization technique (Suzuki, 2013; Liu et al., 2018; Boyd et al., 2011). More specifically, the function $f$ is replaced with its first-order Taylor expansion at the point $\mathbf { z } ^ { k }$ by adding a Bregman divergence term $( \eta _ { k } \mathbf { \dot { / } } 2 ) \lVert \mathbf { z } - \mathbf { z } ^ { k } \rVert _ { 2 } ^ { 2 }$ . As a result, problem (46) becomes
+
+$$
+\begin{array} { r l } { \underset { \mathbf { z } } { \mathrm { m i n i m i z e } } } & { ( \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) ) ^ { T } ( \mathbf { z } - \mathbf { z } ^ { k } ) + \displaystyle \frac { \eta _ { k } } { 2 } \| \mathbf { z } - \mathbf { z } ^ { k } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } } \\ & { + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { b } ^ { \prime } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \sum _ { i = 1 } ^ { P Q } \| \mathbf { z } - \mathbf { c } _ { i } ^ { \prime } \| _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+whose solution is given by
+
+$$
+\mathbf { z } ^ { k + 1 } = \frac { \eta _ { k } \mathbf { z } ^ { k } + \rho \mathbf { a } ^ { \prime } + \rho \mathbf { b } ^ { \prime } + \rho \sum _ { i = 1 } ^ { P Q } \mathbf { c } _ { i } ^ { \prime } - \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) } { \eta _ { k } + ( 2 + P Q ) \rho } .
+$$
+
+# D PROOF OF PROPOSITION 3
+
+We start by converting problem (20) into the ADMM form
+
+$$
+\begin{array} { r l } { \underset { \delta , \mathbf { z } } { \mathrm { m i n i m i z e } } } & { \ f ( \mathbf { x } _ { 0 } + \mathbf { z } ) + g ( \mathbf { z } ) + \gamma D ( \delta ) + h ( \delta ) + g ( \delta ) } \\ { \mathrm { s u b j e c t \ t o } } & { \ \delta = \mathbf { z } , } \end{array}
+$$
+
+where $\mathbf { z }$ and $\delta$ are optimization variables, $g ( \delta )$ is an indicator function with respect to the constraint $\{ \delta _ { i } = 0$ , if $i \in \mathcal { S } _ { \sigma } \bar  \}$ , and $h ( \delta )$ is the other indicator function with respect to the other constraints $( \mathbf { x } _ { 0 } + \pmb { \delta } ) \in [ 0 , 1 ] ^ { n }$ , $\| \pmb { \delta } \| _ { \infty } \le \epsilon$ .
+
+The augmented Lagrangian of problem (20) is given by
+
+$$
+L ( \boldsymbol { \delta } , \mathbf { z } , \mathbf { u } ) = f ( \mathbf { z } + \mathbf { x } _ { 0 } ) + g ( \mathbf { z } ) + \gamma D ( \boldsymbol { \delta } ) + h ( \boldsymbol { \delta } ) + g ( \boldsymbol { \delta } ) + \mathbf { u } ^ { T } ( \boldsymbol { \delta } - \mathbf { z } ) + \frac { \rho } { 2 } \| \boldsymbol { \delta } - \mathbf { z } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\mathbf { u }$ is the Lagrangian multiplier.
+
+ADMM yields the following alternating steps
+
+$$
+\begin{array} { r l } & { \delta ^ { k + 1 } = \underset { \delta } { \arg \operatorname* { m i n } } L ( \delta , \mathbf { z } ^ { k } , \mathbf { u } ^ { k } ) } \\ & { \mathbf { z } ^ { k + 1 } = \underset { \mathbf { z } } { \arg \operatorname* { m i n } } L ( \delta ^ { k + 1 } , \mathbf { z } , \mathbf { u } ^ { k } ) } \\ & { \mathbf { u } ^ { k + 1 } = \mathbf { u } ^ { k } + \rho ( \delta ^ { k + 1 } - \mathbf { z } ^ { k + 1 } ) . } \end{array}
+$$
+
+$\delta$ -step Suppose $D ( \delta ) = \| \delta \| _ { 2 } ^ { 2 }$ , problem (51) becomes
+
+$$
+\begin{array} { r l } { \underset { \delta } { \mathrm { m i n i m i z e } } } & { \gamma \| \pmb { \delta } \| _ { 2 } ^ { 2 } + \frac { \rho } { 2 } \| \pmb { \delta } - \mathbf { a } \| _ { 2 } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { ( \mathbf { x } _ { 0 } + \pmb { \delta } ) \in [ 0 , 1 ] ^ { n } , \| \pmb { \delta } \| _ { \infty } \leq \epsilon } \\ & { \delta _ { i } = 0 , \mathrm { i f ~ } i \in \mathcal { S } _ { \sigma } , } \end{array}
+$$
+
+where $\mathbf { a } : = \mathbf { z } ^ { k } - \mathbf { u } ^ { k } / \rho$ . Problem (54) can be decomposed elementwise
+
+$$
+\begin{array} { r l } { \underset { \delta _ { i } } { \mathrm { m i n i m i z e } } } & { \frac { 2 \gamma + \rho } { \rho } \delta _ { i } ^ { 2 } - 2 a _ { i } \delta _ { i } + a _ { i } ^ { 2 } = \frac { 2 \gamma + \rho } { \rho } \left( \delta _ { i } - \frac { \rho } { 2 \gamma + \rho } a _ { i } \right) ^ { 2 } } \\ { \mathrm { s u b j e c t \ t o } } & { \left( [ \mathbf { x } _ { 0 } ] _ { i } + \delta _ { i } \right) \in [ 0 , 1 ] , ~ | \delta _ { i } | \leq \epsilon } \\ & { \delta _ { i } = 0 , ~ \mathrm { i f } ~ i \in \mathcal { S } _ { \sigma } . } \end{array}
+$$
+
+The solution to problem (55) is then given by
+
+$$
+[ \delta ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { 0 } & { i \in { \mathcal { S } } _ { \sigma } } \\ { \operatorname* { m i n } \{ 1 - [ { \mathbf { x } } _ { 0 } ] _ { i } , \epsilon \} } & { \frac { \rho } { 2 \gamma + \rho } a _ { i } > \operatorname* { m i n } \{ 1 - [ { \mathbf { x } } _ { 0 } ] _ { i } , \epsilon \} , i \notin { \mathcal { S } } _ { \sigma } } \\ { \operatorname* { m a x } \{ - [ { \mathbf { x } } _ { 0 } ] _ { i } , - \epsilon \} } & { \frac { \rho } { 2 \gamma + \rho } a _ { i } < \operatorname* { m a x } \{ - [ { \mathbf { x } } _ { 0 } ] _ { i } , - \epsilon \} , i \notin { \mathcal { S } } _ { \sigma } } \\ { \frac { \rho } { 2 \gamma + \rho } a _ { i } } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+$\mathbf { z }$ -step Problem (52) yields
+
+$$
+\begin{array} { r l } { \underset { \mathbf { z } } { \mathrm { m i n i m i z e } } } & { { } f ( \mathbf { x } _ { 0 } + \mathbf { z } ) + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { { } z _ { i } = 0 , \mathrm { ~ i f ~ } i \in \mathcal { S } _ { \sigma } , } \end{array}
+$$
+
+where $\mathbf { a } ^ { \prime } = \delta ^ { k + 1 } + \mathbf { u } ^ { k } / \rho$ . We solve problem (57) using the linearization technique (Suzuki, 2013; Liu et al., 2018; Boyd et al., 2011),
+
+$$
+\begin{array} { l l } { \mathrm { m i n i m i z e } } & { ( \nabla f ( \mathbf { x } _ { 0 } + \mathbf { z } ^ { k } ) ) ^ { T } ( \mathbf { z } - \mathbf { z } ^ { k } ) + \displaystyle \frac { \eta _ { k } } { 2 } \| \mathbf { z } - \mathbf { z } ^ { k } \| _ { 2 } ^ { 2 } + \displaystyle \frac { \rho } { 2 } \| \mathbf { z } - \mathbf { a } ^ { \prime } \| _ { 2 } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { z _ { i } = 0 , \mathrm { ~ i f ~ } i \in \mathcal { S } _ { \sigma } , } \end{array}
+$$
+
+where $\eta _ { k }$ is a decaying parameter associated with the Bregman divergence term $\| \mathbf { z } - \mathbf { z } ^ { k } \| _ { 2 } ^ { 2 }$ . In problems (57) and (58), only variables $\left\{ z _ { i } \right\}$ satisfying $i \notin S _ { \sigma }$ are unknown. The solution to problem (58) is then given by
+
+$$
+\begin{array} { r } { [ \mathbf { z } ^ { k + 1 } ] _ { i } = \left\{ \begin{array} { l l } { 0 } & { i \in \mathcal { S } _ { \sigma } } \\ { \frac { \eta _ { k } [ \mathbf { z } ^ { k } ] _ { i } + \rho [ \mathbf { a } ^ { \prime } ] _ { i } - [ \nabla f ( \mathbf { z } ^ { k } + \mathbf { x } _ { 0 } ) ] _ { i } } { \eta _ { k } + \rho } } & { i \notin \mathcal { S } _ { \sigma } . } \end{array} \right. } \end{array}
+$$
+
+# E ADVERSARIAL SALIENCY MAP (ASM) AND CLASS ACTIVATION MAPPING (CAM)
+
+$\mathrm { A S M } ( \mathbf { x } , t ) \in \mathbb { R } ^ { d }$ is defined by the forward derivative of a neural network given the input sample $\mathbf { x }$ and the target label $t$ (Papernot et al., 2016b)
+
+$$
+\begin{array} { r } { \mathrm { A S M } ( \mathbf { x } , t ) [ i ] = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } < 0 \mathrm { o r } \sum _ { j \neq t } \frac { \partial Z ( \mathbf { x } ) _ { j } } { \partial \mathbf { x } _ { i } } > 0 } \\ { \left( \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } \right) \left| \sum _ { j \neq t } \frac { \partial Z ( \mathbf { x } ) _ { j } } { \partial \mathbf { x } _ { i } } \right| } & { \mathrm { o t h e r w i s e } , } \end{array} \right. } \end{array}
+$$
+
+where $Z ( \mathbf { x } ) _ { j }$ is the $j$ th element of logits $Z ( \mathbf { x } )$ , representing the output before the last softmax layer in DNNs. If there exist many classes in a dataset (e.g., 1000 classes in ImageNet), then computing $\textstyle \sum _ { j \neq t } { \frac { \partial Z ( \mathbf { x } ) _ { j } } { \partial \mathbf { x } _ { i } } }$ t ∂Z(x)j∂x is intensive. To circumvent the scalability issue of ASM, we focus on the logit change with respect to the true label $t _ { 0 }$ and the target label $t$ only. More specifically, we consider three quantities, $\begin{array} { r } { \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } , - \frac { \partial Z ( \mathbf { x } ) _ { 0 } } { \partial \mathbf { x } _ { i } } } \end{array}$ , and $\begin{array} { r } { \left( \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } \right) \left| \sum _ { j \neq t } \frac { \partial Z ( \mathbf { x } ) _ { j } } { \partial \mathbf { x } _ { i } } \right| , } \end{array}$ , which correspond to a) promotion of the score of the target label $t$ , b) suppression of the classification score of the true label $t _ { 0 }$ , and c) a dual role on suppression and promotion. As a result, we modify (60) as
+
+$$
+\begin{array} { r } { \mathrm { A S M } ( \mathbf { x } , t ) [ i ] = \left\{ \begin{array} { l l } { 0 } & { \mathrm { ~ i f ~ } \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } < 0 \mathrm { ~ o r ~ } \frac { \partial Z ( \mathbf { x } ) _ { t _ { 0 } } } { \partial \mathbf { x } _ { i } } > 0 } \\ { \left( \frac { \partial Z ( \mathbf { x } ) _ { t } } { \partial \mathbf { x } _ { i } } \right) \left| \frac { \partial Z ( \mathbf { x } ) _ { t _ { 0 } } } { \partial \mathbf { x } _ { i } } \right| } & { \mathrm { ~ o t h e r w i s e } . } \end{array} \right. } \end{array}
+$$
+
+CAM allows us to visualize the perturbation of adversaries on predicted class scores given any pair of image and object label, and highlights the discriminative object regions detected by CNNs (Zhou et al., 2016). In Fig. A2, we show ASM and the discriminative regions identified by CAM on several ImageNet samples.
+
+![](images/85cc534c3cb14fcb9e8d40dfbbcc15cf7698922f10fafc4150609cc6375154d4.jpg)  
+Figure A2: (a) Overlay ASM and $\mathbf { B } _ { \mathrm { A S M } } \circ \delta$ on top of image with the true and the target label. From left to right: original image, ASM (darker color represents larger value of ASM score), $\mathbf { B } _ { \mathrm { A S M } } \circ \delta$ under our attack, and $\mathbf { B } _ { \mathrm { A S M } } \circ \pmb { \delta }$ under C&W attack. Here $\nu$ in $\mathbf { B } _ { \mathrm { A S M } }$ is set by the 90th percentile of ASM scores. (b) From left to right: original image, CAM of original label, and perturbations with target label generated from the StrAttack and C&W attack, respectively.
+
+# F EXPERIMENT SETUP AND PARAMETER SETTING
+
+In this work, we consider targeted adversarial attacks since they are believed stronger than untargeted attacks. For targeted attacks, we have different methods to choose the target labels. The average case selects the target label randomly among all the labels that are not the correct label. The best case performs attacks using all incorrect labels, and report the target label that is the least difficult to attack. The worst case performs attacks using all incorrect labels, and report the target label which is the most difficult to attack.
+
+In our experiments, two networks are trained for MNIST and CIFAR-10, respectively, and a pretrained network is utilized for ImageNet. The model architectures for MNIST and CIFAR-10 are the same, both with four convolutional layers, two max pooling layers, two fully connected layers and a softmax layer. It can achieve $9 9 . 5 \%$ and $80 \%$ accuracy on MNIST and CIFAR-10, respectively. For ImageNet, a pre-trained Inception v3 network (Szegedy et al., 2016) is applied which can achieve $96 \%$ top-5 accuracy. All experiments are conducted on machines with NVIDIA GTX 1080 TI GPUs.
+
+The implementations of FGM and IFGM are based on the CleverHans package (Papernot et al., 2016a). The key distortion parameter $\epsilon$ is determined by a fine-grained grid search. For IFGM, we perform 10 FGM iterations and the distortion parameter $\epsilon ^ { \prime }$ is set to $\epsilon / 1 0$ for effectiveness as shown in Tramèr et al. (2018). The implementation of the C&W attack is based on the opensource code provided by Carlini & Wagner (2017). The maximum iteration number is set to 1000 and it has 9 binary search steps.
+
+In the StrAttack, the group size for MNIST and CIFAR-10 is $2 \times 2$ and its stride is set to 2 if the non-overlapping mask is used, otherwise the group size is $3 \times 3$ and stride is 2. The group size for ImageNet is $1 3 \times 1 3$ and its stride is set to 13. In ADMM, the parameter $\rho$ achieves a trade-off between the convergence rate and the convergence value. A larger $\rho$ could make ADMM converging faster but usually leads to perturbations with larger $\ell _ { p }$ distortion values. A proper configuration of the parameters is suggested as follows: We set the penalty parameter $\rho = 1$ , decaying parameter in (14) $\eta _ { 1 } = 5$ , $\tau = 2$ and $\gamma = 1$ . Moreover, we set $c$ defined in (3) to 0.5 for MNIST, 0.25 for CIFAR-10, and 2.5 for ImageNet. Refined attack technique proposed in Sec. 4.2 is applied for all experiments, we set $\sigma$ is equal to $3 \%$ quantile value of non-zero perturbation in $\delta ^ { * }$ . We observe that $73 \%$ of $\delta ^ { * }$ can be retrained to a $\sigma$ -sparse perturbation successfully which proof the effective of our refined attack step.
+
+# G SUPPLEMENTARY EXPERIMENTAL RESULTS
+
+![](images/39f973ae40ee9639c50e1d0c0d12ab51daabe2b3424b93e84b701ef9598822b7.jpg)  
+Figure A3: C&W attack vs StrAttack on MNIST with grid size $2 \times 2$ .
+
+Some random choice samples from MNIST (Fig. A3), CIFAR-10 (Fig. A4) and ImageNet (Fig. A5) compare StrAttack with C&W attack. For better sparse visual effect, we only show non-overlapping mask function results here. From these samples, we can discover a consistent phenomenon that our StrAttack is more interested in some particular regions, they usually appear on the objects or their edges in original images, distinctly seen in MNIST (Fig. A3) and ImageNet (Fig. A5).
+
+# H STRATTACK AGAINST DEFENSIVE DISTILLATION AND ADVERSARIALTRAINING
+
+In this section, we present the performance of the StrAttack against defensive distillation (Papernot et al., 2016c) and adversarial training (Tramèr et al., 2018). In defensive distillation, we evaluate the
+
+![](images/1709eb9cc90aba912103f361d377de750a609161f1d9a6fdd84a87d51c2bb3a3.jpg)  
+Figure A4: C&W attack vs StrAttack on CIFAR-10 with grid size $2 \times 2$ .
+
+StrAttack for different temperature parameters on MNIST and CIFAR-10. We generate 9000 adversarial examples with 1000 randomly selected images from MNIST and CIFAR-10, respectively. The attack success rates of the StrAttack for different temperatures $T$ are all $100 \%$ . The reason is that distillation at temperature $T$ makes the logits approximately $T$ times larger but does not change the relative values of logits. The StrAttack which works on the relative values of logits does not fail.
+
+We further use the StrAttack to break DNNs training on adversarial examples (Tramèr et al., 2018) with their correct labels on MNIST. The StrAttack is performed on three neural networks: the first network is unprotected, the second is obtained by retraining with $9 0 0 0 \mathrm { C } \& \mathrm { { W } }$ adversarial examples, and the third network is retained with 9000 adversarial examples crafted by the StrAttack. The success rate and distortions on the three networks are shown in Table A1. The StrAttack can break all three networks with $100 \%$ success rate. However, adversarial training shows certain defense effects as an increase on the $\ell _ { 1 }$ or $\ell _ { 2 }$ distortion on the latter two networks over the unprotected network is observed.
+
+Table A1: StrAttack against adversarial training on MNIST   
+
+<table><tr><td rowspan="2">Adversarial training</td><td colspan="3">Best case</td><td colspan="3">Averagecase</td><td colspan="3">Worst case</td></tr><tr><td>ASR</td><td>l1</td><td>l2</td><td>ASR</td><td>l1</td><td>l2</td><td>ASR</td><td>l1</td><td>l2</td></tr><tr><td>None</td><td>100</td><td>10.9</td><td>1.51</td><td>100</td><td>18.05</td><td>2.16</td><td>100</td><td>26.9</td><td>2.81</td></tr><tr><td>C&amp;W</td><td>100</td><td>16.1</td><td>1.87</td><td>100</td><td>25.1</td><td>2.58</td><td>100</td><td>34.2</td><td>3.26</td></tr><tr><td>structured</td><td>100</td><td>15.6</td><td>1.86</td><td>100</td><td>25.1</td><td>2.61</td><td>100</td><td>34.6</td><td>3.31</td></tr></table>
+
+![](images/054d38529f162de9cc5fb7bf9a29dfeeb97413bd5d3c4b9bfc005f20fa05d6ac.jpg)  
+Figure A5: C&W attack vs StrAttack on ImageNet with grid size $1 3 \times 1 3$ .

parse/train/Hy6GHpkCW/Hy6GHpkCW.md

@@ -0,0 +1,367 @@
+# A NEURAL REPRESENTATION OF SKETCH DRAWINGS
+
+David Ha   
+Google Brain   
+hadavid@google.com   
+Douglas Eck   
+Google Brain   
+deck@google.com
+
+# ABSTRACT
+
+We present sketch-rnn, a recurrent neural network (RNN) able to construct stroke-based drawings of common objects. The model is trained on a dataset of human-drawn images representing many different classes. We outline a framework for conditional and unconditional sketch generation, and describe new robust training methods for generating coherent sketch drawings in a vector format.
+
+# 1 INTRODUCTION
+
+Recently, there have been major advancements in generative modelling of images using neural networks as a generative tool. Generative Adversarial Networks (GANs) (Goodfellow, 2016), Variational Inference (VI) (Kingma & Welling, 2013), and Autoregressive (AR) (Reed et al., 2017) models have become popular tools in this fast growing area. Most of the work thus far has been targeted towards modelling low resolution, pixel images. Humans, however, do not understand the world as a grid of pixels, but rather develop abstract concepts to represent what we see. From a young age, we develop the ability to communicate what we see by drawing on paper with a pencil or crayon. In this way we learn to express a sequential, vector representation of an image as a short sequence of strokes. In this paper we investigate an alternative to traditional pixel image modelling approaches, and propose a generative model for vector images.
+
+![](images/941a5c666d5a87047158428a458c9dac6bdd673a6a74f06f01c223c7f4e0ca16.jpg)  
+Figure 1: Latent space interpolation of various vector images produced by our model (left). Interpolation of two different Kanji characters (亀 書) as sequence of strokes (right)
+
+Our goal is to train machines to draw and generalize abstract concepts in a manner similar to humans. In this work, as a first step towards this goal, we train our model on a dataset of hand-drawn sketches, each represented as a sequence of motor actions controlling a pen: which direction to move, when to lift the pen up, and when to stop drawing. In doing so, we created a model that potentially has many applications, from assisting the creative process of an artist, to helping teach students how to draw.
+
+This paper makes the following contributions: We outline a framework for both unconditional and conditional generation of vector images composed of a sequence of lines. Our recurrent neural network-based generative model is capable of producing sketches of common objects in a vector format. We develop a training procedure unique to vector images to make the training more robust. In the conditional generation model, we explore the latent space developed by the model to represent a vector image. We also discuss creative applications of our methodology. We make available a dataset of 50 million hand drawn vector images to encourage further development of generative modelling for vector images, and also release an implementation of our model as an open source project.1
+
+# 2 RELATED WORK
+
+There is a long history of work related to algorithms that mimic painters. One such work is Portrait Drawing by Paul the Robot (Tresset & Fol Leymarie, 2013; Xie et al., 2012), where an underlying algorithm controlling a mechanical robot arm sketches lines on a canvas with a programmable artistic style to mimic a given digitized portrait of a person. Reinforcement Learning based-approaches (Xie et al., 2012) have been developed to discover a set of paint brush strokes that can best represent a given input photograph. These prior works generally attempt to mimic digitized photographs, rather than develop generative models of vector images.
+
+Neural Network-based approaches have been developed for generative models of images, although the majority of neural network-related research on image generation deal with pixel images (Goodfellow, 2016; Isola et al., 2016; Kaae Sønderby et al., 2016; Kingma et al., 2016; Reed et al., 2017; White, 2016). There has been relatively little work done on vector image generation using neural networks. An earlier work (Simhon & Dudek, 2004) makes use of Hidden Markov Models to synthesize lines and curves of a human sketch. More recent work (Graves, 2013) on handwriting generation with Recurrent Neural Networks laid the groundwork for utilizing Mixture Density Networks (Bishop, 1994) to generate continuous data points. Recent works of this approach attempted to generate vectorized Kanji characters (Ha, 2015; Zhang et al., 2016) by modelling Chinese characters as a sequence of pen stroke actions.
+
+The approach outlined in this work allows one to explore the latent space representation of vector images. For instance, we can use our model to interpolate between two Kanji characters in Figure 1 by first encoding the characters, represented as a sequence of strokes, into a latent space of embedding vectors. Previous work (Bowman et al., 2015) outlined a methodology to combine Sequence-toSequence models with a Variational Autoencoder to model natural English sentences in latent vector space. A related work (Lake et al., 2015), utilizes probabilistic program induction, rather than neural networks, to perform one-shot modelling of the Omniglot dataset containing images of symbols.
+
+One of the factors limiting research development in the space of generative vector drawings is the lack of publicly available datasets. Previously, the Sketch dataset (Eitz et al., 2012), consisting of 20K vector sketches, was used to explore feature extraction techniques. A subsequent work, the Sketchy dataset (Sangkloy et al., 2016), provided 70K vector sketches along with corresponding pixel images for various classes. This allowed for a larger-scale exploration of human sketches. ShadowDraw (Lee et al., 2011) is an interactive system that predicts what a finished drawing looks like based on a set of incomplete brush strokes from the user while the sketch is being drawn. ShadowDraw used a dataset of 30K raster images combined with extracted vectorized features. In this work, we use a much larger dataset of 50 million vector sketches that is made publicly available.
+
+# 3 METHODOLOGY
+
+# 3.1 DATASET
+
+We constructed QuickDraw, a dataset of 50 million vector drawings obtained from Quick, Draw! (Jongejan et al., 2016), an online game where the players are asked to draw objects belonging to a particular object class in less than 20 seconds. QuickDraw consists of hundreds of classes of common objects. Each class of QuickDraw is a dataset of 70K training samples, in addition to 2.5K validation and 2.5K test samples.
+
+We use a data format that represents a sketch as a set of pen stroke actions. This representation is an extension of the format used in (Graves, 2013). Our format extends the binary pen stroke event into a multi-state event. In this data format, the initial coordinate of the drawing is located at the origin.
+
+A sketch is a list of points, and each point is a vector consisting of 5 elements: $( \Delta x , \Delta y , p _ { 1 } , p _ { 2 } , p _ { 3 } )$ . The first two elements are the offset distance in the $\mathbf { X }$ and y directions of the pen from the previous point. The last 3 elements represents a binary one-hot vector of 3 possible states. The first pen state, $p _ { 1 }$ , indicates that the pen is currently touching the paper, and that a line will be drawn connecting the next point with the current point. The second pen state, $p _ { 2 }$ , indicates that the pen will be lifted from the paper after the current point, and that no line will be drawn next. The final pen state, $p _ { 3 }$ , indicates that the drawing has ended, and subsequent points, including the current point, will not be rendered.
+
+# 3.2 SKETCH-RNN
+
+![](images/d28393eb9a12e71ca8d8c00eef5d747f634fa02a5bb5261f05049c0bdd69e6eb.jpg)  
+Figure 2: Schematic diagram of sketch-rnn.
+
+Our model is a Sequence-to-Sequence Variational Autoencoder (VAE), similar to the architecture described in (Bowman et al., 2015; Kingma & Welling, 2013). Our encoder is a bidirectional RNN (Schuster et al., 1997) that takes in a sketch as an input, and outputs a latent vector of size $N _ { z }$ . Specifically, we feed the sketch sequence, $S$ , and also the same sketch sequence in reverse order, $S _ { \mathrm { r e v e r s e } }$ , into the two encoding RNNs of the bidirectional RNN, to obtain two final hidden states:
+
+$$
+h _ { \right. } = { \mathrm { e n c o d e } } _ { \right. } ( S ) , \ h _ { \left. } = { \mathrm { e n c o d e } } _ { \left. } ( S _ { \mathrm { r e v e r s e } } ) , \ h = [ \ h _ { \right. } \ ; \ h _ { \left. } ] .
+$$
+
+We take this final concatenated hidden state, $h$ , and project it into two vectors $\mu$ and $\hat { \sigma }$ , each of size $N _ { z }$ , using a fully connected layer. We convert $\hat { \sigma }$ into a non-negative standard deviation parameter $\sigma$ using an exponential operation. We use $\mu$ and $\sigma$ , along with $\mathcal { N } ( 0 , I )$ , a vector of IID Gaussian variables of size $N _ { z }$ , to construct a random vector, $z \in \mathbb { R } ^ {  { N _ { z } } }$ , as in the approach for a VAE:
+
+$$
+\mu = W _ { \mu } h + b _ { \mu } , \hat { \sigma } = W _ { \sigma } h + b _ { \sigma } , \sigma = \exp \Big ( \frac { \hat { \sigma } } { 2 } \Big ) , z = \mu + \sigma \odot \mathcal { N } ( 0 , I ) .
+$$
+
+Under this encoding scheme, the latent vector $z$ is not a deterministic output for a given input sketch, but a random vector conditioned on the input sketch.
+
+Our decoder is an autoregressive RNN that samples output sketches conditional on a given latent vector $z$ . The initial hidden states $h _ { 0 }$ , and optional cell states $c _ { 0 }$ (if applicable) of the decoder RNN is the output of a single layer network: $[ \ : h _ { 0 } ; \ : \overline { { { c _ { 0 } } } } \ : ] = \operatorname { t a n h } ( W _ { z } z + b _ { z } ) \ :$
+
+At each step $i$ of the decoder RNN, we feed the previous point, $S _ { i - 1 }$ and the latent vector $z$ in as a concatenated input $x _ { i }$ , where $S _ { 0 }$ is defined as $( 0 , 0 , 1 , 0 , 0 )$ . The output at each time step are the parameters for a probability distribution of the next data point $S _ { i }$ . In Equation 3, we model $( \Delta x , \Delta y )$ as a Gaussian mixture model (GMM) with $M$ normal distributions as in (Bishop, 1994; Graves, 2013), and $( q _ { 1 } , q _ { 2 } , q _ { 3 } )$ as a categorical distribution to model the ground truth data $\left( p _ { 1 } , p _ { 2 } , p _ { 3 } \right)$ , where $( q _ { 1 } + q _ { 2 } + q _ { 3 } = 1 )$ as done in (Ha, 2015) and (Zhang et al., 2016). Unlike (Graves, 2013), our generated sequence is conditioned from a latent code $z$ sampled from our encoder, which is trained end-to-end alongside the decoder.
+
+$$
+p ( \Delta x , \Delta y ) = \sum _ { j = 1 } ^ { M } { \Pi _ { j } \ N } ( \Delta x , \Delta y \mid \mu _ { x , j } , \mu _ { y , j } , \sigma _ { x , j } , \sigma _ { y , j } , \rho _ { x y , j } ) , \mathrm { ~ w h e r e ~ } \sum _ { j = 1 } ^ { M } { \Pi _ { j } } = 1 \mathrm { ~ a n d ~ } ( \mathrm { R e } ( \Delta x , \Delta y ) )
+$$
+
+$\mathcal N ( x , y | \mu _ { x } , \mu _ { y } , \sigma _ { x } , \sigma _ { y } , \rho _ { x y } )$ is the probability distribution function for a bivariate normal distribution. Each of the $M$ bivariate normal distributions consist of five parameters: $( \mu _ { x } , \mu _ { y } , \sigma _ { x } , \sigma _ { y } , \rho _ { x y } )$ , where $\mu _ { x }$ and $\mu _ { y }$ are the means, $\sigma _ { x }$ and $\sigma _ { y }$ are the standard deviations, and $\rho _ { x y }$ is the correlation parameter of each bivariate normal distribution. An additional vector $\Pi$ of length $M$ , also a categorical distribution, are the mixture weights of the Gaussian mixture model. Hence the size of the output vector $y$ is $5 M + M + 3$ , which includes the 3 logits needed to generate $( q _ { 1 } , q _ { 2 } , q _ { 3 } )$ .
+
+The next hidden state of the RNN, generated with its forward operation, projects into the output vector $y _ { i }$ using a fully-connected layer:
+
+$$
+\begin{array} { r } { x _ { i } = \left[ S _ { i - 1 } ~ ; ~ z \right] , ~ \left[ h _ { i } ~ ; ~ c _ { i } ~ \right] = ~ \mathtt { f o r w a r d } ( x _ { i } , ~ \left[ h _ { i - 1 } ~ ; ~ c _ { i - 1 } \right] ) , ~ y _ { i } = W _ { y } h _ { i } + b _ { y } , ~ y _ { i } \in \mathbb { R } ^ { 6 M + 3 } . } \end{array}
+$$
+
+The vector $y _ { i }$ is broken down into the parameters of the probability distribution of the next data point: $\big [ \left( \hat { \Pi } \mu _ { x } \mu _ { y } \hat { \sigma } _ { x } \hat { \sigma } _ { y } \hat { \rho } _ { x y } \right) _ { 1 } \left( \hat { \Pi } \mu _ { x } \mu _ { y } \hat { \sigma } _ { x } \hat { \sigma } _ { y } \hat { \rho } _ { x y } \right) _ { 2 } \ldots \left( \hat { \Pi } \mu _ { x } \mu _ { y } \hat { \sigma } _ { x } \hat { \sigma } _ { y } \hat { \rho } _ { x y } \right) _ { M } \left( \hat { q } _ { 1 } \hat { q } _ { 2 } \hat { q } _ { 3 } \right) \big ] = y _ { i } .$ (5) As in (Graves, 2013), we apply exp and tanh operations to ensure the standard deviation values are non-negative, and that the correlation value is between $^ { - 1 }$ and 1:
+
+$$
+\sigma _ { x } = \exp ( \hat { \sigma } _ { x } ) , \ \sigma _ { y } = \exp ( \hat { \sigma } _ { y } ) , \ \rho _ { x y } = \operatorname { t a n h } ( \hat { \rho } _ { x y } ) .
+$$
+
+The probabilities for the categorical distributions are calculated using the outputs as logit value
+
+$$
+q _ { k } = \frac { \exp ( \hat { q } _ { k } ) } { \sum _ { j = 1 } ^ { 3 } \exp ( \hat { q } _ { j } ) } , k \in \{ 1 , ~ 2 , ~ 3 \} , ~ \Pi _ { k } = \frac { \exp ( \hat { \Pi } _ { k } ) } { \sum _ { j = 1 } ^ { M } \exp ( \hat { \Pi } _ { j } ) } , k \in \{ 1 , ~ \ldots , ~ { \cal M } \} .
+$$
+
+A key challenge is to train our model to know when to stop drawing. Because the probabilities of the three pen stroke events are highly unbalanced, the model becomes more difficult to train. The probability of a $p _ { 1 }$ event is much higher than $p _ { 2 }$ , and the $p _ { 3 }$ event will only happen once per drawing. The approach developed in (Ha, 2015) and later followed by (Zhang et al., 2016) was to use different weightings for each pen event when calculating the losses, such as a hand-tuned weighting of $( 1 , 1 0 , 1 0 0 )$ . We find this inelegant approach to be inadequate for our dataset of diverse images.
+
+We develop a simpler, more robust approach that works well for a broad class of sketch drawing data. In our approach, all sequences are generated to a length of $N _ { \mathrm { m a x } }$ where $N _ { \mathrm { m a x } }$ is the length of the longest sketch in our training dataset. In principle $N _ { \mathrm { m a x } }$ can be considered a hyper parameter. As the length of $S$ is usually shorter than $N _ { \mathrm { m a x } }$ , we set $S _ { i }$ to be $( 0 , 0 , 0 , 0 , 1 )$ for $i > N _ { s }$ . We discuss the training in detail in the next section.
+
+After training, we can sample sketches from our model. During the sampling process, we generate the parameters for both GMM and categorical distributions at each time step, and sample an outcome $S _ { i } ^ { \prime }$ for that time step. Unlike the training process, we feed the sampled outcome $S _ { i } ^ { \prime }$ as input for the next time step. We continue to sample until $p _ { 3 } = 1$ , or when we have reached $i = N _ { \operatorname* { m a x } }$ . Like the encoder, the sampled output is not deterministic, but a random sequence, conditioned on the input latent vector $z$ . We can control the level of randomness we would like our samples to have during the sampling process by introducing a temperature parameter $\tau$ :
+
+$$
+\hat { q } _ { k } \to \frac { \hat { q } _ { k } } { \tau } , \hat { \Pi } _ { k } \to \frac { \hat { \Pi } _ { k } } { \tau } , \sigma _ { x } ^ { 2 } \to \sigma _ { x } ^ { 2 } \tau , \sigma _ { y } ^ { 2 } \to \sigma _ { y } ^ { 2 } \tau .
+$$
+
+We can scale the softmax parameters of the categorial distribution and also the $\sigma$ parameters of the bivariate normal distribution by a temperature parameter $\tau$ , to control the level of randomness in our samples. $\tau$ is typically set between 0 and 1. In the limiting case as $\tau  0$ , our model becomes deterministic and samples will consist of the most likely point in the probability density function. Figure 3 illustrates of effect of sampling sketches with various temperature parameters.
+
+# 3.3 UNCONDITIONAL GENERATION
+
+![](images/f696899c1ed98946685075eec02fb01a5394625b737c4268bf119a071c335733.jpg)  
+Figure 3: Unconditional generation of firetrucks, yoga poses, gardens and owls with varying $\tau$
+
+As a special case, we can also train our model to generate sketches unconditionally, where we only train the decoder RNN module, without any input or latent vectors. By removing the encoder, the decoder RNN as a standalone model is an autoregressive model without latent variables. In this use case, the initial hidden states and cell states of the decoder RNN are initialized to zero. The inputs $x _ { i }$ of the decoder RNN at each time step is only $S _ { i - 1 }$ or $S _ { i - 1 } ^ { \prime }$ , as we do not need to concatenate a latent vector $z$ . In Figure 3, we sample various sketch images generated unconditionally by varying the temperature parameter from $\tau = 0 . 2$ at the top in blue, to $\tau = 0 . 9$ at the bottom in red.
+
+# 3.4 TRAINING
+
+Our training procedure follows the approach of the Variational Autoencoder (Kingma & Welling, 2013), where the loss function is the sum of two terms: the Reconstruction Loss, $L _ { R }$ , and the Kullback-Leibler Divergence Loss, $L _ { K L }$ . We train our model to optimize this two-part loss function. The Reconstruction loss term, described in Equation 9, maximizes the log-likehood of the generated probability distribution to explain the training data $S$ . We can calculate this reconstruction loss, $L _ { R }$ , using the generated parameters of the pdf and the training data $S$ . $L _ { R }$ is composed of the sum of the log loss of the offset terms $( \Delta x , \Delta y )$ , $L _ { s }$ , and the log loss of the pen state terms $( p _ { 1 } , p _ { 2 } , p _ { 3 } ) , L _ { p }$ :
+
+$$
+\begin{array} { l } { { { \cal L } _ { s } = \displaystyle - \frac { 1 } { N _ { \mathrm { m a x } } } \sum _ { i = 1 } ^ { N _ { s } } \log \Big ( \sum _ { j = 1 } ^ { M } \Pi _ { j , i } \mathscr { N } ( \Delta x _ { i } , \Delta y _ { i } \mid \mu _ { x , j , i } , \mu _ { y , j , i } , \sigma _ { x , j , i } , \sigma _ { y , j , i } , \rho _ { x y , j , i } ) \Big ) } } \\ { { { \cal L } _ { p } = \displaystyle - \frac { 1 } { N _ { \mathrm { m a x } } } \sum _ { i = 1 } ^ { N _ { \mathrm { m a x } } } \sum _ { k = 1 } ^ { 3 } p _ { k , i } \log ( q _ { k , i } ) , \ L _ { R } = L _ { s } + L _ { p } . } } \end{array}
+$$
+
+Note that we discard the pdf parameters modelling the $( \Delta x , \Delta y )$ points beyond $N _ { s }$ when calculating $L _ { s }$ , while $L _ { p }$ is calculated using all of the pdf parameters modelling the $\left( p _ { 1 } , p _ { 2 } , p _ { 3 } \right)$ points until $N _ { \mathrm { m a x } }$ . Both terms are normalized by the total sequence length $N _ { \mathrm { m a x } }$ . We found this methodology of loss calculation to be more robust and allows the model to easily learn when it should stop drawing, unlike the earlier mentioned method of assigning importance weightings to $p _ { 1 } , p _ { 2 }$ , and $p _ { 3 }$ .
+
+The Kullback-Leibler (KL) divergence loss term measures the difference between the distribution of our latent vector $z$ , to that of an IID Gaussian vector with zero mean and unit variance. Optimizing for this loss term allows us to minimize this difference. We use the result in (Kingma $\&$ Welling, 2013), and calculate the KL loss term, $L _ { K L }$ , normalized by number of dimensions $N _ { z }$ of $z$ :
+
+$$
+L _ { K L } = - \frac { 1 } { 2 N _ { z } } \Big ( 1 + \hat { \sigma } - \mu ^ { 2 } - \exp ( \hat { \sigma } ) \Big ) .
+$$
+
+The loss function in Equation 11 is a weighted sum of both the $L _ { R }$ and $L _ { K L }$ loss terms:
+
+$$
+L o s s = L _ { R } + w _ { K L } L _ { K L } .
+$$
+
+There is a tradeoff between optimizing for one term over the other. As $w _ { K L }  0$ , our model approaches a pure autoencoder, sacrificing the ability to enforce a prior over our latent space while obtaining better reconstruction loss metrics. Note that for unconditional generation, where our model is the standalone decoder, there will be no $L _ { K L }$ term as we only optimize for $L _ { R }$ .
+
+![](images/0fd54f56dfc6335ffef4f02ea70c198459fd81cb71d7072401332f0c6af7a695.jpg)  
+Figure 4: Tradeoff between $L _ { R }$ and $L _ { K L }$ , for two models trained on single class datasets (left). Validation Loss Graph for models trained on the Yoga dataset using various $w _ { K L }$ (right
+
+Figure 4 illustrates the tradeoff between different settings of $w _ { K L }$ and the resulting $L _ { R }$ and $L _ { K L }$ metrics on the test set, along with the $L _ { R }$ metric on a standalone decoder RNN for comparison. As the unconditional model does not receive any prior information about the entire sketch it needs to generate, the $L _ { R }$ metric for the standalone decoder model serves as an upper bound for various conditional models using a latent vector.
+
+# 4 EXPERIMENTS
+
+We conduct several experiments with sketch-rnn for both conditional and unconditional vector image generation. We train sketch-rnn on various QuickDraw classes using various settings for $w _ { K L }$ and record the breakdown of losses. To experiment with a diverse set of classes with varying complexities, we select the cat, pig, face, firetruck, garden, owl, mosquito and yoga class. We also experiment on multi-class datasets by concatenating different classes together to form (cat, pig) and (crab, face, pig, rabbit). The results for test set evaluation on various datasets are displayed in Table 1.
+
+The sketch-rnn model treats the RNN cell as an abstract component. In our experiments, we use Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) as the encoder RNN. For the decoder RNN, we use HyperLSTM, as this type of RNN cell excels at sequence generation tasks (Ha et al., 2017). The ability for HyperLSTM to spontaneously augment its own weights enables it to adapt to many different regimes in a large diverse dataset. Please see the Appendix for more details.
+
+<table><tr><td>Dataset</td><td>WKL</td><td>= 1.00</td><td></td><td>WKL = 0.50</td><td>WKL</td><td>= 0.25</td><td>Decoder Only</td></tr><tr><td></td><td>LR</td><td>LKL</td><td>LR</td><td>LKL</td><td>LR</td><td>LKL</td><td>LR</td></tr><tr><td>cat</td><td>-0.98</td><td>0.29</td><td>-1.33</td><td>0.70</td><td>-1.46</td><td>1.01</td><td>-0.57</td></tr><tr><td>pig</td><td>-1.14</td><td>0.22</td><td>-1.37</td><td>0.49</td><td>-1.52</td><td>0.80</td><td>-0.82</td></tr><tr><td>cat, pig</td><td>-1.02</td><td>0.22</td><td>-1.24</td><td>0.49</td><td>-1.50</td><td>0.98</td><td>-0.75</td></tr><tr><td>crab,face,pig,rabbit</td><td>-0.91</td><td>0.22</td><td>-1.04</td><td>0.40</td><td>-1.47</td><td>1.17</td><td>-0.67</td></tr><tr><td>face</td><td>-1.13</td><td>0.27</td><td>-1.55</td><td>0.71</td><td>-1.90</td><td>1.44</td><td>-0.73</td></tr><tr><td>firetruck</td><td>-1.24</td><td>0.22</td><td>-1.26</td><td>0.24</td><td>-1.78</td><td>1.10</td><td>-0.90</td></tr><tr><td>garden</td><td>-0.79</td><td>0.20</td><td>-0.81</td><td>0.25</td><td>-0.99</td><td>0.54</td><td>-0.62</td></tr><tr><td>owl</td><td>-0.93</td><td>0.20</td><td>-1.03</td><td>0.34</td><td>-1.29</td><td>0.77</td><td>-0.66</td></tr><tr><td>mosquito</td><td>-0.67</td><td>0.30</td><td>-1.02</td><td>0.66</td><td>-1.41</td><td>1.54</td><td>-0.34</td></tr><tr><td>yoga</td><td>-0.80</td><td>0.24</td><td>-1.07</td><td>0.55</td><td>-1.51</td><td>1.33</td><td>-0.48</td></tr></table>
+
+Table 1: Loss figures $L _ { R }$ and $L _ { K L }$ ) for various $w _ { K L }$ settings.
+
+The relative loss numbers are consistent with our expectations. We see that the reconstruction loss term $L _ { R }$ decreases as we relax the $w _ { K L }$ parameter controlling the weight for the KL loss term, and meanwhile the KL loss term $L _ { R }$ increases as a result. The $L _ { R }$ for the conditional model is strictly less than the unconditional, standalone decoder model. In Figure 4 (right), we plot validation-set loss graphs for on the yoga class for models with various $w _ { K L }$ settings. As $L _ { R }$ decreases, the $L _ { K L }$ term tends to increase due to the tradeoff between $L _ { R }$ and $L _ { K L }$ .
+
+# 4.1 CONDITIONAL RECONSTRUCTION
+
+We qualitatively assess the reconstructed sketch $S ^ { \prime }$ given an input sketch $S$ . In Figure 5 (left), we sample several reconstructions at various levels of temperature $\tau$ using a model trained on the single cat class, starting at 0.01 on the left and linearly increasing to 1.0 on the right. The reconstructed cat sketches have similar properties as the input image, and occasionally add or remove details such as a whisker, a mouth, a nose, or the orientation of the tail.
+
+![](images/7dacd33d6e318f898b53332f59eae5eb29864aead3b2fd214f39831f11d3d78e.jpg)  
+Figure 5: Conditional generation of cats (left) and pigs (right).
+
+When presented with a non-standard image of a cat, such as a cat’s face with three eyes, the reconstructed cat only has two eyes. If we input a sketch from another image class, such a toothbrush, the model seemingly generate sketches with similar orientation and properties as the toothbrush input image, but with some cat-like features such as cat ears, whiskers or feet. We perform a similar experiment with a model trained on the pig class, as shown in Figure 5 (right).
+
+# 4.2 LATENT SPACE INTERPOLATION
+
+By interpolating between latent vectors, we can visualize how one image morphs into another image by visualizing the reconstructions of the interpolations. As we enforce a Gaussian prior on the latent space, we expect fewer gaps in the space between two encoded latent vectors. We expect a model trained using a higher $w _ { K L }$ setting to produce images that are closer to the data manifold given a spherically interpolated (White, 2016) latent vector $z$ , compared to another model trained with a lower wKL.
+
+![](images/952656a9deee8795e3fd805634af165d863f5ad1d8559f623ff7f192d7d2976c.jpg)  
+Figure 6: Latent space interpolation between cat and pig using with various $w _ { K L }$ settings (left). Sketch Drawing Analogies (right).
+
+To demonstrate this, we train several models using various $w _ { K L }$ , on a dataset consisting of both cat and pigs, and we encode two distinct images from the test set - a cat face and a full pig. Figure 6 (left) shows the reconstructed images from the interpolated latent vectors between the two original images. As expected, models trained with higher $w _ { K L }$ produce more coherent interpolated images.
+
+# 4.3 SKETCH DRAWING ANALOGIES
+
+The interpolation example in Figure 6 (left) suggests that the latent vector $z$ encode conceptual features of a sketch. Can we use these features to augment other sketches without such features – for example, adding a body to a cat’s head? Indeed, we find that sketch drawing analogies are possible for models trained with low $L _ { K L }$ numbers. Given the smoothness of the latent space, where any interpolated vector between two latent vectors results in a coherent sketch, we can perform vector arithmetic on the latent vectors encoded from different sketches and explore how the model organizes the latent space to represent different concepts in the manifold of generated sketches.
+
+For example, as shown in Figure 6 (right), we can subtract the latent vector of an encoded pig head from the latent vector of a full pig, to arrive at a vector that represents a body. Adding this difference to the latent vector of a cat head results in a full cat (i.e. cat head $^ +$ body $=$ full cat). We repeat the experiment to remove the body of a full pig. These drawing analogies allow us to explore how the model organizes its latent space to represent different concepts in the manifold of generated sketches.
+
+![](images/0182bb7c1611ea15459e55b331cd79713529168c90ec71dccdcd279207d62404.jpg)  
+4.4 PREDICTING DIFFERENT ENDINGS OF INCOMPLETE SKETCHES
+
+Figure 7: sketch-rnn predicting possible endings of various incomplete sketches (the red lines).
+
+We can use sketch-rnn to finish an incomplete sketch. By using the decoder RNN as a standalone model, we can generate a sketch that is conditioned on the previous points. We use the decoder RNN to first encode an incomplete sketch into a hidden state $h$ . Afterwards, we generate the remaining points of the sketch using $h$ as the initial hidden state. We show results in Figure 7 using decoder-only models trained on individual classes, and sample completions by setting $\tau = 0 . 8$ .
+
+# 5 APPLICATIONS AND FUTURE WORK
+
+We believe sketch-rnn will enable many creative applications. Even the decoder-only model trained on various classes can assist the creative process of an artist by suggesting many possible ways of finishing a sketch, helping artists expand their imagination. In the conditional model, exploring the latent space between different objects can potentially enable artists to find interesting intersections and relationships between different drawings. Even in the simplest use, pattern designers can apply sketch-rnn to generate a large number of similar, but unique designs for textile or wallpaper prints.
+
+As we saw earlier in Section 4.1, a model trained to draw pigs can be made to draw pig-like trucks if given an input sketch of a truck. We can extend this result to applications that might help creative designers come up with abstract designs that can resonate more with their target audience. For instance, in Figure 8 (right), we feed sketches of four different chairs into our cat-drawing model to produce four “chair-like cats”. We can even interpolate between the four images to explore the latent space of chair-like cats, and select from a large grid of generated designs.
+
+![](images/01984930a9a6baed64318e856f5337d4d02cfa7c4120f93cf261122d7ef23045.jpg)  
+Figure 8: Generating similar, but unique sketches based on a single human sketch in the box (left). Latent space of generated cats conditioned on sketch drawings of chairs (right).
+
+A model trained on higher quality sketches may find its way into educational applications that can help teach students how to draw. Even with the simple sketches in QuickDraw, the authors of this work have become much more proficient at drawing animals, insects, and various sea creatures after conducting these experiments. A related application is to encode a crude, poorly sketched drawing and generate more aesthetically looking reproductions by using a model trained with a high $w _ { K L }$ setting and sampling with a low temperature $\tau$ to produce a more coherent version of the drawing. In the future, we can also investigate augmenting the latent vector in the direction that maximizes the aesthetics of the drawing by incorporating user-rating data into the training process.
+
+Combining hybrid variations of sequence-generation models with unsupervised, cross-domain pixel image generation models, such as Image-to-Image models (Dong et al., 2017; Kim et al., 2017; Liu et al., 2017), is another exciting direction that we can explore. We can already combine this model with supervised, cross-domain models such as Pix2Pix (Isola et al., 2016), to occasionally generate photo realistic cat images from generated sketches of cats. The opposite direction of converting a photograph of a cat into an unrealistic, but similar looking sketch of a cat composed of a minimal number of lines seems to be a more interesting problem.
+
+# 6 CONCLUSION
+
+In this work, we develop a methodology to model sketch drawings using recurrent neural networks. sketch-rnn is able to generate possible ways to finish an existing, but unfinished sketch drawing. Our model can also encode existing sketches into a latent vector, and generate similar looking sketches conditioned on the latent space. We demonstrate what it means to interpolate between two different sketches by interpolating between its latent space, and also show that we can manipulate attributes of a sketch by augmenting the latent space. We demonstrate the importance of enforcing a prior distribution on the latent vector for coherent vector image generation during interpolation. By making available a large dataset of sketch drawings, we hope to encourage further research and development in the area of generative vector image modelling.
+
+# 7 ACKNOWLEDGEMENTS
+
+We thank Ian Johnson, Jonas Jongejan, Martin Wattenberg, Mike Schuster, Thomas Deselaers, Ben Poole, Kyle Kastner, Junyoung Chung and Kyle McDonald for their help with this project. This work was done as part of the Google Brain Residency program (g.co/brainresidency).
+
+# REFERENCES
+
+Jimmy L. Ba, Jamie R. Kiros, and Geoffrey E. Hinton. Layer normalization. NIPS, 2016.
+
+Christopher M. Bishop. Mixture density networks. Technical Report, 1994. URL http:// publications.aston.ac.uk/373/.
+
+Samuel R. Bowman, Luke Vilnis, Oriol Vinyals, Andrew M. Dai, Rafal Józefowicz, and Samy Bengio. Generating Sentences from a Continuous Space. CoRR, abs/1511.06349, 2015. URL http://arxiv.org/abs/1511.06349.
+
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+
+# A APPENDIX
+
+# A.1 DATASET DETAILS
+
+![](images/c36584dcc88515e06684c5609daac7ea4e5a8497368a58e0e49c0a8b8b0c0813.jpg)  
+Figure 9: Example sketch drawings from QuickDraw dataset.
+
+The data from QuickDraw (Jongejan et al., 2016) expands daily, and every so often new classes are added to the game. As such, the QuickDraw dataset now consists of hundreds of classes, from 75 classes initially, in Table 2. In total, there are $\sim 5 0$ million sketches in the released dataset, although for the purpose of constructing an organized dataset for research purposes, we have limited the number of sketches in each class.
+
+<table><tr><td rowspan=1 colspan=1>alarm clock</td><td rowspan=1 colspan=1>ambulance</td><td rowspan=1 colspan=1>angel</td><td rowspan=1 colspan=1>ant</td><td rowspan=1 colspan=1>barn</td></tr><tr><td rowspan=1 colspan=1>basket</td><td rowspan=1 colspan=1>bee</td><td rowspan=1 colspan=1>bicycle</td><td rowspan=1 colspan=1>book</td><td rowspan=1 colspan=1>bridge</td></tr><tr><td rowspan=1 colspan=1>bulldozer</td><td rowspan=1 colspan=1>bus</td><td rowspan=1 colspan=1>butterfly</td><td rowspan=1 colspan=1>cactus</td><td rowspan=1 colspan=1>castle</td></tr><tr><td rowspan=1 colspan=1>cat</td><td rowspan=1 colspan=1>chair</td><td rowspan=1 colspan=1>couch</td><td rowspan=1 colspan=1>crab</td><td rowspan=1 colspan=1>cruise ship</td></tr><tr><td rowspan=1 colspan=1>dolphin</td><td rowspan=1 colspan=1>duck</td><td rowspan=1 colspan=1>elephant</td><td rowspan=1 colspan=1>eye</td><td rowspan=1 colspan=1>face</td></tr><tr><td rowspan=1 colspan=1>fan</td><td rowspan=1 colspan=1>fire hydrant</td><td rowspan=1 colspan=1>firetruck</td><td rowspan=1 colspan=1>flamingo</td><td rowspan=1 colspan=1>flower</td></tr><tr><td rowspan=1 colspan=1>garden</td><td rowspan=1 colspan=1>hand</td><td rowspan=1 colspan=1>hedgehog</td><td rowspan=1 colspan=1>helicopter</td><td rowspan=1 colspan=1>kangaroo</td></tr><tr><td rowspan=1 colspan=1>key</td><td rowspan=1 colspan=1>lighthouse</td><td rowspan=1 colspan=1>lion</td><td rowspan=1 colspan=1>map</td><td rowspan=1 colspan=1>mermaid</td></tr><tr><td rowspan=1 colspan=1>octopus</td><td rowspan=1 colspan=1>owl</td><td rowspan=1 colspan=1>paintbrush</td><td rowspan=1 colspan=1>palm tree</td><td rowspan=1 colspan=1>parrot</td></tr><tr><td rowspan=1 colspan=1>passport</td><td rowspan=1 colspan=1>peas</td><td rowspan=1 colspan=1>penguin</td><td rowspan=1 colspan=1>pig</td><td rowspan=1 colspan=1>pineapple</td></tr><tr><td rowspan=1 colspan=1>postcard</td><td rowspan=1 colspan=1>power outlet</td><td rowspan=1 colspan=1>rabbit</td><td rowspan=1 colspan=1>radio</td><td rowspan=1 colspan=1>rain</td></tr><tr><td rowspan=1 colspan=1>rhinoceros</td><td rowspan=1 colspan=1>roller coaster</td><td rowspan=1 colspan=1>sandwich</td><td rowspan=1 colspan=1>scorpion</td><td rowspan=1 colspan=1>sea turtle</td></tr><tr><td rowspan=1 colspan=1>sheep</td><td rowspan=1 colspan=1>skull</td><td rowspan=1 colspan=1>snail</td><td rowspan=1 colspan=1>snowflake</td><td rowspan=1 colspan=1>speedboat</td></tr><tr><td rowspan=1 colspan=1>spider</td><td rowspan=1 colspan=1>strawberry</td><td rowspan=1 colspan=1>swan</td><td rowspan=1 colspan=1>swing set</td><td rowspan=1 colspan=1>tennis racquet</td></tr><tr><td rowspan=1 colspan=1>the mona lisa</td><td rowspan=1 colspan=1>toothbrush</td><td rowspan=1 colspan=1>truck</td><td rowspan=1 colspan=1>whale</td><td rowspan=1 colspan=1>windmill</td></tr></table>
+
+Table 2: Initial 75 QuickDraw classes used for this work.
+
+Each class consists of 70K training samples and 2.5K validation and test samples. Stroke simplification using the Ramer–Douglas–Peucker algorithm (Douglas & Peucker, 1973) with a parameter of $\epsilon = 2 . 0$ has been applied to simplify the lines. The data was originally recorded in pixel-dimensions, so we normalized the offsets $( \Delta x , \Delta y )$ using a single scaling factor. This scaling factor was calculated to adjust the offsets in the training set to have a standard deviation of 1. For simplicity, we do not normalize the offsets $( \Delta x , \Delta y )$ to have zero mean, since the means are already relatively small. Figure 10 shows a training example before normalization of $( \Delta x , \Delta y )$ data columns.
+
+![](images/45118061d03ad7220c3b78b0feeafc147a6628870f4276a6898ec0ff91a61c4a.jpg)  
+Figure 10: A sample sketch, as a sequence of $( \Delta x , \Delta y , p _ { 1 } , p _ { 2 } , p _ { 3 } )$ points and in rendered form. In the rendered sketch, the line color corresponds to the sequential stroke ordering.
+
+# A.2 TRAINING DETAILS
+
+As a recap from the main text, we defined the Reconstruction loss term $L _ { R }$ as:
+
+$$
+\begin{array} { l } { { \displaystyle { \cal L } _ { s } = - \frac { 1 } { N _ { \mathrm { m a x } } } \sum _ { i = 1 } ^ { N _ { s } } \log \Big ( \sum _ { j = 1 } ^ { M } \Pi _ { j , i } \mathcal { N } ( \Delta x _ { i } , \Delta y _ { i } \mid \mu _ { x , j , i } , \mu _ { y , j , i } , \sigma _ { x , j , i } , \sigma _ { y , j , i } , \rho _ { x y , j , i } ) \Big ) } } \\ { { \displaystyle { \cal L } _ { p } = - \frac { 1 } { N _ { \mathrm { m a x } } } \sum _ { i = 1 } ^ { N _ { \mathrm { m a x } } } \sum _ { k = 1 } ^ { 3 } p _ { k , i } \log ( q _ { k , i } ) } } \\ { { \displaystyle { \cal L } _ { R } = { \cal L } _ { s } + { \cal L } _ { p } . } } \end{array}
+$$
+
+We also defined the KL loss term $L _ { K L }$ as:
+
+$$
+L _ { K L } = - \frac { 1 } { 2 N _ { z } } \Big ( 1 + \hat { \sigma } - \mu ^ { 2 } - \exp ( \hat { \sigma } ) \Big ) .
+$$
+
+The loss function in Equation 14 is a weighted sum of both the $L _ { R }$ and $L _ { K L }$ loss terms:
+
+$$
+L o s s = L _ { R } + w _ { K L } L _ { K L } .
+$$
+
+While the loss function in Equation 14 can be used during training, we find that annealing the KL term in the loss function (Equation 15) produced better results. This modification is only used for model training, and the original loss function in Equation 14 is still used to evaluate validation and test sets, and for early stopping.
+
+$$
+\begin{array} { c } { \eta _ { s t e p } = 1 - ( 1 - \eta _ { m i n } ) R ^ { s t e p } } \\ { L o s s _ { t r a i n } = L _ { R } + w _ { K L } \eta _ { s t e p } \operatorname* { m a x } ( L _ { K L } , K L _ { m i n } ) } \end{array}
+$$
+
+We find that annealing the KL loss term generally results in better losses. Annealing the $L _ { K L }$ term in the loss function directs the optimizer to first focus more on the reconstruction term in Equation 12, which is the more difficult loss term of the model to optimize for, before having to deal with optimizing for the KL loss term in Equation 13, a far simpler expression in comparison. This approach has been used in (Bowman et al., 2015; Kaae Sønderby et al., 2016; Kingma et al., 2016). Our annealing term $\eta _ { s t e p }$ starts at $\eta _ { m i n }$ (typically 0 or 0.01) at training step 0, and converges to 1 for large training steps. $R$ is a term close to, but less than 1.
+
+If the distribution of $z$ is close enough to $\mathcal { N } ( 0 , I )$ , we can sample sketches from the decoder using randomly sampled $z$ from $\mathcal { N } ( 0 , I )$ as the input. In practice, we find that going from a larger $L _ { K L }$ value $\left( L _ { K L } > 1 . 0 \right)$ ) to a smaller $L _ { K L }$ value of 0.3 generally results in a substantial increase in the quality of sampled images using randomly sampled $z \sim \mathcal { N } ( 0 , I )$ . However, going from $L _ { K L } = 0 . 3$ to $L _ { K L }$ values closer to zero does not lead to any further noticeable improvements. Hence we find it useful to put a floor on $L _ { K L }$ in the loss function by enforcing $\operatorname* { m a x } ( L _ { K L } , K L _ { m i n } )$ in Equation 15.
+
+The $K L _ { m i n }$ term inside the max operator is typically set to a small value such as 0.10 to 0.50. This term will encourage the optimizer to put less focus on optimizing for the KL loss term $L _ { K L }$ once it is low enough, so we can obtain better metrics for the reconstruction loss term $L _ { R }$ . This approach is similar to the approach described in (Kingma et al., 2016) as free bits, where they apply the max operator separately inside each dimension of the latent vector $z$ .
+
+# A.3 MODEL CONFIGURATION
+
+Our encoder and decoder RNNs consist of 512 and 2048 nodes respectively. In our model, we use $M = 2 0$ mixture components for the decoder RNN. The latent vector $z$ has $N _ { z } = 1 2 8$ dimensions. We apply Layer Normalization (Ba et al., 2016) to our model, and during training apply recurrent dropout [9] with a keep probability of $90 \%$ . We train the model with batch sizes of 100 samples, using Adam (Kingma & Ba, 2015) with a learning rate of 0.0001 and gradient clipping of 1.0. All models are trained with $K L _ { m i n } = 0 . 2 0 , R = 0 . 9 9 9 9 9$ . During training, we perform simple data augmentation by multiplying the offset columns $( \Delta x , \Delta y )$ by two IID random factors chosen uniformly between 0.90 and 1.10. Unless mentioned otherwise, all experiments are conducted with $w _ { K L } = 1 . 0 0$ .
+
+# A.4 MODEL LIMITATIONS
+
+Although sketch-rnn can model a large variety of sketch drawings, there are several limitations in the current approach we wish to highlight. For most single-class datasets, sketch-rnn is capable of modelling sketches up to around 300 data points. The model becomes increasingly difficult to train beyond this length. For our dataset, we applied the Ramer–Douglas–Peucker algorithm (Douglas & Peucker, 1973) to simplify the strokes of the sketch data to less than 200 data points while still keeping most of the important visual information of each sketch.
+
+![](images/517eda5c5623d275776c6bf6cbf13101058e1b15a02a02c9d1312e227a44ed9f.jpg)  
+Figure 11: Unconditional generated sketches of frogs, cats, and crabs at $\tau = 0 . 8$
+
+For more complicated classes of images, such as mermaids or lobsters, the reconstruction loss metrics are not as good compared to simpler classes such as ants, faces or firetrucks. The models trained on these more challenging image classes tend to draw smoother, more circular line segments that do not resemble individual sketches, but rather resemble an averaging of many sketches in the training set. We can see some of this artifact in the frog class, in Figure 11. This smoothness may be analogous to the blurriness effect produced by a Variational Autoencoder (Kingma & Welling, 2013) that is trained on pixel images. Depending on the use case of the model, smooth circular lines can be viewed as aesthetically pleasing and a desirable property.
+
+![](images/5565a659dc5857b2f9eedcd3fd65982f624a52d0b980c1d0a8bc03a922945f15.jpg)  
+Figure 12: Unconditional generations from model trained on 75 classes (left), and from model trained on crab, face, pig and rabbit classes (right).
+
+While both conditional and unconditional models are capable of training on datasets consisting of several classes, such as (cat, pig), and (crab, face, pig, rabbit), sketch-rnn is ineffective at modelling a large number of classes simultaneously. In Figure 12, we sample sketches using an unconditional model trained on 75 classes, and a model trained on 4 classes. The samples generated from the 75-class model are incoherent, with individual sketches displaying features from multiple classes. The four-class unconditional model usually generates samples of a single class, but occasionally also combines features from multiple classes. In the future, we will explore incorporating class information outside of the latent space to handle the modelling of a large number of classes simultaneously.
+
+# A.5 MULTI-SKETCH DRAWING INTERPOLATION
+
+![](images/ddb2eba428729354e835510798299809b1eb3550e424f24bfb014306688b1257.jpg)  
+Figure 13: Example of conditional generated sketches with single class models. Latent space interpolation from left to right, and then top to bottom.
+
+In addition to interpolating between two sketches, like in Figure13, we can also visualize the interpolation between four sketches in latent space to gain further insight from the model. In this section we show more examples conditionally generated with sketch-rnn. We take four generated images, place them on four corners of a grid, and populate the rest of the grid using the interpolation of the latent vectors at the corners. Figure 14 shows two examples of this four-way interpolation, using models trained on both (cat, pig) classes, and face class. All samples generated with $\tau = 0 . 1$ .
+
+![](images/1699457d5a3968730bbd2c9e227caf1f62a328e07b008c4234dd38caa8adb31a.jpg)  
+Figure 14: Example input sketches and sketch-rnn generated reproductions (Top). Latent space interpolation between the four reproduced sketches (Bottom).
+
+The left most figure of Figure 15 visualizes the interpolation between a full pig, a rabbit’s head, a crab, and a face, using a model trained on these four classes. In certain parts of the space between a crab and a face is a rabbit’s head, and we see that the ears of the rabbit becomes the crab’s claws. Applying the model on the yoga class, it is interesting to see how one yoga position slowly transitions to another via a set of interpolated yoga positions generated by the model. For visual effect, we also interpolate between four distinct colors, and color each sketch using a unique interpolated color.
+
+![](images/9b8a2ecca92266c71df8e543b82f3f3692d5a38b2ac52ad2a90b26ea747bfdd0.jpg)  
+Figure 15: Interpolation of (pig, rabbit, crab and face), yoga poses, mosquitoes and mermaids. We also interpolate between four distinct colors for visual effect.
+
+We also construct latent space interpolation examples for the mosquito class and the mermaid class, in the last two grids Figure 15. We see that the model can interpolate between concepts such as style of wings, leg counts, and orientation. In Figure 16 below, we show more interpolation examples of other classes from the dataset.
+
+![](images/fbd3ecbf4be308733e73c68488ee1ab0a69f1b82eee7f2559bb06353a6bcd2ee.jpg)  
+Figure 16: Latent space interpolation between four generated gardens, owls, cats, and firetrucks.
+
+A.6 WHICH LOSS CONTROLS IMAGE COHERENCY?
+
+We would like to question the relative importance of the reconstruction loss term $L _ { R }$ , relative to the KL loss term $L _ { K L }$ , when our goal is to produce higher quality image reconstructions. While our reconstruction loss term $L _ { R }$ optimizes for the log-likelihood of the set of strokes that make up a sketch, this metric alone does not give us any guarantee that a model with a lower $L _ { R }$ number will produce higher quality reconstructions compared to a model with a higher $L _ { R }$ number.
+
+For example, imagine a simple sketch of an face, $\circledddot$ , where most of the data points of $S$ are be used to represent the head, and only a minority of points represent facial features such as the eyes and mouth. It is possible to reconstruct the face with incoherent facial features, and yet still score a lower $L _ { R }$ number compared to another reconstruction with a coherent and similar face, if the edges around the incoherent face are generated more precisely.
+
+In Figure 17, we compare the reconstructed images generated using models trained with various $w _ { K L }$ settings. In the first three examples from the left, we train our model on a dataset consisting of four image classes (crab, face, pig, rabbit). We deliberately sketch input drawings that contain features of two classes, such as a rabbit with a pig mouth and pig tail, a person with animal ears, and a rabbit with crab claws. We see that the model trained using higher $w _ { K L }$ weights, tend to generate sketches with features of a single class that look more coherent, despite having lower $L _ { K L }$ numbers. For instance, the model with $w _ { K L } = 1 . 0 0$ omit pig features, animal ears, and crab claws from its reconstructions. In contrast, the model with $w _ { K L } = 0 . 2 5$ , with higher $L _ { K L }$ , but lower $L _ { R }$ numbers tries to keep both inconsistent features, while generating sketches that look less coherent.
+
+In the last three examples in Figure 17, we repeat the experiment on models trained on single-class images, and see similar results even when we deliberately choose input samples from the test set with noisier lines.
+
+If we look at the interpolations produced in the latent space interpolation examples from Section 4.2 in the main text, models with better KL loss terms also generate more meaningful reconstructions from the interpolated space between two latent vectors. This suggests the latent vector for models with lower $L _ { K L }$ control more meaningful parts of the drawings, such as controlling whether the sketch is an animal head only or a full animal with a body, or whether to draw a cat head or a pig head. Altering such latent vectors can allow us to directly manipulate these animal features. Conversely, altering the latent codes of models with higher $L _ { K L }$ results in scattered movement of individual line segments, rather than alterations of meaningful conceptual features of the animal.
+
+This result is consistent with incoherent reconstructions seen in Figure 17. With a lower $L _ { K L }$ , the model is likely to generate coherent images given any random $z$ . Even with a non-standard, or noisy, input image, the model will still encode a $z$ that produces coherent images. For models with lower $L _ { K L }$ numbers, the encoded latent vectors contain conceptual features belonging to the input image, while for models with higher $L _ { K L }$ numbers, the latent vectors merely encode information about specific line segments. This observation suggests that when using sketch-rnn on a new dataset, we should first try different $w _ { K L }$ settings to evaluate the tradeoff between $L _ { R }$ and $L _ { K L }$ , and then choose a setting for $w _ { K L }$ (and $K L _ { m i n . }$ ) that best suit our requirements.
+
+![](images/024e0ff3fb7fd1a2232a98a7fed9cd4b259669e911b8fbaea37546292b54b650.jpg)  
+Figure 17: Reconstructions of sketch drawings using models with various $w _ { K L }$ settings.

parse/train/Hy6GHpkCW/Hy6GHpkCW_content_list.json

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+    {
+        "type": "text",
+        "text": "A NEURAL REPRESENTATION OF SKETCH DRAWINGS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            98,
+            815,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Ha   \nGoogle Brain   \nhadavid@google.com   \nDouglas Eck   \nGoogle Brain   \ndeck@google.com ",
+        "bbox": [
+            183,
+            145,
+            362,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            522,
+            145,
+            671,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            223,
+            544,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present sketch-rnn, a recurrent neural network (RNN) able to construct stroke-based drawings of common objects. The model is trained on a dataset of human-drawn images representing many different classes. We outline a framework for conditional and unconditional sketch generation, and describe new robust training methods for generating coherent sketch drawings in a vector format. ",
+        "bbox": [
+            232,
+            257,
+            766,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            357,
+            336,
+            372
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, there have been major advancements in generative modelling of images using neural networks as a generative tool. Generative Adversarial Networks (GANs) (Goodfellow, 2016), Variational Inference (VI) (Kingma & Welling, 2013), and Autoregressive (AR) (Reed et al., 2017) models have become popular tools in this fast growing area. Most of the work thus far has been targeted towards modelling low resolution, pixel images. Humans, however, do not understand the world as a grid of pixels, but rather develop abstract concepts to represent what we see. From a young age, we develop the ability to communicate what we see by drawing on paper with a pencil or crayon. In this way we learn to express a sequential, vector representation of an image as a short sequence of strokes. In this paper we investigate an alternative to traditional pixel image modelling approaches, and propose a generative model for vector images. ",
+        "bbox": [
+            173,
+            390,
+            826,
+            529
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/941a5c666d5a87047158428a458c9dac6bdd673a6a74f06f01c223c7f4e0ca16.jpg",
+        "image_caption": [
+            "Figure 1: Latent space interpolation of various vector images produced by our model (left). Interpolation of two different Kanji characters (亀 書) as sequence of strokes (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "Our goal is to train machines to draw and generalize abstract concepts in a manner similar to humans. In this work, as a first step towards this goal, we train our model on a dataset of hand-drawn sketches, each represented as a sequence of motor actions controlling a pen: which direction to move, when to lift the pen up, and when to stop drawing. In doing so, we created a model that potentially has many applications, from assisting the creative process of an artist, to helping teach students how to draw. ",
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+        "type": "text",
+        "text": "This paper makes the following contributions: We outline a framework for both unconditional and conditional generation of vector images composed of a sequence of lines. Our recurrent neural network-based generative model is capable of producing sketches of common objects in a vector format. We develop a training procedure unique to vector images to make the training more robust. In the conditional generation model, we explore the latent space developed by the model to represent a vector image. We also discuss creative applications of our methodology. We make available a dataset of 50 million hand drawn vector images to encourage further development of generative modelling for vector images, and also release an implementation of our model as an open source project.1 ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There is a long history of work related to algorithms that mimic painters. One such work is Portrait Drawing by Paul the Robot (Tresset & Fol Leymarie, 2013; Xie et al., 2012), where an underlying algorithm controlling a mechanical robot arm sketches lines on a canvas with a programmable artistic style to mimic a given digitized portrait of a person. Reinforcement Learning based-approaches (Xie et al., 2012) have been developed to discover a set of paint brush strokes that can best represent a given input photograph. These prior works generally attempt to mimic digitized photographs, rather than develop generative models of vector images. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neural Network-based approaches have been developed for generative models of images, although the majority of neural network-related research on image generation deal with pixel images (Goodfellow, 2016; Isola et al., 2016; Kaae Sønderby et al., 2016; Kingma et al., 2016; Reed et al., 2017; White, 2016). There has been relatively little work done on vector image generation using neural networks. An earlier work (Simhon & Dudek, 2004) makes use of Hidden Markov Models to synthesize lines and curves of a human sketch. More recent work (Graves, 2013) on handwriting generation with Recurrent Neural Networks laid the groundwork for utilizing Mixture Density Networks (Bishop, 1994) to generate continuous data points. Recent works of this approach attempted to generate vectorized Kanji characters (Ha, 2015; Zhang et al., 2016) by modelling Chinese characters as a sequence of pen stroke actions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The approach outlined in this work allows one to explore the latent space representation of vector images. For instance, we can use our model to interpolate between two Kanji characters in Figure 1 by first encoding the characters, represented as a sequence of strokes, into a latent space of embedding vectors. Previous work (Bowman et al., 2015) outlined a methodology to combine Sequence-toSequence models with a Variational Autoencoder to model natural English sentences in latent vector space. A related work (Lake et al., 2015), utilizes probabilistic program induction, rather than neural networks, to perform one-shot modelling of the Omniglot dataset containing images of symbols. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One of the factors limiting research development in the space of generative vector drawings is the lack of publicly available datasets. Previously, the Sketch dataset (Eitz et al., 2012), consisting of 20K vector sketches, was used to explore feature extraction techniques. A subsequent work, the Sketchy dataset (Sangkloy et al., 2016), provided 70K vector sketches along with corresponding pixel images for various classes. This allowed for a larger-scale exploration of human sketches. ShadowDraw (Lee et al., 2011) is an interactive system that predicts what a finished drawing looks like based on a set of incomplete brush strokes from the user while the sketch is being drawn. ShadowDraw used a dataset of 30K raster images combined with extracted vectorized features. In this work, we use a much larger dataset of 50 million vector sketches that is made publicly available. ",
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+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 DATASET ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We constructed QuickDraw, a dataset of 50 million vector drawings obtained from Quick, Draw! (Jongejan et al., 2016), an online game where the players are asked to draw objects belonging to a particular object class in less than 20 seconds. QuickDraw consists of hundreds of classes of common objects. Each class of QuickDraw is a dataset of 70K training samples, in addition to 2.5K validation and 2.5K test samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use a data format that represents a sketch as a set of pen stroke actions. This representation is an extension of the format used in (Graves, 2013). Our format extends the binary pen stroke event into a multi-state event. In this data format, the initial coordinate of the drawing is located at the origin. ",
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+    {
+        "type": "text",
+        "text": "A sketch is a list of points, and each point is a vector consisting of 5 elements: $( \\Delta x , \\Delta y , p _ { 1 } , p _ { 2 } , p _ { 3 } )$ . The first two elements are the offset distance in the $\\mathbf { X }$ and y directions of the pen from the previous point. The last 3 elements represents a binary one-hot vector of 3 possible states. The first pen state, $p _ { 1 }$ , indicates that the pen is currently touching the paper, and that a line will be drawn connecting the next point with the current point. The second pen state, $p _ { 2 }$ , indicates that the pen will be lifted from the paper after the current point, and that no line will be drawn next. The final pen state, $p _ { 3 }$ , indicates that the drawing has ended, and subsequent points, including the current point, will not be rendered. ",
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+    {
+        "type": "text",
+        "text": "3.2 SKETCH-RNN ",
+        "text_level": 1,
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+    {
+        "type": "image",
+        "img_path": "images/d28393eb9a12e71ca8d8c00eef5d747f634fa02a5bb5261f05049c0bdd69e6eb.jpg",
+        "image_caption": [
+            "Figure 2: Schematic diagram of sketch-rnn. "
+        ],
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+        "text": "Our model is a Sequence-to-Sequence Variational Autoencoder (VAE), similar to the architecture described in (Bowman et al., 2015; Kingma & Welling, 2013). Our encoder is a bidirectional RNN (Schuster et al., 1997) that takes in a sketch as an input, and outputs a latent vector of size $N _ { z }$ . Specifically, we feed the sketch sequence, $S$ , and also the same sketch sequence in reverse order, $S _ { \\mathrm { r e v e r s e } }$ , into the two encoding RNNs of the bidirectional RNN, to obtain two final hidden states: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d825974a8b71219ba5dc4b10cf86aed8a7e893ec31b85d18a60c1e890c90706e.jpg",
+        "text": "$$\nh _ { \\right. } = { \\mathrm { e n c o d e } } _ { \\right. } ( S ) , \\ h _ { \\left. } = { \\mathrm { e n c o d e } } _ { \\left. } ( S _ { \\mathrm { r e v e r s e } } ) , \\ h = [ \\ h _ { \\right. } \\ ; \\ h _ { \\left. } ] .\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We take this final concatenated hidden state, $h$ , and project it into two vectors $\\mu$ and $\\hat { \\sigma }$ , each of size $N _ { z }$ , using a fully connected layer. We convert $\\hat { \\sigma }$ into a non-negative standard deviation parameter $\\sigma$ using an exponential operation. We use $\\mu$ and $\\sigma$ , along with $\\mathcal { N } ( 0 , I )$ , a vector of IID Gaussian variables of size $N _ { z }$ , to construct a random vector, $z \\in \\mathbb { R } ^ {  { N _ { z } } }$ , as in the approach for a VAE: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efba65927a9440e81b76bb921eaa5504e80cd716ee9dfac067d37a934b666d00.jpg",
+        "text": "$$\n\\mu = W _ { \\mu } h + b _ { \\mu } , \\hat { \\sigma } = W _ { \\sigma } h + b _ { \\sigma } , \\sigma = \\exp \\Big ( \\frac { \\hat { \\sigma } } { 2 } \\Big ) , z = \\mu + \\sigma \\odot \\mathcal { N } ( 0 , I ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Under this encoding scheme, the latent vector $z$ is not a deterministic output for a given input sketch, but a random vector conditioned on the input sketch. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our decoder is an autoregressive RNN that samples output sketches conditional on a given latent vector $z$ . The initial hidden states $h _ { 0 }$ , and optional cell states $c _ { 0 }$ (if applicable) of the decoder RNN is the output of a single layer network: $[ \\ : h _ { 0 } ; \\ : \\overline { { { c _ { 0 } } } } \\ : ] = \\operatorname { t a n h } ( W _ { z } z + b _ { z } ) \\ :$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "At each step $i$ of the decoder RNN, we feed the previous point, $S _ { i - 1 }$ and the latent vector $z$ in as a concatenated input $x _ { i }$ , where $S _ { 0 }$ is defined as $( 0 , 0 , 1 , 0 , 0 )$ . The output at each time step are the parameters for a probability distribution of the next data point $S _ { i }$ . In Equation 3, we model $( \\Delta x , \\Delta y )$ as a Gaussian mixture model (GMM) with $M$ normal distributions as in (Bishop, 1994; Graves, 2013), and $( q _ { 1 } , q _ { 2 } , q _ { 3 } )$ as a categorical distribution to model the ground truth data $\\left( p _ { 1 } , p _ { 2 } , p _ { 3 } \\right)$ , where $( q _ { 1 } + q _ { 2 } + q _ { 3 } = 1 )$ as done in (Ha, 2015) and (Zhang et al., 2016). Unlike (Graves, 2013), our generated sequence is conditioned from a latent code $z$ sampled from our encoder, which is trained end-to-end alongside the decoder. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ca30dad6af90c64b012efadf3e77dc8aa8b11a1e4ef23eda1dc77c805f499429.jpg",
+        "text": "$$\np ( \\Delta x , \\Delta y ) = \\sum _ { j = 1 } ^ { M } { \\Pi _ { j } \\ N } ( \\Delta x , \\Delta y \\mid \\mu _ { x , j } , \\mu _ { y , j } , \\sigma _ { x , j } , \\sigma _ { y , j } , \\rho _ { x y , j } ) , \\mathrm { ~ w h e r e ~ } \\sum _ { j = 1 } ^ { M } { \\Pi _ { j } } = 1 \\mathrm { ~ a n d ~ } ( \\mathrm { R e } ( \\Delta x , \\Delta y ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "$\\mathcal N ( x , y | \\mu _ { x } , \\mu _ { y } , \\sigma _ { x } , \\sigma _ { y } , \\rho _ { x y } )$ is the probability distribution function for a bivariate normal distribution. Each of the $M$ bivariate normal distributions consist of five parameters: $( \\mu _ { x } , \\mu _ { y } , \\sigma _ { x } , \\sigma _ { y } , \\rho _ { x y } )$ , where $\\mu _ { x }$ and $\\mu _ { y }$ are the means, $\\sigma _ { x }$ and $\\sigma _ { y }$ are the standard deviations, and $\\rho _ { x y }$ is the correlation parameter of each bivariate normal distribution. An additional vector $\\Pi$ of length $M$ , also a categorical distribution, are the mixture weights of the Gaussian mixture model. Hence the size of the output vector $y$ is $5 M + M + 3$ , which includes the 3 logits needed to generate $( q _ { 1 } , q _ { 2 } , q _ { 3 } )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The next hidden state of the RNN, generated with its forward operation, projects into the output vector $y _ { i }$ using a fully-connected layer: ",
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+    {
+        "type": "equation",
+        "img_path": "images/864ed23d2c582dabd5b26f2e2937bd5610a8ff43a3914fcdbc196fe9557c9c48.jpg",
+        "text": "$$\n\\begin{array} { r } { x _ { i } = \\left[ S _ { i - 1 } ~ ; ~ z \\right] , ~ \\left[ h _ { i } ~ ; ~ c _ { i } ~ \\right] = ~ \\mathtt { f o r w a r d } ( x _ { i } , ~ \\left[ h _ { i - 1 } ~ ; ~ c _ { i - 1 } \\right] ) , ~ y _ { i } = W _ { y } h _ { i } + b _ { y } , ~ y _ { i } \\in \\mathbb { R } ^ { 6 M + 3 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The vector $y _ { i }$ is broken down into the parameters of the probability distribution of the next data point: $\\big [ \\left( \\hat { \\Pi } \\mu _ { x } \\mu _ { y } \\hat { \\sigma } _ { x } \\hat { \\sigma } _ { y } \\hat { \\rho } _ { x y } \\right) _ { 1 } \\left( \\hat { \\Pi } \\mu _ { x } \\mu _ { y } \\hat { \\sigma } _ { x } \\hat { \\sigma } _ { y } \\hat { \\rho } _ { x y } \\right) _ { 2 } \\ldots \\left( \\hat { \\Pi } \\mu _ { x } \\mu _ { y } \\hat { \\sigma } _ { x } \\hat { \\sigma } _ { y } \\hat { \\rho } _ { x y } \\right) _ { M } \\left( \\hat { q } _ { 1 } \\hat { q } _ { 2 } \\hat { q } _ { 3 } \\right) \\big ] = y _ { i } .$ (5) As in (Graves, 2013), we apply exp and tanh operations to ensure the standard deviation values are non-negative, and that the correlation value is between $^ { - 1 }$ and 1: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/861ba290d541144cc4a70b54aca573fb0301da33ba93c83feb35d7c5877e3eaf.jpg",
+        "text": "$$\n\\sigma _ { x } = \\exp ( \\hat { \\sigma } _ { x } ) , \\ \\sigma _ { y } = \\exp ( \\hat { \\sigma } _ { y } ) , \\ \\rho _ { x y } = \\operatorname { t a n h } ( \\hat { \\rho } _ { x y } ) .\n$$",
+        "text_format": "latex",
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The probabilities for the categorical distributions are calculated using the outputs as logit value ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ec72f38c91d056431eb0df76c1d92f5aa581fd6f2c3f71b9a0dffcf2b2ab361.jpg",
+        "text": "$$\nq _ { k } = \\frac { \\exp ( \\hat { q } _ { k } ) } { \\sum _ { j = 1 } ^ { 3 } \\exp ( \\hat { q } _ { j } ) } , k \\in \\{ 1 , ~ 2 , ~ 3 \\} , ~ \\Pi _ { k } = \\frac { \\exp ( \\hat { \\Pi } _ { k } ) } { \\sum _ { j = 1 } ^ { M } \\exp ( \\hat { \\Pi } _ { j } ) } , k \\in \\{ 1 , ~ \\ldots , ~ { \\cal M } \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A key challenge is to train our model to know when to stop drawing. Because the probabilities of the three pen stroke events are highly unbalanced, the model becomes more difficult to train. The probability of a $p _ { 1 }$ event is much higher than $p _ { 2 }$ , and the $p _ { 3 }$ event will only happen once per drawing. The approach developed in (Ha, 2015) and later followed by (Zhang et al., 2016) was to use different weightings for each pen event when calculating the losses, such as a hand-tuned weighting of $( 1 , 1 0 , 1 0 0 )$ . We find this inelegant approach to be inadequate for our dataset of diverse images. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We develop a simpler, more robust approach that works well for a broad class of sketch drawing data. In our approach, all sequences are generated to a length of $N _ { \\mathrm { m a x } }$ where $N _ { \\mathrm { m a x } }$ is the length of the longest sketch in our training dataset. In principle $N _ { \\mathrm { m a x } }$ can be considered a hyper parameter. As the length of $S$ is usually shorter than $N _ { \\mathrm { m a x } }$ , we set $S _ { i }$ to be $( 0 , 0 , 0 , 0 , 1 )$ for $i > N _ { s }$ . We discuss the training in detail in the next section. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "After training, we can sample sketches from our model. During the sampling process, we generate the parameters for both GMM and categorical distributions at each time step, and sample an outcome $S _ { i } ^ { \\prime }$ for that time step. Unlike the training process, we feed the sampled outcome $S _ { i } ^ { \\prime }$ as input for the next time step. We continue to sample until $p _ { 3 } = 1$ , or when we have reached $i = N _ { \\operatorname* { m a x } }$ . Like the encoder, the sampled output is not deterministic, but a random sequence, conditioned on the input latent vector $z$ . We can control the level of randomness we would like our samples to have during the sampling process by introducing a temperature parameter $\\tau$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0c9d61cb87c8cbb671cf9e83e6cad4c862c91a030623db6cab30e9a17fa6b2d8.jpg",
+        "text": "$$\n\\hat { q } _ { k } \\to \\frac { \\hat { q } _ { k } } { \\tau } , \\hat { \\Pi } _ { k } \\to \\frac { \\hat { \\Pi } _ { k } } { \\tau } , \\sigma _ { x } ^ { 2 } \\to \\sigma _ { x } ^ { 2 } \\tau , \\sigma _ { y } ^ { 2 } \\to \\sigma _ { y } ^ { 2 } \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We can scale the softmax parameters of the categorial distribution and also the $\\sigma$ parameters of the bivariate normal distribution by a temperature parameter $\\tau$ , to control the level of randomness in our samples. $\\tau$ is typically set between 0 and 1. In the limiting case as $\\tau  0$ , our model becomes deterministic and samples will consist of the most likely point in the probability density function. Figure 3 illustrates of effect of sampling sketches with various temperature parameters. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 UNCONDITIONAL GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/f696899c1ed98946685075eec02fb01a5394625b737c4268bf119a071c335733.jpg",
+        "image_caption": [
+            "Figure 3: Unconditional generation of firetrucks, yoga poses, gardens and owls with varying $\\tau$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As a special case, we can also train our model to generate sketches unconditionally, where we only train the decoder RNN module, without any input or latent vectors. By removing the encoder, the decoder RNN as a standalone model is an autoregressive model without latent variables. In this use case, the initial hidden states and cell states of the decoder RNN are initialized to zero. The inputs $x _ { i }$ of the decoder RNN at each time step is only $S _ { i - 1 }$ or $S _ { i - 1 } ^ { \\prime }$ , as we do not need to concatenate a latent vector $z$ . In Figure 3, we sample various sketch images generated unconditionally by varying the temperature parameter from $\\tau = 0 . 2$ at the top in blue, to $\\tau = 0 . 9$ at the bottom in red. ",
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+    {
+        "type": "text",
+        "text": "3.4 TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our training procedure follows the approach of the Variational Autoencoder (Kingma & Welling, 2013), where the loss function is the sum of two terms: the Reconstruction Loss, $L _ { R }$ , and the Kullback-Leibler Divergence Loss, $L _ { K L }$ . We train our model to optimize this two-part loss function. The Reconstruction loss term, described in Equation 9, maximizes the log-likehood of the generated probability distribution to explain the training data $S$ . We can calculate this reconstruction loss, $L _ { R }$ , using the generated parameters of the pdf and the training data $S$ . $L _ { R }$ is composed of the sum of the log loss of the offset terms $( \\Delta x , \\Delta y )$ , $L _ { s }$ , and the log loss of the pen state terms $( p _ { 1 } , p _ { 2 } , p _ { 3 } ) , L _ { p }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/d9a42fcbb0b70b035cfe1a364b800e6831503124fcdca62ec750631d15b53ffd.jpg",
+        "text": "$$\n\\begin{array} { l } { { { \\cal L } _ { s } = \\displaystyle - \\frac { 1 } { N _ { \\mathrm { m a x } } } \\sum _ { i = 1 } ^ { N _ { s } } \\log \\Big ( \\sum _ { j = 1 } ^ { M } \\Pi _ { j , i } \\mathscr { N } ( \\Delta x _ { i } , \\Delta y _ { i } \\mid \\mu _ { x , j , i } , \\mu _ { y , j , i } , \\sigma _ { x , j , i } , \\sigma _ { y , j , i } , \\rho _ { x y , j , i } ) \\Big ) } } \\\\ { { { \\cal L } _ { p } = \\displaystyle - \\frac { 1 } { N _ { \\mathrm { m a x } } } \\sum _ { i = 1 } ^ { N _ { \\mathrm { m a x } } } \\sum _ { k = 1 } ^ { 3 } p _ { k , i } \\log ( q _ { k , i } ) , \\ L _ { R } = L _ { s } + L _ { p } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that we discard the pdf parameters modelling the $( \\Delta x , \\Delta y )$ points beyond $N _ { s }$ when calculating $L _ { s }$ , while $L _ { p }$ is calculated using all of the pdf parameters modelling the $\\left( p _ { 1 } , p _ { 2 } , p _ { 3 } \\right)$ points until $N _ { \\mathrm { m a x } }$ . Both terms are normalized by the total sequence length $N _ { \\mathrm { m a x } }$ . We found this methodology of loss calculation to be more robust and allows the model to easily learn when it should stop drawing, unlike the earlier mentioned method of assigning importance weightings to $p _ { 1 } , p _ { 2 }$ , and $p _ { 3 }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Kullback-Leibler (KL) divergence loss term measures the difference between the distribution of our latent vector $z$ , to that of an IID Gaussian vector with zero mean and unit variance. Optimizing for this loss term allows us to minimize this difference. We use the result in (Kingma $\\&$ Welling, 2013), and calculate the KL loss term, $L _ { K L }$ , normalized by number of dimensions $N _ { z }$ of $z$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b2b9fa989ea4edab5440e567254008677028fa305240896e0fe4a99b292e1241.jpg",
+        "text": "$$\nL _ { K L } = - \\frac { 1 } { 2 N _ { z } } \\Big ( 1 + \\hat { \\sigma } - \\mu ^ { 2 } - \\exp ( \\hat { \\sigma } ) \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The loss function in Equation 11 is a weighted sum of both the $L _ { R }$ and $L _ { K L }$ loss terms: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9e249d25d7d036481dabda6ba12e3487b444d4c3ddf655108dbdec2a1e3598f.jpg",
+        "text": "$$\nL o s s = L _ { R } + w _ { K L } L _ { K L } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "There is a tradeoff between optimizing for one term over the other. As $w _ { K L }  0$ , our model approaches a pure autoencoder, sacrificing the ability to enforce a prior over our latent space while obtaining better reconstruction loss metrics. Note that for unconditional generation, where our model is the standalone decoder, there will be no $L _ { K L }$ term as we only optimize for $L _ { R }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fd54f56dfc6335ffef4f02ea70c198459fd81cb71d7072401332f0c6af7a695.jpg",
+        "image_caption": [
+            "Figure 4: Tradeoff between $L _ { R }$ and $L _ { K L }$ , for two models trained on single class datasets (left). Validation Loss Graph for models trained on the Yoga dataset using various $w _ { K L }$ (right "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 4 illustrates the tradeoff between different settings of $w _ { K L }$ and the resulting $L _ { R }$ and $L _ { K L }$ metrics on the test set, along with the $L _ { R }$ metric on a standalone decoder RNN for comparison. As the unconditional model does not receive any prior information about the entire sketch it needs to generate, the $L _ { R }$ metric for the standalone decoder model serves as an upper bound for various conditional models using a latent vector. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We conduct several experiments with sketch-rnn for both conditional and unconditional vector image generation. We train sketch-rnn on various QuickDraw classes using various settings for $w _ { K L }$ and record the breakdown of losses. To experiment with a diverse set of classes with varying complexities, we select the cat, pig, face, firetruck, garden, owl, mosquito and yoga class. We also experiment on multi-class datasets by concatenating different classes together to form (cat, pig) and (crab, face, pig, rabbit). The results for test set evaluation on various datasets are displayed in Table 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The sketch-rnn model treats the RNN cell as an abstract component. In our experiments, we use Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) as the encoder RNN. For the decoder RNN, we use HyperLSTM, as this type of RNN cell excels at sequence generation tasks (Ha et al., 2017). The ability for HyperLSTM to spontaneously augment its own weights enables it to adapt to many different regimes in a large diverse dataset. Please see the Appendix for more details. ",
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+    {
+        "type": "table",
+        "img_path": "images/a04839ddc2eefff1bacacf2825d63d7fad0d246999cdb29a18ac501e807f494c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Loss figures $L _ { R }$ and $L _ { K L }$ ) for various $w _ { K L }$ settings. "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td>WKL</td><td>= 1.00</td><td></td><td>WKL = 0.50</td><td>WKL</td><td>= 0.25</td><td>Decoder Only</td></tr><tr><td></td><td>LR</td><td>LKL</td><td>LR</td><td>LKL</td><td>LR</td><td>LKL</td><td>LR</td></tr><tr><td>cat</td><td>-0.98</td><td>0.29</td><td>-1.33</td><td>0.70</td><td>-1.46</td><td>1.01</td><td>-0.57</td></tr><tr><td>pig</td><td>-1.14</td><td>0.22</td><td>-1.37</td><td>0.49</td><td>-1.52</td><td>0.80</td><td>-0.82</td></tr><tr><td>cat, pig</td><td>-1.02</td><td>0.22</td><td>-1.24</td><td>0.49</td><td>-1.50</td><td>0.98</td><td>-0.75</td></tr><tr><td>crab,face,pig,rabbit</td><td>-0.91</td><td>0.22</td><td>-1.04</td><td>0.40</td><td>-1.47</td><td>1.17</td><td>-0.67</td></tr><tr><td>face</td><td>-1.13</td><td>0.27</td><td>-1.55</td><td>0.71</td><td>-1.90</td><td>1.44</td><td>-0.73</td></tr><tr><td>firetruck</td><td>-1.24</td><td>0.22</td><td>-1.26</td><td>0.24</td><td>-1.78</td><td>1.10</td><td>-0.90</td></tr><tr><td>garden</td><td>-0.79</td><td>0.20</td><td>-0.81</td><td>0.25</td><td>-0.99</td><td>0.54</td><td>-0.62</td></tr><tr><td>owl</td><td>-0.93</td><td>0.20</td><td>-1.03</td><td>0.34</td><td>-1.29</td><td>0.77</td><td>-0.66</td></tr><tr><td>mosquito</td><td>-0.67</td><td>0.30</td><td>-1.02</td><td>0.66</td><td>-1.41</td><td>1.54</td><td>-0.34</td></tr><tr><td>yoga</td><td>-0.80</td><td>0.24</td><td>-1.07</td><td>0.55</td><td>-1.51</td><td>1.33</td><td>-0.48</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The relative loss numbers are consistent with our expectations. We see that the reconstruction loss term $L _ { R }$ decreases as we relax the $w _ { K L }$ parameter controlling the weight for the KL loss term, and meanwhile the KL loss term $L _ { R }$ increases as a result. The $L _ { R }$ for the conditional model is strictly less than the unconditional, standalone decoder model. In Figure 4 (right), we plot validation-set loss graphs for on the yoga class for models with various $w _ { K L }$ settings. As $L _ { R }$ decreases, the $L _ { K L }$ term tends to increase due to the tradeoff between $L _ { R }$ and $L _ { K L }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 CONDITIONAL RECONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            597,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We qualitatively assess the reconstructed sketch $S ^ { \\prime }$ given an input sketch $S$ . In Figure 5 (left), we sample several reconstructions at various levels of temperature $\\tau$ using a model trained on the single cat class, starting at 0.01 on the left and linearly increasing to 1.0 on the right. The reconstructed cat sketches have similar properties as the input image, and occasionally add or remove details such as a whisker, a mouth, a nose, or the orientation of the tail. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7dacd33d6e318f898b53332f59eae5eb29864aead3b2fd214f39831f11d3d78e.jpg",
+        "image_caption": [
+            "Figure 5: Conditional generation of cats (left) and pigs (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When presented with a non-standard image of a cat, such as a cat’s face with three eyes, the reconstructed cat only has two eyes. If we input a sketch from another image class, such a toothbrush, the model seemingly generate sketches with similar orientation and properties as the toothbrush input image, but with some cat-like features such as cat ears, whiskers or feet. We perform a similar experiment with a model trained on the pig class, as shown in Figure 5 (right). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 LATENT SPACE INTERPOLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            437,
+            117
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By interpolating between latent vectors, we can visualize how one image morphs into another image by visualizing the reconstructions of the interpolations. As we enforce a Gaussian prior on the latent space, we expect fewer gaps in the space between two encoded latent vectors. We expect a model trained using a higher $w _ { K L }$ setting to produce images that are closer to the data manifold given a spherically interpolated (White, 2016) latent vector $z$ , compared to another model trained with a lower wKL. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/952656a9deee8795e3fd805634af165d863f5ad1d8559f623ff7f192d7d2976c.jpg",
+        "image_caption": [
+            "Figure 6: Latent space interpolation between cat and pig using with various $w _ { K L }$ settings (left). Sketch Drawing Analogies (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate this, we train several models using various $w _ { K L }$ , on a dataset consisting of both cat and pigs, and we encode two distinct images from the test set - a cat face and a full pig. Figure 6 (left) shows the reconstructed images from the interpolated latent vectors between the two original images. As expected, models trained with higher $w _ { K L }$ produce more coherent interpolated images. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 SKETCH DRAWING ANALOGIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            446,
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+            462
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The interpolation example in Figure 6 (left) suggests that the latent vector $z$ encode conceptual features of a sketch. Can we use these features to augment other sketches without such features – for example, adding a body to a cat’s head? Indeed, we find that sketch drawing analogies are possible for models trained with low $L _ { K L }$ numbers. Given the smoothness of the latent space, where any interpolated vector between two latent vectors results in a coherent sketch, we can perform vector arithmetic on the latent vectors encoded from different sketches and explore how the model organizes the latent space to represent different concepts in the manifold of generated sketches. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For example, as shown in Figure 6 (right), we can subtract the latent vector of an encoded pig head from the latent vector of a full pig, to arrive at a vector that represents a body. Adding this difference to the latent vector of a cat head results in a full cat (i.e. cat head $^ +$ body $=$ full cat). We repeat the experiment to remove the body of a full pig. These drawing analogies allow us to explore how the model organizes its latent space to represent different concepts in the manifold of generated sketches. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0182bb7c1611ea15459e55b331cd79713529168c90ec71dccdcd279207d62404.jpg",
+        "image_caption": [
+            "4.4 PREDICTING DIFFERENT ENDINGS OF INCOMPLETE SKETCHES"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            695,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 7: sketch-rnn predicting possible endings of various incomplete sketches (the red lines). ",
+        "bbox": [
+            171,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We can use sketch-rnn to finish an incomplete sketch. By using the decoder RNN as a standalone model, we can generate a sketch that is conditioned on the previous points. We use the decoder RNN to first encode an incomplete sketch into a hidden state $h$ . Afterwards, we generate the remaining points of the sketch using $h$ as the initial hidden state. We show results in Figure 7 using decoder-only models trained on individual classes, and sample completions by setting $\\tau = 0 . 8$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 APPLICATIONS AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We believe sketch-rnn will enable many creative applications. Even the decoder-only model trained on various classes can assist the creative process of an artist by suggesting many possible ways of finishing a sketch, helping artists expand their imagination. In the conditional model, exploring the latent space between different objects can potentially enable artists to find interesting intersections and relationships between different drawings. Even in the simplest use, pattern designers can apply sketch-rnn to generate a large number of similar, but unique designs for textile or wallpaper prints. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As we saw earlier in Section 4.1, a model trained to draw pigs can be made to draw pig-like trucks if given an input sketch of a truck. We can extend this result to applications that might help creative designers come up with abstract designs that can resonate more with their target audience. For instance, in Figure 8 (right), we feed sketches of four different chairs into our cat-drawing model to produce four “chair-like cats”. We can even interpolate between the four images to explore the latent space of chair-like cats, and select from a large grid of generated designs. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/01984930a9a6baed64318e856f5337d4d02cfa7c4120f93cf261122d7ef23045.jpg",
+        "image_caption": [
+            "Figure 8: Generating similar, but unique sketches based on a single human sketch in the box (left). Latent space of generated cats conditioned on sketch drawings of chairs (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A model trained on higher quality sketches may find its way into educational applications that can help teach students how to draw. Even with the simple sketches in QuickDraw, the authors of this work have become much more proficient at drawing animals, insects, and various sea creatures after conducting these experiments. A related application is to encode a crude, poorly sketched drawing and generate more aesthetically looking reproductions by using a model trained with a high $w _ { K L }$ setting and sampling with a low temperature $\\tau$ to produce a more coherent version of the drawing. In the future, we can also investigate augmenting the latent vector in the direction that maximizes the aesthetics of the drawing by incorporating user-rating data into the training process. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Combining hybrid variations of sequence-generation models with unsupervised, cross-domain pixel image generation models, such as Image-to-Image models (Dong et al., 2017; Kim et al., 2017; Liu et al., 2017), is another exciting direction that we can explore. We can already combine this model with supervised, cross-domain models such as Pix2Pix (Isola et al., 2016), to occasionally generate photo realistic cat images from generated sketches of cats. The opposite direction of converting a photograph of a cat into an unrealistic, but similar looking sketch of a cat composed of a minimal number of lines seems to be a more interesting problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            318,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we develop a methodology to model sketch drawings using recurrent neural networks. sketch-rnn is able to generate possible ways to finish an existing, but unfinished sketch drawing. Our model can also encode existing sketches into a latent vector, and generate similar looking sketches conditioned on the latent space. We demonstrate what it means to interpolate between two different sketches by interpolating between its latent space, and also show that we can manipulate attributes of a sketch by augmenting the latent space. We demonstrate the importance of enforcing a prior distribution on the latent vector for coherent vector image generation during interpolation. By making available a large dataset of sketch drawings, we hope to encourage further research and development in the area of generative vector image modelling. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            279,
+            398,
+            295
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Ian Johnson, Jonas Jongejan, Martin Wattenberg, Mike Schuster, Thomas Deselaers, Ben Poole, Kyle Kastner, Junyoung Chung and Kyle McDonald for their help with this project. This work was done as part of the Google Brain Residency program (g.co/brainresidency). ",
+        "bbox": [
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+            352
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            373,
+            287,
+            388
+        ],
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+    },
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 DATASET DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            137,
+            348,
+            151
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/c36584dcc88515e06684c5609daac7ea4e5a8497368a58e0e49c0a8b8b0c0813.jpg",
+        "image_caption": [
+            "Figure 9: Example sketch drawings from QuickDraw dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            172,
+            761,
+            445
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The data from QuickDraw (Jongejan et al., 2016) expands daily, and every so often new classes are added to the game. As such, the QuickDraw dataset now consists of hundreds of classes, from 75 classes initially, in Table 2. In total, there are $\\sim 5 0$ million sketches in the released dataset, although for the purpose of constructing an organized dataset for research purposes, we have limited the number of sketches in each class. ",
+        "bbox": [
+            174,
+            484,
+            825,
+            554
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/9219d8a9023f47f9aeeb3766ee4a2524c7113dbf3a820d9e642e4ab4b1f5b0fe.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Initial 75 QuickDraw classes used for this work. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>alarm clock</td><td rowspan=1 colspan=1>ambulance</td><td rowspan=1 colspan=1>angel</td><td rowspan=1 colspan=1>ant</td><td rowspan=1 colspan=1>barn</td></tr><tr><td rowspan=1 colspan=1>basket</td><td rowspan=1 colspan=1>bee</td><td rowspan=1 colspan=1>bicycle</td><td rowspan=1 colspan=1>book</td><td rowspan=1 colspan=1>bridge</td></tr><tr><td rowspan=1 colspan=1>bulldozer</td><td rowspan=1 colspan=1>bus</td><td rowspan=1 colspan=1>butterfly</td><td rowspan=1 colspan=1>cactus</td><td rowspan=1 colspan=1>castle</td></tr><tr><td rowspan=1 colspan=1>cat</td><td rowspan=1 colspan=1>chair</td><td rowspan=1 colspan=1>couch</td><td rowspan=1 colspan=1>crab</td><td rowspan=1 colspan=1>cruise ship</td></tr><tr><td rowspan=1 colspan=1>dolphin</td><td rowspan=1 colspan=1>duck</td><td rowspan=1 colspan=1>elephant</td><td rowspan=1 colspan=1>eye</td><td rowspan=1 colspan=1>face</td></tr><tr><td rowspan=1 colspan=1>fan</td><td rowspan=1 colspan=1>fire hydrant</td><td rowspan=1 colspan=1>firetruck</td><td rowspan=1 colspan=1>flamingo</td><td rowspan=1 colspan=1>flower</td></tr><tr><td rowspan=1 colspan=1>garden</td><td rowspan=1 colspan=1>hand</td><td rowspan=1 colspan=1>hedgehog</td><td rowspan=1 colspan=1>helicopter</td><td rowspan=1 colspan=1>kangaroo</td></tr><tr><td rowspan=1 colspan=1>key</td><td rowspan=1 colspan=1>lighthouse</td><td rowspan=1 colspan=1>lion</td><td rowspan=1 colspan=1>map</td><td rowspan=1 colspan=1>mermaid</td></tr><tr><td rowspan=1 colspan=1>octopus</td><td rowspan=1 colspan=1>owl</td><td rowspan=1 colspan=1>paintbrush</td><td rowspan=1 colspan=1>palm tree</td><td rowspan=1 colspan=1>parrot</td></tr><tr><td rowspan=1 colspan=1>passport</td><td rowspan=1 colspan=1>peas</td><td rowspan=1 colspan=1>penguin</td><td rowspan=1 colspan=1>pig</td><td rowspan=1 colspan=1>pineapple</td></tr><tr><td rowspan=1 colspan=1>postcard</td><td rowspan=1 colspan=1>power outlet</td><td rowspan=1 colspan=1>rabbit</td><td rowspan=1 colspan=1>radio</td><td rowspan=1 colspan=1>rain</td></tr><tr><td rowspan=1 colspan=1>rhinoceros</td><td rowspan=1 colspan=1>roller coaster</td><td rowspan=1 colspan=1>sandwich</td><td rowspan=1 colspan=1>scorpion</td><td rowspan=1 colspan=1>sea turtle</td></tr><tr><td rowspan=1 colspan=1>sheep</td><td rowspan=1 colspan=1>skull</td><td rowspan=1 colspan=1>snail</td><td rowspan=1 colspan=1>snowflake</td><td rowspan=1 colspan=1>speedboat</td></tr><tr><td rowspan=1 colspan=1>spider</td><td rowspan=1 colspan=1>strawberry</td><td rowspan=1 colspan=1>swan</td><td rowspan=1 colspan=1>swing set</td><td rowspan=1 colspan=1>tennis racquet</td></tr><tr><td rowspan=1 colspan=1>the mona lisa</td><td rowspan=1 colspan=1>toothbrush</td><td rowspan=1 colspan=1>truck</td><td rowspan=1 colspan=1>whale</td><td rowspan=1 colspan=1>windmill</td></tr></table>",
+        "bbox": [
+            250,
+            570,
+            748,
+            787
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Each class consists of 70K training samples and 2.5K validation and test samples. Stroke simplification using the Ramer–Douglas–Peucker algorithm (Douglas & Peucker, 1973) with a parameter of $\\epsilon = 2 . 0$ has been applied to simplify the lines. The data was originally recorded in pixel-dimensions, so we normalized the offsets $( \\Delta x , \\Delta y )$ using a single scaling factor. This scaling factor was calculated to adjust the offsets in the training set to have a standard deviation of 1. For simplicity, we do not normalize the offsets $( \\Delta x , \\Delta y )$ to have zero mean, since the means are already relatively small. Figure 10 shows a training example before normalization of $( \\Delta x , \\Delta y )$ data columns. ",
+        "bbox": [
+            173,
+            825,
+            826,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/45118061d03ad7220c3b78b0feeafc147a6628870f4276a6898ec0ff91a61c4a.jpg",
+        "image_caption": [
+            "Figure 10: A sample sketch, as a sequence of $( \\Delta x , \\Delta y , p _ { 1 } , p _ { 2 } , p _ { 3 } )$ points and in rendered form. In the rendered sketch, the line color corresponds to the sequential stroke ordering. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            98,
+            825,
+            219
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            280,
+            354,
+            295
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As a recap from the main text, we defined the Reconstruction loss term $L _ { R }$ as: ",
+        "bbox": [
+            173,
+            306,
+            686,
+            321
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/435c04dad98d5d1066caafb906462c8727e20fe998b14d5902da206b83fbb118.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal L } _ { s } = - \\frac { 1 } { N _ { \\mathrm { m a x } } } \\sum _ { i = 1 } ^ { N _ { s } } \\log \\Big ( \\sum _ { j = 1 } ^ { M } \\Pi _ { j , i } \\mathcal { N } ( \\Delta x _ { i } , \\Delta y _ { i } \\mid \\mu _ { x , j , i } , \\mu _ { y , j , i } , \\sigma _ { x , j , i } , \\sigma _ { y , j , i } , \\rho _ { x y , j , i } ) \\Big ) } } \\\\ { { \\displaystyle { \\cal L } _ { p } = - \\frac { 1 } { N _ { \\mathrm { m a x } } } \\sum _ { i = 1 } ^ { N _ { \\mathrm { m a x } } } \\sum _ { k = 1 } ^ { 3 } p _ { k , i } \\log ( q _ { k , i } ) } } \\\\ { { \\displaystyle { \\cal L } _ { R } = { \\cal L } _ { s } + { \\cal L } _ { p } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            329,
+            759,
+            438
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also defined the KL loss term $L _ { K L }$ as: ",
+        "bbox": [
+            174,
+            444,
+            450,
+            459
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/695c80f0dd9316175853a5a089883da4c8e9dae19fffe814815e5b989d77504c.jpg",
+        "text": "$$\nL _ { K L } = - \\frac { 1 } { 2 N _ { z } } \\Big ( 1 + \\hat { \\sigma } - \\mu ^ { 2 } - \\exp ( \\hat { \\sigma } ) \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            465,
+            633,
+            497
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The loss function in Equation 14 is a weighted sum of both the $L _ { R }$ and $L _ { K L }$ loss terms: ",
+        "bbox": [
+            173,
+            503,
+            748,
+            520
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5dbdd0e924f6f4848fe57810b038719d4da8423207269af7a72e175ff1edc0ab.jpg",
+        "text": "$$\nL o s s = L _ { R } + w _ { K L } L _ { K L } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            526,
+            584,
+            542
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "While the loss function in Equation 14 can be used during training, we find that annealing the KL term in the loss function (Equation 15) produced better results. This modification is only used for model training, and the original loss function in Equation 14 is still used to evaluate validation and test sets, and for early stopping. ",
+        "bbox": [
+            173,
+            551,
+            825,
+            608
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccf389811c38ade38ac0dadb09ccefb3c48d86ff7c2aed89c304e8a9930fc470.jpg",
+        "text": "$$\n\\begin{array} { c } { \\eta _ { s t e p } = 1 - ( 1 - \\eta _ { m i n } ) R ^ { s t e p } } \\\\ { L o s s _ { t r a i n } = L _ { R } + w _ { K L } \\eta _ { s t e p } \\operatorname* { m a x } ( L _ { K L } , K L _ { m i n } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            614,
+            666,
+            654
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We find that annealing the KL loss term generally results in better losses. Annealing the $L _ { K L }$ term in the loss function directs the optimizer to first focus more on the reconstruction term in Equation 12, which is the more difficult loss term of the model to optimize for, before having to deal with optimizing for the KL loss term in Equation 13, a far simpler expression in comparison. This approach has been used in (Bowman et al., 2015; Kaae Sønderby et al., 2016; Kingma et al., 2016). Our annealing term $\\eta _ { s t e p }$ starts at $\\eta _ { m i n }$ (typically 0 or 0.01) at training step 0, and converges to 1 for large training steps. $R$ is a term close to, but less than 1. ",
+        "bbox": [
+            173,
+            659,
+            825,
+            757
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "If the distribution of $z$ is close enough to $\\mathcal { N } ( 0 , I )$ , we can sample sketches from the decoder using randomly sampled $z$ from $\\mathcal { N } ( 0 , I )$ as the input. In practice, we find that going from a larger $L _ { K L }$ value $\\left( L _ { K L } > 1 . 0 \\right)$ ) to a smaller $L _ { K L }$ value of 0.3 generally results in a substantial increase in the quality of sampled images using randomly sampled $z \\sim \\mathcal { N } ( 0 , I )$ . However, going from $L _ { K L } = 0 . 3$ to $L _ { K L }$ values closer to zero does not lead to any further noticeable improvements. Hence we find it useful to put a floor on $L _ { K L }$ in the loss function by enforcing $\\operatorname* { m a x } ( L _ { K L } , K L _ { m i n } )$ in Equation 15. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            848
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The $K L _ { m i n }$ term inside the max operator is typically set to a small value such as 0.10 to 0.50. This term will encourage the optimizer to put less focus on optimizing for the KL loss term $L _ { K L }$ once it is low enough, so we can obtain better metrics for the reconstruction loss term $L _ { R }$ . This approach is similar to the approach described in (Kingma et al., 2016) as free bits, where they apply the max operator separately inside each dimension of the latent vector $z$ . ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 MODEL CONFIGURATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            393,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our encoder and decoder RNNs consist of 512 and 2048 nodes respectively. In our model, we use $M = 2 0$ mixture components for the decoder RNN. The latent vector $z$ has $N _ { z } = 1 2 8$ dimensions. We apply Layer Normalization (Ba et al., 2016) to our model, and during training apply recurrent dropout [9] with a keep probability of $90 \\%$ . We train the model with batch sizes of 100 samples, using Adam (Kingma & Ba, 2015) with a learning rate of 0.0001 and gradient clipping of 1.0. All models are trained with $K L _ { m i n } = 0 . 2 0 , R = 0 . 9 9 9 9 9$ . During training, we perform simple data augmentation by multiplying the offset columns $( \\Delta x , \\Delta y )$ by two IID random factors chosen uniformly between 0.90 and 1.10. Unless mentioned otherwise, all experiments are conducted with $w _ { K L } = 1 . 0 0$ . ",
+        "bbox": [
+            173,
+            130,
+            826,
+            256
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 MODEL LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            275,
+            367,
+            290
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Although sketch-rnn can model a large variety of sketch drawings, there are several limitations in the current approach we wish to highlight. For most single-class datasets, sketch-rnn is capable of modelling sketches up to around 300 data points. The model becomes increasingly difficult to train beyond this length. For our dataset, we applied the Ramer–Douglas–Peucker algorithm (Douglas & Peucker, 1973) to simplify the strokes of the sketch data to less than 200 data points while still keeping most of the important visual information of each sketch. ",
+        "bbox": [
+            174,
+            301,
+            825,
+            386
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/517eda5c5623d275776c6bf6cbf13101058e1b15a02a02c9d1312e227a44ed9f.jpg",
+        "image_caption": [
+            "Figure 11: Unconditional generated sketches of frogs, cats, and crabs at $\\tau = 0 . 8$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            398,
+            776,
+            522
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For more complicated classes of images, such as mermaids or lobsters, the reconstruction loss metrics are not as good compared to simpler classes such as ants, faces or firetrucks. The models trained on these more challenging image classes tend to draw smoother, more circular line segments that do not resemble individual sketches, but rather resemble an averaging of many sketches in the training set. We can see some of this artifact in the frog class, in Figure 11. This smoothness may be analogous to the blurriness effect produced by a Variational Autoencoder (Kingma & Welling, 2013) that is trained on pixel images. Depending on the use case of the model, smooth circular lines can be viewed as aesthetically pleasing and a desirable property. ",
+        "bbox": [
+            173,
+            565,
+            826,
+            678
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/5565a659dc5857b2f9eedcd3fd65982f624a52d0b980c1d0a8bc03a922945f15.jpg",
+        "image_caption": [
+            "Figure 12: Unconditional generations from model trained on 75 classes (left), and from model trained on crab, face, pig and rabbit classes (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            690,
+            761,
+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "While both conditional and unconditional models are capable of training on datasets consisting of several classes, such as (cat, pig), and (crab, face, pig, rabbit), sketch-rnn is ineffective at modelling a large number of classes simultaneously. In Figure 12, we sample sketches using an unconditional model trained on 75 classes, and a model trained on 4 classes. The samples generated from the 75-class model are incoherent, with individual sketches displaying features from multiple classes. The four-class unconditional model usually generates samples of a single class, but occasionally also combines features from multiple classes. In the future, we will explore incorporating class information outside of the latent space to handle the modelling of a large number of classes simultaneously. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 MULTI-SKETCH DRAWING INTERPOLATION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            250,
+            521,
+            263
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ddb2eba428729354e835510798299809b1eb3550e424f24bfb014306688b1257.jpg",
+        "image_caption": [
+            "Figure 13: Example of conditional generated sketches with single class models. Latent space interpolation from left to right, and then top to bottom. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            282,
+            825,
+            388
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In addition to interpolating between two sketches, like in Figure13, we can also visualize the interpolation between four sketches in latent space to gain further insight from the model. In this section we show more examples conditionally generated with sketch-rnn. We take four generated images, place them on four corners of a grid, and populate the rest of the grid using the interpolation of the latent vectors at the corners. Figure 14 shows two examples of this four-way interpolation, using models trained on both (cat, pig) classes, and face class. All samples generated with $\\tau = 0 . 1$ . ",
+        "bbox": [
+            173,
+            443,
+            826,
+            526
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1699457d5a3968730bbd2c9e227caf1f62a328e07b008c4234dd38caa8adb31a.jpg",
+        "image_caption": [
+            "Figure 14: Example input sketches and sketch-rnn generated reproductions (Top). Latent space interpolation between the four reproduced sketches (Bottom). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            540,
+            826,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The left most figure of Figure 15 visualizes the interpolation between a full pig, a rabbit’s head, a crab, and a face, using a model trained on these four classes. In certain parts of the space between a crab and a face is a rabbit’s head, and we see that the ears of the rabbit becomes the crab’s claws. Applying the model on the yoga class, it is interesting to see how one yoga position slowly transitions to another via a set of interpolated yoga positions generated by the model. For visual effect, we also interpolate between four distinct colors, and color each sketch using a unique interpolated color. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b8a2ecca92266c71df8e543b82f3f3692d5a38b2ac52ad2a90b26ea747bfdd0.jpg",
+        "image_caption": [
+            "Figure 15: Interpolation of (pig, rabbit, crab and face), yoga poses, mosquitoes and mermaids. We also interpolate between four distinct colors for visual effect. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            198,
+            828,
+            325
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also construct latent space interpolation examples for the mosquito class and the mermaid class, in the last two grids Figure 15. We see that the model can interpolate between concepts such as style of wings, leg counts, and orientation. In Figure 16 below, we show more interpolation examples of other classes from the dataset. ",
+        "bbox": [
+            173,
+            373,
+            826,
+            429
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/fbd3ecbf4be308733e73c68488ee1ab0a69f1b82eee7f2559bb06353a6bcd2ee.jpg",
+        "image_caption": [
+            "Figure 16: Latent space interpolation between four generated gardens, owls, cats, and firetrucks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            438,
+            828,
+            565
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.6 WHICH LOSS CONTROLS IMAGE COHERENCY? ",
+        "bbox": [
+            174,
+            618,
+            545,
+            633
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We would like to question the relative importance of the reconstruction loss term $L _ { R }$ , relative to the KL loss term $L _ { K L }$ , when our goal is to produce higher quality image reconstructions. While our reconstruction loss term $L _ { R }$ optimizes for the log-likelihood of the set of strokes that make up a sketch, this metric alone does not give us any guarantee that a model with a lower $L _ { R }$ number will produce higher quality reconstructions compared to a model with a higher $L _ { R }$ number. ",
+        "bbox": [
+            174,
+            645,
+            825,
+            715
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For example, imagine a simple sketch of an face, $\\circledddot$ , where most of the data points of $S$ are be used to represent the head, and only a minority of points represent facial features such as the eyes and mouth. It is possible to reconstruct the face with incoherent facial features, and yet still score a lower $L _ { R }$ number compared to another reconstruction with a coherent and similar face, if the edges around the incoherent face are generated more precisely. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            791
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Figure 17, we compare the reconstructed images generated using models trained with various $w _ { K L }$ settings. In the first three examples from the left, we train our model on a dataset consisting of four image classes (crab, face, pig, rabbit). We deliberately sketch input drawings that contain features of two classes, such as a rabbit with a pig mouth and pig tail, a person with animal ears, and a rabbit with crab claws. We see that the model trained using higher $w _ { K L }$ weights, tend to generate sketches with features of a single class that look more coherent, despite having lower $L _ { K L }$ numbers. For instance, the model with $w _ { K L } = 1 . 0 0$ omit pig features, animal ears, and crab claws from its reconstructions. In contrast, the model with $w _ { K L } = 0 . 2 5$ , with higher $L _ { K L }$ , but lower $L _ { R }$ numbers tries to keep both inconsistent features, while generating sketches that look less coherent. ",
+        "bbox": [
+            173,
+            799,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the last three examples in Figure 17, we repeat the experiment on models trained on single-class images, and see similar results even when we deliberately choose input samples from the test set with noisier lines. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            145
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "If we look at the interpolations produced in the latent space interpolation examples from Section 4.2 in the main text, models with better KL loss terms also generate more meaningful reconstructions from the interpolated space between two latent vectors. This suggests the latent vector for models with lower $L _ { K L }$ control more meaningful parts of the drawings, such as controlling whether the sketch is an animal head only or a full animal with a body, or whether to draw a cat head or a pig head. Altering such latent vectors can allow us to directly manipulate these animal features. Conversely, altering the latent codes of models with higher $L _ { K L }$ results in scattered movement of individual line segments, rather than alterations of meaningful conceptual features of the animal. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            263
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "This result is consistent with incoherent reconstructions seen in Figure 17. With a lower $L _ { K L }$ , the model is likely to generate coherent images given any random $z$ . Even with a non-standard, or noisy, input image, the model will still encode a $z$ that produces coherent images. For models with lower $L _ { K L }$ numbers, the encoded latent vectors contain conceptual features belonging to the input image, while for models with higher $L _ { K L }$ numbers, the latent vectors merely encode information about specific line segments. This observation suggests that when using sketch-rnn on a new dataset, we should first try different $w _ { K L }$ settings to evaluate the tradeoff between $L _ { R }$ and $L _ { K L }$ , and then choose a setting for $w _ { K L }$ (and $K L _ { m i n . }$ ) that best suit our requirements. ",
+        "bbox": [
+            173,
+            270,
+            826,
+            383
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/024e0ff3fb7fd1a2232a98a7fed9cd4b259669e911b8fbaea37546292b54b650.jpg",
+        "image_caption": [
+            "Figure 17: Reconstructions of sketch drawings using models with various $w _ { K L }$ settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            392,
+            826,
+            676
+        ],
+        "page_idx": 15
+    }
+]

parse/train/N1i6BJzouX4/N1i6BJzouX4.md

@@ -0,0 +1,261 @@
+# Rectifying the Shortcut Learning of Background for Few-Shot Learning
+
+Xu Luo1, Longhui Wei2, Liangjian Wen1, Jinrong Yang4, Lingxi Xie3, Zenglin $\mathbf { \bar { X } u } ^ { 6 , 7 * }$ , Qi $\mathbf { T i a n ^ { 5 * } }$
+
+1University of Electronic Science and Technology of China   
+2University of Science and Technology of China 3Tsinghua University   
+4Huazhong University of Science and Technology 5Xidian University   
+6Harbin Institute of Technology Shenzhen 7Pengcheng Laboratory   
+Frank.Luox@outlook.com,{weilh2568,zenglin}@gmail.com
+
+# Abstract
+
+The category gap between training and evaluation has been characterised as one of the main obstacles to the success of Few-Shot Learning (FSL). In this paper, we for the first time empirically identify image background, common in realistic images, as a shortcut knowledge helpful for in-class classification but ungeneralizable beyond training categories in FSL. A novel framework, COSOC, is designed to tackle this problem by extracting foreground objects in images at both training and evaluation without any extra supervision. Extensive experiments carried on inductive FSL tasks demonstrate the effectiveness of our approaches.
+
+# 1 Introduction
+
+Through observing a few samples at a glance, humans can accurately identify brand-new objects. This advantage comes from years of experiences accumulated by the human vision system. Inspired by such learning capabilities, Few-Shot Learning (FSL) is developed to tackle the problem of learning from limited data [24, 53]. At training, FSL models absorb knowledge from a large-scale dataset; later at evaluation, the learned knowledge is leveraged to solve a series of downstream classification tasks, each of which contains very few support (training) images from brand-new categories.
+
+The category gap between training and evaluation has been considered as one of the core issues in FSL [10]. Intuitively, the prior knowledge of old categories learned at training may not be applicable to novel ones. [62] consider solving this problem from a causal perspective. Their backdoor adjustment method, however, adjusts the prior knowledge in a black-box manner and cannot tell which specific prior knowledge is harmful and should be suppressed.
+
+In this paper, we identify image background as one specific harmful source knowledge for FSL. Empirical studies in [56] suggest that there exists spurious correlations between background and category of images (e.g., birds usually stand on branches, and shells often lie on the beaches; see Fig. 1), which serves as a shortcut knowledge for modern CNN-based vision systems to learn. It is further revealed that background knowledge has positive impact on the performance of in-class classification tasks. As illustrated in the simple example of Fig. 1, images from the same category are more likely to share similar background, making it possible for background knowledge to generalize from training to testing in common classification tasks. For FSL, however, the category gap produces brand-new foreground, background and their combinations at evaluation. The correlations learned at training thus may not be able to generalize and would probably mislead the predictions. We take empirical investigations on the role of image foreground and background in FSL, revealing how image background drastically affects the learning and evaluation of FSL in a negative way.
+
+![](images/46ca30684b2417029fe7d6b2f941a7266a99865d3dd331920f761b747db523e1.jpg)  
+Figure 1: An illustrative example that demonstrates why background information is useful for regular classification but harmful for few-shot learning.
+
+Since the background is harmful, it would be good if we could force the model to concentrate on foreground objects at both training and evaluation, but this is not easy since we do not have any prior knowledge of the entity and position of the foreground objects in images. When humans are going to recognize foreground objects of images from the same class, they usually look for a shared local pattern that appears in the majority of images, and recognize patches with this pattern as foreground. This inspires us to design a novel framework, COSOC, to extract foreground of images for both training and evaluation of FSL by seeking shared patterns among images. The approach does not depend on any additional fine-grained supervisions such as bounding boxes or pixel-level labelings.
+
+The procedure of foreground extraction of images in the training set is implemented before training. The corresponding algorithm, named Clustering-based Object Seeker (COS), first pre-trains a feature extractor on the training set using contrative learning, which has an outstanding performance, shown empirically in a later section, on the task of discriminating between ground-truth foreground objects. The feature extractor then maps random crops of images—candidates of foreground objects—into a well-shaped feature space. This is followed by runing a clustering algorithm on all of the features of the same class, imitating the procedure of seeking shared local patterns inspired by human behavior. Each cropped patch is then assigned a foreground score according to its distance to the nearest cluster centroid, for determining a sampling probability of that patch in the later formal training of FSL models. For evaluation, we develop Shared Object Concentrator (SOC), an algorithm that applies iterative feature matching within the support set, looking for one crop per image at one time that is most likely to be foreground. The sorted averaging features of obtained crops are further leveraged to match crops of query images so that foreground crops have higher matching scores. A weighted sum of matching scores are finally calculated as classification logits of each query sample. Compared to other potential foreground extracting algorithms such as saliency-based methods, our COS and SOC algorithms have additional capability of capturing shared, inter-image information, performing better in complicated, multi-object scenery. Our methods also have flexibility of dynamically assigning beliefs (probabilities) to all candidate foreground objects, relieving the risk of overconfidence.
+
+Our contributions can be summarized as follows. i) By conducting empirical studies on the role of image foreground and background in FSL, we reveal that image background serves as a source of shortcut knowledge which harms the evaluation performance. ii) To solve this problem, we propose COSOC, a framework combining COS and SOC, which can draw the model’s attention to image foreground at both training and evaluation. iii) Extensive experiments for non-transductive FSL tasks demonstrate the effectiveness of our method.
+
+# 2 Related Works
+
+Few-shot Image Classification. Plenty of previous work tackled few-shot learning in meta-learning framework [18, 50], where a model learns experience about how to solve few-shot learning tasks by tackling pseudo few-shot classification tasks constructed from the training set. Existing methods that ultilize meta-learning can be generally divided into three groups: (1) Optimization-based methods learn the experience of how to optimize the model given few training samples. This kind of methods either meta-learn a good model initialization point [12, 45, 39, 70, 20] or the whole optimization process [40, 58, 34, 27] or both [3, 36]. (2) Hallucination-based methods [16, 54, 46, 67, 25, 9, 26, 37] learn to augment similar support samples in few-shot tasks, thus can greatly alleviate the low-shot problem. (3) Metric-based methods [53, 48, 49, 61, 59] learn to map images into a metric feature space and classify query images by computing feature distances to support images. Among them, several recent works [19, 63, 57, 10] intended to seek correspondence between images either by attention or meta-filter, in order to obtain a more reasonable similarity measure. Our SOC algorithm in one-shot setting is in spirit similar to these methods, in that we both apply pair-wise feature alignment between support and query images, implicitly removing backgrounds that are more likely to be dissimilar across images. SOC differs in multi-shot setting, where potentially useful shared inter-image information in support set exists and can be captured by our SOC algorithm.
+
+The Influence of Background. A body of prior work studied the impact of image background on learning-based vision systems from different perspectives. [52] showed initial evidence of the existence of background correlations and how it influences the predictions of vision models. [64, 43] analyzed background dependence for object detection. Another relevant work [4] utilized camera traps for investigating how performance drops when adapting classifiers to unseen cameras with novel backgrounds. They explore the effect of class-independent background (i.e., background changes from training to testing while categories remain the same) on classification performance. Although the problem is also concerned with image background, no shortcut learning of background exists under this setting. This is because under each training camera trap, the classifier must distinguish different categories with background fixed, causing the background knowledge being not useful for predictions of training images. Instead, the learning signal during training pushes the classifier towards ignoring each specific background. The difficulties under this setting lie in the domain shift challenge—the classifier is confident to handle previously existing backgrounds, but lost in novel backgrounds. More recently, [56] systematically explore the role of image background in modern deep-learning-based vision systems through well-designed experiments. The results give clear evidence on the existence of background correlations and identify it as a positive shortcut knowledge for models to learn. Our results, on the contrary, identify background correlations as a negative knowledge in the context of few-shot learning.
+
+Contrastive Learning. Recent success on contrastive learning of visual representations has greatly promoted the development of unsupervised learning [6, 17, 15, 5]. The promising performance of contrastive learning relies on the instance-level discrimination loss which maximizes agreement between transformed views of the same image and minimizes agreement between transformed views of different images. Recently there have been some attempts [30, 13, 10, 33, 35, 31] at integrating contrastive learning into the framework of FSL. Although achieving good results, these work struggle to have an in-depth understanding of why contrastive learning has positive effects on FSL. Our work takes a step forward, revealing the advantages of contrastive learning over supervised FSL models in identifying core objects of images.
+
+# 3 Empirical Investigation
+
+Problem Definition. Few-shot learning consists of a training set $\mathcal { D } _ { B }$ and an evaluation set $\mathcal { D } _ { v }$ which share no overlapping classes. $\mathcal { D } _ { B }$ contains a large amount of labeled data and is usually used at first to train a backbone network $f _ { \theta } ( \cdot )$ . After training, a set of $N$ -way $K$ -shot classification tasks $\mathcal { T } = \{ ( \boldsymbol { S } _ { \tau } , \boldsymbol { \mathcal { Q } } _ { \tau } ) \} _ { \tau = 1 } ^ { N _ { T } }$ are constructed, each by first sampling $N$ classes in $D _ { v }$ and then sampling $K$ and $M$ images from each class to constitute $S _ { \tau }$ and $\mathcal { Q } _ { \tau }$ , respectively. In each task $\tau$ , given the learned backbone fθ(·) and a small support set Sτ = {(xτk,n, yτk,n)}K,Nk,n=1 consisting of $K$ images $x _ { k , n } ^ { \tau }$ and corresponding labels $y _ { k , n } ^ { \tau }$ from each of $N$ classes, a few-shot classification algorithm is designed to classify $M N$ images from the query set $\mathcal { Q } _ { \tau } = \{ ( x _ { m n } ^ { \tau } ) \} _ { m , n = 1 } ^ { M , N }$ .
+
+Preparation. To investigate the role of background and foreground in FSL, we need ground-truth image foreground for comparison. However, it is time-consuming to label the whole dataset. Thus we select only a subset $\mathcal { D } _ { \mathrm { n e w } } = ( \mathcal { D } _ { B } , \mathcal { D } _ { v } )$ of miniImageNet [53] and crop each image manually according to the largest rectangular bounding box that contains the foreground object. We denote the uncropped version of the subset as ( $\mathcal { D } _ { B }$ -Ori, $\mathcal { D } _ { v }$ -Ori), and the cropped foreground version as $( \mathcal { D } _ { B } \mathrm { - F G } , \mathbf { \bar { \mathcal { D } } } _ { v } \mathbf { - F G }$ ). Two well-known FSL baselines are selected in our empirical studies: Cosine
+
+![](images/745128dbbb1ff084c514b47b51cba1da332833c2bb78aa51acaa8934edc52c59.jpg)  
+Figure 2: 5-way 5-shot FSL performance on different variants of training and evaluation datasets detailed in Sec. 3. (a) Empirical exploration of image foreground and background in FSL using two models: PN and CC. (b) Comparison between CC and Exemplar trained on the full training set of miniImageNet and evaluated on $\mathcal { D } _ { v }$ -Ori and $\mathcal { D } _ { v }$ -FG.
+
+Classifier (CC) [14] and Prototypical Networks (PN) [48]. See Appendix A for details of constructing $\mathcal { D } _ { \mathrm { n e w } }$ and a formal introdcution of CC and PN.
+
+# 3.1 The Role of Foreground and Background in Few-Shot Image Classification
+
+Fig. 2(a) shows the average of 5-way 5-shot classification accuracy obtained by training CC and PN on $\mathcal { D } _ { B }$ -Ori and $\mathcal { D } _ { B }$ -FG, and evaluating on $\mathcal { D } _ { v }$ -Ori and $\mathcal { D } _ { v }$ -FG, respectively. See Appendix F for additional 5-way 1-shot experiments.
+
+Category gap disables generalization of background knowledge. It can be first noticed that, under any condition, the performance is consistently and significantly improved if background is removed at the evaluation stage (switch from $\mathcal { D } _ { v }$ -Ori to $\mathcal { D } _ { v }$ -FG). The result implies that background at the evaluation stage in FSL is harmful. This is the opposite of that reported in [56] which shows background helps improve on the performance of traditional classification task, where no category gap exists between training and evaluation. Thus we can infer that the class/distribution gap in FSL disables generalization of background knowledge and degrades performance.
+
+Removing background at training prevents shortcut learning. When only foreground is given at evaluation $\mathcal { D } _ { v }$ -FG), the models trained with only foreground ( $\mathcal { D } _ { B }$ -FG) perform much better than those trained with original images $( \mathcal { D } _ { B } \mathrm { - O r i } )$ . This indicates that models trained with original images may not pay enough attention to the foreground object that really matters for classification. Background information at training serves as a shortcut for models to learn and cannot generalize to brand-new classes. In contrast, models trained with only foreground "learn to compare" different objects—a desirable ability for reliable generalization to downstream few-shot learning tasks with out-of-domain classes.
+
+Training with background helps models to handle complex scenes. When evaluating on $\mathcal { D } _ { v }$ -Ori, the models trained with original dataset $\mathcal { D } _ { B }$ -Ori are slightly better than those with foreground dataset $\mathcal { D } _ { B }$ -FG. We attribute this to a sort of domain shift: models trained with $\mathcal { D } _ { B }$ -FG never meet images with complex background and do not know how to handle it. In Appendix D.1 we further verify the assertion by showing evaluation accuracy of each class under the above two training situations. Note that since we apply random crop augmentation at training, domain shift does not exist if the models are instead trained on $\mathcal { D } _ { B }$ -Ori and evaluated on $\mathcal { D } _ { v }$ -FG.
+
+Simple fusion sampling combines advantages of both sides. One may wish to cut off shortcut learning of background while maintaining adaptability of model to complex scenes. A simple solution may be fusion sampling: given an image as input, choose its foreground version with probability $p$ , and its original version with probability $1 - p$ . We simply set $p$ equal to 0.5. We denote the dataset using this sampling strategy as $\mathcal { D } _ { B }$ -Fuse. As observed in Fig. 2(a), models trained this way indeed combine advantages of both sides: achieving relatively good performance on both $\mathcal { D } _ { v }$ -Ori and $\mathcal { D } _ { v }$ -FG. In Appendix C, we compare the training curves of PN trained on three versions of datasets to further investigate the effectiveness of fusion sampling.
+
+The above analysis provides new inspiration for how to improve FSL further: (1) Fusion sampling of foreground and original images could be applied to training. (2) Since background information disturbs evaluation, it is needed to focus on foreground objects or assign image patches, that are more likely to be foreground, a larger weight for classification. Therefore, a foreground object identification mechanism is required at both training (for fusion sampling) and evaluation.
+
+# 3.2 Contrastive Learning is Good at Identifying Objects
+
+In this subsection, we reveal the potential of contrastive learning in identifying foreground objects, which we will use later for foreground extraction. Given one transformed view of one image, contrastive learning tends to distinguish another transformed view of that same image from thousands of views of other images. A more detailed introduction of contrastive learning is given in Appendix B. The two augmented views of the same image always cover the same object, but probably with different parts, sizes and color. To discriminate two augmented patches from thousands of other image patches, the model has to learn to identify the key discriminative information of the object under varying environment. In this manner, semantic relations among crops of images are explicitly modeled, thereby clustering semantically similar contents automatically. The features of different images are pushed away, while those of similar objects in different images are pulled closer. Thus it is reasonable to speculate that contrastive learning may enable models with better identification of centered foreground object.
+
+To verify this, we train CC and contrastive learning models on the whole training set of miniImageNet $\mathcal { D } _ { B }$ -Full) and compare their accuracy on $\mathcal { D } _ { v }$ -Ori and $\mathcal { D } _ { v }$ -FG. The contrastive learning method we use is Exemplar [68], a modified version of MoCo [17]. Fig 2(b) shows that, while the evaluation accuracy of Exemplar on $\mathcal { D } _ { v }$ -Ori is slightly worse than that of CC, Exemplar performs much better when only foreground of images are given at evaluation, affirming that contrastive learning indeed has a better discriminative ability of single centered object. In Appendix D.2, we provide a more in-depth analysis of why contrastive learning has such properties and infer that the shape bias and viewpoint invariance may play an important role.
+
+# 4 Rectifying the Shortcut Learning of Background
+
+Given the analysis in the previous section, we wish to focus more on image foreground both at training and evaluation. Inspired by how humans recognise foreground objects, we propose COSOC, a framework ultilizing contrastive learning to draw the model’s attention to the foreground objects of images.
+
+# 4.1 Clustering-based Object Seeker (COS) with Fusion Sampling for Training
+
+Since contrastive learning is good at discriminating foreground objects, we utilize it to extract foreground objects before training. The first step is to pre-train a backbone $f _ { \theta } ( \cdot )$ on the training set $\mathcal { D } _ { B }$ using Exemplar [68]. Then a clustering-based algorithm is used to extract "objects" identified by the pre-trained model. The basic idea is that features of foreground objects in images within one class extracted by contrastive learalgorithm; see a simple example in ng models are similar, therig. 3. All images within the $i$ by can be i-th class in $\mathcal { D } _ { B }$ ified via a form a set $\{ \mathbf { x } _ { n } ^ { i } \} _ { n = 1 } ^ { N }$ $i$   
+objects in one class is detailed as follows:
+
+1) For each image ${ \bf x } _ { n }$ , we randomly crop it $L$ times to obtain $L$ image patches $\{ \mathbf { p } _ { n , m } \} _ { m = 1 } ^ { L }$ . Each image patch $\mathbf { p } _ { n , m }$ is then passed through the pre-trained model $f _ { \theta }$ and we get a normalized feature vector vn,m $\begin{array} { r } { \mathbf { v } _ { n , m } = \frac { f _ { \theta } ( \mathbf { p } _ { n , m } ) } { | | f _ { \theta } ( \mathbf { p } _ { n , m } ) | | _ { 2 } } \in \mathbb { R } ^ { d } } \end{array}$ .
+
+2) We run a clustering algorithm $\mathcal { A }$ on all features vectors of the class and obtain $H$ clusters $\{ \mathbf { z } _ { j } \} _ { j = 1 } ^ { H } = \mathcal { A } ( \{ \mathbf { v } _ { n , m } \} _ { n , m = 1 } ^ { \bar { N } , L ^ { - } } )$ , where $\mathbf { z } _ { j }$ is the feature centroid of the $j$ -th cluster.
+
+3) We say an image ${ \bf x } _ { n } \in { \bf z } _ { j }$ , if there exists $k \in [ L ]$ s.t. ${ \bf v } _ { n , k } \in { \bf z } _ { j }$ , where $[ L ] = \{ 1 , 2 , \dots , L \}$ . Let $\begin{array} { r } { l ( \mathbf { z } _ { j } ) = \frac { \# \{ \mathbf { x } | \mathbf { x } \in \mathbf { z } _ { j } \} } { N } } \end{array}$ be the proportion of images in the class that belong to $\mathbf { z } _ { j }$ . If $l ( \mathbf { z } _ { j } )$ is small, then the cluster $\mathbf { z } _ { j }$ is not representative for the whole class and is possibly background. Thus we remove all the cluremaining ers cl $\mathbf { z }$ witters $l ( \mathbf { z } ) < \gamma$ , where represe $\gamma$ is a threshold that controls the generality of clusters. The “objects” of the class that we are looking for. $h$ $\{ \mathbf { z } _ { j } \} _ { j = \alpha _ { 1 } } ^ { \alpha _ { h } }$
+
+![](images/7920e4d487b562d01c46a9f315f77b58b065d7f7621741046f41a9f7bda24e5b.jpg)  
+Figure 3: Simplified schematic illustration of COS algorithm. We show how we obtain foreground objects from three exemplified images. The value under each crop denotes its foreground score.
+
+4) The foreground score of image patch $p _ { n , m }$ is defined as $\begin{array} { r } { s _ { n , m } = 1 - \operatorname* { m i n } _ { j \in [ h ] } | | \mathbf { v } _ { n , m } - \mathbf { z } _ { \alpha _ { j } } | | _ { 2 } / \eta } \end{array}$ , where $\eta \ : = \ : \operatorname* { m a x } _ { n , m } \operatorname* { m i n } _ { j \in [ h _ { c } ] } | | \mathbf { v } _ { n , m } - \mathbf { z } _ { \alpha _ { j } } | | _ { 2 }$ is used to normalize the score into $[ 0 , 1 ]$ . Then top- $\mathbf { \nabla } \cdot \mathbf { k }$ scores of each image xn are obtained as {sn,m}βkm=β1 $\{ s _ { n , m } \} _ { m = \beta _ { 1 } } ^ { \beta _ { k } } = \operatorname { T o p k } ( s _ { n , m } )$ . The corresponding patches {pn,m} km=β1 are seen as possible crops of the foreground object in image ${ \bf x } _ { n }$ , and the foreground scores {sn,m}βkm=β as the confidence. We then use it as prior knowledge to rectify the shortcut learning of background for FSL models.
+
+The training strategy resembles fusion sampling introduced before. For an image ${ \bf x } _ { n }$ , the probability that we choose the original version is $1 - \operatorname* { m a x } _ { i \in [ k ] } s _ { n , \beta _ { i } }$ , and the probability of choosing $\mathbf { p } _ { n , \beta _ { j } }$ from top- $\mathbf { \nabla } \cdot \mathbf { k }$ patches is $\big ( s _ { n , \beta _ { j } } / \sum _ { i \in [ k ] } s _ { n , \beta _ { i } } \big ) \cdot \operatorname* { m a x } _ { i \in [ k ] } s _ { n , \beta _ { i } }$ . Then we adjust the chosen image patch and make sure that the least area proportion to the original image keeps as a constant. We use this strategy to train a backbone $f _ { \theta } ( \cdot )$ using a FSL algorithm.
+
+# 4.2 Few-shot Evaluation with Shared Object Concentrator (SOC)
+
+As discussed before, if the foreground crop of the image is used at evaluation, the performance of FSL model will be boosted by a large margin, serving as an upper bound of the model performance. To approach this upper bound, we propose SOC algorithm to capture foreground objects by seeking shared contents among support images of the same class and query images.
+
+![](images/80351e67f7854e28e64b79bed52f3ba918f8d433473dc457fce81a0209cc47e2.jpg)  
+Figure 4: The overall pipeline of step 1 in SOC. Points in one color represent features of crops from one image. The red points are $\omega _ { 1 } , \omega _ { 2 }$ and $\omega _ { 3 }$ .
+
+Step 1: Shared Content Searching within Each Class. For each image $\mathbf { x } _ { k }$ within one class $c$ from support set $S _ { \tau }$ , we randomly crop it $V$ times and obtain corresponding candidates $\{ \mathbf { p } _ { k , n } \} _ { n = 1 , . . , V }$ . Each patch $\mathbf { p } _ { k , n }$ is individually sent to the learned backbone $f _ { \theta }$ to obtain a normalized feature vector ${ \mathbf v } _ { k , n }$ . Thus we have totally $K \times V$ feature vectors within a class $c$ Our goal is to obtain a feature vector $\omega _ { 1 }$ that
+
+contains maximal shared information of all images in class $c$ . Ideally, $\omega _ { 1 }$ represents the centroid o the most similar $K$ image patches, each from one image, which can be formulated as
+
+$$
+\begin{array} { r l } & { \displaystyle \boldsymbol { \omega } _ { 1 } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbf { v } _ { k , \lambda _ { o p t } ( k ) } , } \\ & { \lambda _ { o p t } = \arg \operatorname* { m a x } _ { 1 \leq i < j \leq K } \cos ( \mathbf { v } _ { i , \lambda ( i ) } , \mathbf { v } _ { j , \lambda ( j ) } ) , } \\ & { \quad \quad \lambda { \in } [ K ] ^ { [ V ] } \mathbf { 1 } _ { 1 \leq i < j \leq K } } \end{array}
+$$
+
+where $\cos ( \cdot , \cdot )$ denotes cosine similarity and $[ K ] ^ { [ V ] }$ denotes the set of functions that take $[ K ]$ as domain and $[ V ]$ as range. While $\lambda _ { o p t }$ can be obtained by enumerating all possible combinations of image patches, the computation complexity of this brute-force method is $\mathcal { O } ( V ^ { K } )$ , which is computation prohibitive when $V$ or $K$ is large. Thus when the computation is not affordable, we turn to use a simplified method that leverages iterative optimization. Instead of seeking for the closest image patches, we directly optimize $\omega _ { 1 }$ so that the sum of minimum distance to patches of each image is minimized, i.e.,
+
+$$
+\omega _ { 1 } = \underset { \omega \in \mathcal { R } ^ { d } } { \arg \operatorname* { m a x } } \sum _ { k = 1 } ^ { K } \underset { n } { \operatorname* { m a x } } [ \cos ( \omega , \mathbf { v } _ { k , n } ) ] ,
+$$
+
+which can be achieved by iterative optimization algorithms. We apply SGD in our experiments. After optimization, we remove the patch of each image that is most similar to $\omega _ { 1 }$ , and obtain $K \times ( V - 1 )$ feature vectors. Then we repeatedly implement the above optimization process until no features are left, as shown in Fig. 4. We eventually obtain $V$ sorted feature vectors $\{ \omega _ { n } \} _ { n = 1 } ^ { V }$ , which we use to represent the class $c$ . As for the case where shot $K = 1$ , there is no shared inter-image information inside class, so similar to the handling in PN and DeepEMD [63] , we just skip step 1 and use the original $V$ feature vectors.
+
+Step 2: Feature Matching for Concentrating on Foreground Object of Query Images. Once the foreground class representations are identified, the next step is to use them to implicitly concentrate on foreground of query images by feature matching. For each image $\mathbf { x }$ in the query set $\mathcal { Q } _ { \tau }$ , we also randomly crop it for $V$ times and obtain $V$ candidate features $\{ \mu _ { n } \} _ { n = 1 } ^ { V }$ . For each class $c$ , we have $V$ sorted representative feature vectors $\{ \omega _ { n } \} _ { n = 1 } ^ { V }$ obtained in step 1. We then match the most similar patches between query features and class features, i.e.,
+
+$$
+s _ { 1 } = \operatorname* { m a x } _ { 1 \leq i , j \leq V } [ \alpha ^ { j - 1 } \mathrm { c o s } ( \pmb { \mu } _ { i } , \pmb { \omega } _ { j } ) ] ,
+$$
+
+where $\alpha \leq 1$ is an importance factor. Thus the weight $\alpha ^ { j - 1 }$ decreases exponentially in index $n - 1$ , indicating a decreased belief of each vector representing foreground. Similarly, the two matched class features are removed and the above process repeats until no features left. Finally, the score of $c$ is obtained as a weighted sum of all similarities, i.e., $\begin{array} { r } { S _ { c } = \sum _ { n = 1 } ^ { V } \beta ^ { n - 1 } s _ { n } } \end{array}$ , where $\beta \leq 1$ $\mathbf { x }$ w.r.t. is another importance factor controlling the belief of each crop being foreground objects. In this way, features matched earlier—thus more likely to be foreground—will have higher contributions to the score. The predicted class of $\mathbf { x }$ is the one with the highest score.
+
+# 5 Experiments
+
+# 5.1 Experiment Setup
+
+Dataset. We adopt two benchmark datasets which are the most representative in few-shot learning. The first is miniImageNet [53], a small subset of ILSVRC-12 [44] that contains 600 images within each of the 100 categories. The categories are split into 64, 16, 20 classes for training, validation and evaluation, respectively. The second dataset, tieredImageNet [41], is a much larger subset of ILSVRC12 and is more challenging. It is constructed by choosing 34 super-classes with 608 categories. The super-classes are split into 20, 6, 8 super-classes which ensures separation between training and evaluation categories. The final dataset contains 351, 97, 160 classes for training, validation and evaluation, respectively. On both datasets, the input image size is $8 4 \times 8 4$ for fair comparison.
+
+Evaluation Protocols. We follow the 5-way 5-shot (1-shot) FSL evaluation setting. Specifically, 2000 tasks, each contains 15 testing images and 5 (1) training images per class, are randomly sampled from the evaluation set $\mathcal { D } _ { v }$ and the average classification accuracy is computed. This is repeated 5 times and the mean of the average accuracy with $9 5 \%$ confidence intervals is reported.
+
+Implementation Details. The backbone we use throughout the article is ResNet-12, which is widely used in few-shot learning. We use Pytorch [38] to implement all our experiments on two NVIDIA 1080Ti GPUs. We train the model using SGD with cosine learning rate schedule without restart to reduce the number of hyperparameters (Which epochs to decay the learning rate). The initial learning rate for training Exemplar is 0.1, and for CC is 0.005. The batch size for Exemplar, CC are 256 and 128, respectively. For miniImageNet, we train Exemplar for 150k iterations, and train CC for $^ \mathrm { 6 k }$ iterations. For tieredImageNet, we train Exemplar for approximately $9 0 0 \mathrm { k }$ iterations, and train CC for $1 2 0 \mathrm { k }$ iterations. We choose $\mathbf { k }$ -means [32] as the clustering algorithm for COS. The threshold $\gamma$ is set to 0.5, and top 3 out of 30 features are chosen per image at the training stage. At the evaluation stage, we crop each image 7 times. The importance factors $\alpha$ and $\beta$ are both set to 0.8.
+
+Table 1: Ablative study on miniImageNet. All models are trained on the full training set of miniImageNet. Since the aim of SOC algorithm is to find foreground objects, it is unnecessary to evaluate SOC on the foreground dataset $\mathcal { D } _ { v }$ -FG. FT means finetuning from Exemplar used in COS.   
+
+<table><tr><td rowspan="2">CC</td><td rowspan="2">FT</td><td rowspan="2">COS</td><td rowspan="2">SOC</td><td colspan="2">Du-Ori</td><td colspan="2">Du-FG</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>√</td><td></td><td></td><td></td><td>62.67 ± 0.32</td><td>80.22 ± 0.24</td><td>66.69 ± 0.32</td><td>82.86 ± 0.19</td></tr><tr><td>√</td><td></td><td>√</td><td></td><td>64.76 ± 0.13</td><td>81.18 ± 0.21</td><td>71.13 ± 0.36</td><td>86.21 ± 0.15</td></tr><tr><td>卜</td><td>√</td><td>√</td><td></td><td>65.05 ± 0.06</td><td>81.16 ± 0.17</td><td>71.36 ± 0.30</td><td>86.20 ± 0.14</td></tr><tr><td>厂</td><td></td><td></td><td>√</td><td>64.41 ± 0.22</td><td>81.54 ± 0.28</td><td></td><td>=</td></tr><tr><td></td><td></td><td>√</td><td>√</td><td>69.29 ± 0.12</td><td>84.94 ± 0.28</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Comparisons with baselines of foreground extractors using saliency detection algorithms on miniImageNet. For fair comparison, all models in the right column at evaluation use multi-cropping. GT means evaluating with ground truth foreground.   
+
+<table><tr><td colspan="3">Used for training</td><td colspan="3">Used forevaluation</td></tr><tr><td>Method</td><td>1-shot</td><td>5-shot</td><td>Method</td><td>1-shot</td><td>5-shot</td></tr><tr><td>CC</td><td>62.67 ± 0.32</td><td>80.22 ± 0.24</td><td>COS</td><td>67.23 ± 0.35</td><td>82.79 ± 0.31</td></tr><tr><td>CC+RBD</td><td>63.24 ± 0.41</td><td>80.45 ± 0.37</td><td>COS+RBD</td><td>67.03 ± 0.52</td><td>82.57 ± 0.27</td></tr><tr><td>CC+MBD</td><td>61.50 ± 0.31</td><td>79.12 ± 0.32</td><td>COS+MBD</td><td>62.98 ± 0.45</td><td>79.56 ± 0.38</td></tr><tr><td>CC+FT</td><td>62.71 ± 0.11</td><td>80.06 ± 0.08</td><td>COS+FT</td><td>64.74 ± 0.28</td><td>80.74 ± 0.13</td></tr><tr><td>CC+COS</td><td>64.76 ± 0.13</td><td>81.18 ± 0.21</td><td>COSOC</td><td>69.28 ± 0.49</td><td>85.16 ± 0.42</td></tr><tr><td>1</td><td>1</td><td>=</td><td>COS+GT</td><td>72.71 ± 0.57</td><td>87.43 ± 0.36</td></tr></table>
+
+# 5.2 Model Analysis
+
+In this subsection, we show the effectiveness of each component of our method. Tab. 1 shows the ablation study conducted on miniImageNet.
+
+On the effect of finetuning. Since a feature extractor is pre-trained using contrastive learning in COS, it may help accelerate convergence if we directly finetune from the pre-trained model instead of training from scratch. As shown in line 2-3 in Tab. 1, fintuning gives no improvement on the performance over training from scratch. Thus we adopt finetuning mainly for speeding up convergence $5 \times$ faster).
+
+Effectiveness of COS Algorithm. As observed in Tab. 1, When COS is applied on CC, the performance is improved on both versions of datasets. In Fig. 5, we show the curves of training and validation error of CC during training with and without COS. Both models are trained from scratch and validated on the full miniImageNet. We observe that CC sinks into overfitting: the training accuracy drops to zero, and validation accuracy stops improving before
+
+![](images/71cf6cb98a6373acd448719b594eadf68dc2fbab7cf77278bf966fac3ce2ff3b.jpg)  
+Figure 5: Comparison of training and validation curves between CC with and without COS.
+
+the end of the training. Meanwhile, the COS algorithm helps slow down convergence and prevent training accuracy from reaching zero. This makes validation accuracy comparable at first but higher at the end. Our COS algorithm weakens the “background shortcut” for learning, draws model’s attention on foreground objects, and improves upon generalization.
+
+Effectiveness of SOC Algorithm. The result in Tab. 1 shows that the SOC algorithm is the key to maximally exploit the potential of good object-discrimination ability. The performance even approaches the upper bound performance obtained by evaluating the model on the ground-truth foreground $\mathcal { D } _ { v }$ -FG. One potential unfairness in our SOC algorithm may lie in the use of multicropping, which could possibly lead to performance improvement for other approaches as well. We ablate this concern in Appendix G, as well as in the comparisons to other methods in the later subsections.
+
+Table 3: Comparisons with state-of-the-art models on miniImageNet and tieredImageNet. The average inductive 5-way few-shot classification accuracies with 95 confidence interval are reported. \* indicates methods evaluated using multi-cropping.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">backbone</td><td colspan="2">miniImageNet</td><td colspan="2">tieredImageNet</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>MetaOptNet [22]</td><td>ResNet-12</td><td>62.64 ±0.82</td><td>78.63 ±0.46</td><td>65.99 ± 0.72</td><td>81.56± 0.53</td></tr><tr><td>DC[28]</td><td>ResNet-12</td><td>62.53 ± 0.19</td><td>79.77 ± 0.19</td><td>=</td><td>1</td></tr><tr><td>CTM[25]</td><td>ResNet-18</td><td>64.12 ± 0.82</td><td>80.51 ± 0.13</td><td>68.41 ± 0.39</td><td>84.28 ± 1.73</td></tr><tr><td>CAM[19]</td><td>ResNet-12</td><td>63.85 ± 0.48</td><td>79.44 ± 0.34</td><td>69.89 ± 0.51</td><td>84.23 ± 0.37</td></tr><tr><td>AFHN[26]</td><td>ResNet-18</td><td>62.38 ± 0.72</td><td>78.16 ± 0.56</td><td>=</td><td>1</td></tr><tr><td>DSN [47]</td><td>ResNet-12</td><td>62.64 ± 0.66</td><td>78.83 ± 0.45</td><td>66.22 ± 0.75</td><td>82.79 ± 0.48</td></tr><tr><td>AM3+TRAML[23]</td><td>ResNet-12</td><td>67.10 ± 0.52</td><td>79.54 ± 0.60</td><td></td><td>=</td></tr><tr><td>Net-Cosine [29]</td><td>ResNet-12</td><td>63.85 ± 0.81</td><td>81.57 ± 0.56</td><td></td><td>=</td></tr><tr><td>CA [2]</td><td>WRN-28-10</td><td>65.92 ± 0.60</td><td>82.85 ± 0.55</td><td>74.40 ± 0.68</td><td>86.61 ± 0.59</td></tr><tr><td>MABAS [21]</td><td>ResNet-12</td><td>65.08 ± 0.86</td><td>82.70 ± 0.54</td><td></td><td></td></tr><tr><td>ConsNet [59]</td><td>ResNet-12</td><td>64.89 ±0.23</td><td>79.95 ± 0.17</td><td></td><td>=</td></tr><tr><td>IEPT[66]</td><td>ResNet-12</td><td>67.05 ± 0.44</td><td>82.90 ±0.30</td><td>72.24 ± 0.50</td><td>86.73 ± 0.34</td></tr><tr><td>MELR[11]</td><td>ResNet-12</td><td>67.40 ± 0.43</td><td>83.40 ±0.28</td><td>72.14 ± 0.51</td><td>87.01 ±0.35</td></tr><tr><td>IER-Distill [42]</td><td>ResNet-12</td><td>67.28 ± 0.80</td><td>84.78 ± 0.52</td><td>72.21 ± 0.90</td><td>87.08 ± 0.58</td></tr><tr><td>LDAMF [57]</td><td>ResNet-12</td><td>67.76 ± 0.46</td><td>82.71 ± 0.31</td><td>71.89 ± 0.52</td><td>85.96 ± 0.35</td></tr><tr><td>FRN [55]</td><td>ResNet-12</td><td>66.45 ± 0.19</td><td>82.83 ± 0.13</td><td>72.06 ± 0.22</td><td>86.89 ± 0.14</td></tr><tr><td>Baseline*[7]</td><td>ResNet-12</td><td>63.83 ± 0.67</td><td>81.38 ± 0.41</td><td></td><td></td></tr><tr><td>DeepEMD* [63]</td><td>ResNet-12</td><td>67.63 ± 0.46</td><td>83.47 ± 0.61</td><td>74.29 ± 0.32</td><td>86.98 ± 0.60</td></tr><tr><td>RFS-Distill* [51]</td><td>ResNet-12</td><td>65.02 ± 0.44</td><td>82.04 ± 0.38</td><td>71.52 ± 0.69</td><td>86.03 ± 0.49</td></tr><tr><td>FEAT*[60]</td><td>ResNet-12</td><td>68.03 ±0.38</td><td>82.99 ± 0.31</td><td></td><td>=</td></tr><tr><td>Meta-baseline* [8]</td><td>ResNet-12</td><td>65.31 ± 0.51</td><td>81.26 ± 0.23</td><td>68.62 ± 0.27</td><td>83.74 ±0.18</td></tr><tr><td>COSOC* (ours)</td><td>ResNet-12</td><td>69.28 ± 0.49</td><td>85.16 ± 0.42</td><td>73.57 ± 0.43</td><td>87.57 ± 0.10</td></tr></table>
+
+Note that if we apply only the SOC algorithm on CC, the performance degrades. This indicates that COS and SOC are both necessary: COS provides the discrimination ability of foreground objects and SOC leverages it to maximally boost the performance.
+
+# 5.3 Comparison to Saliency-based Foreground Extractors
+
+There could be other possible ways of extracting foreground objects. A simple yet possibly strong baseline could be running saliency detection to extract the most salient region in an image, followed by cropping to obtain patches without background. We consider comparing with three classical unsupervised saliency methods—RBD [69], FT [1] and MBD [65]. The cropping threshold is specially tuned. For training, fusion sampling with probability 0.5 is used for unsupervised saliency methods. For evaluation, We replace the original images with crops obtained by unsupervised saliency methods directly for classification. Tab. 2 displays the comparisons of performance using different foreground extraction methods applied at training or evaluation. For fair comparison, all methods are trained from scratch, and all compared baselines are evaluated with multi-cropping (i.e. using the average of features obtained from multiple crops for classification)and tested on the same backbone (COS trained).
+
+The results show that: (1) Our method performs consistently much better than the listed unsupervised saliency methods. (2) The performance of different unsupervised saliency methods varies. While RBD gives a small improvement, MBD and FT have negative effect on the performance. The performance severely depends on the effectiveness of unsupervised saliency methods, and is very sensitive to the cropping threshold. Intuitively speaking, saliency detection methods focus on noticeable objects in the image, and might fail when there is another irrelevant salient object in the image (e.g., a man is walking a dog. Dog is the label, but the man is of high saliency). On the contrary, our method focuses on shared objects across images in the same class, thereby avoiding this problem. In addition, our COS algorithm has the ability to dynamically assign foreground scores to different patches, which reduces the risk of overconfidence. One of our main contributions is paving a new way towards improving FSL by rectifying shortcut learning of background, which can be implemented using any effective methods. Given the upper bound with ground truth foreground, we believe there is room to improve and there can be other more effective approaches in the future.
+
+![](images/3ab616a6195363bd563d3bf2e73da6c5f9695a968f315c522360602c0f3b52ac.jpg)  
+Figure 6: Examples of objects obtained with COS from the training set of miniImageNet. The first row shows the original images;the second row shows the picked patch with the highest foreground score.
+
+![](images/7f65b76b5deebbac9324f58291087107217b0b2cc19a6557a4f2fa52a22bee6a.jpg)  
+Figure 7: Visualization examples of the SOC algorithm. The first row displays 5 images that belong to dalmatian and guitar classes respectively from evaluation set of miniImageNet. The second row shows image patches that are picked up from the first round of SOC algorithm. Our method succesfully puts focus on the shared contents/foreground.
+
+# 5.4 Comparison to State-of-the-Arts
+
+Tab. 3 presents 5-way 1-shot and 5-shot classification results on miniImageNet and tieredImageNet. We compare with state-of-the-art few-shot learning methods. For fair comparison, we reimplement some methods, and evaluate them with multi-cropping. See Appendix G for a detailed study on the influence of multi-cropping. Our method achieves state-of-the-art performance under all settings except for 1-shot task on tieredImageNet, on which the performance of our method is slightly worse than CA, which uses WRN-28-10, a deeper backbone, as the feature extractor.
+
+# 5.5 Visualization
+
+Fig. 6 and 7 display visualization examples of the COS and SOC algorithms. See more examples in Appendix H. Thanks to the well-designed mechanism of capturing shared inter-image information, the COS and SOC algorithms are capable of locating foreground patches embodied in complicated, multi-object scenery.
+
+# 6 Conclusion
+
+Few-shot image classification benefits from increasingly more complex network and algorithm design, but little attention has been focused on image itself. In this paper, we reveal that image background serves as a source of harmful knowledge that few-shot learning models easily absorb in. This problem is tackled by our COSOC framework that can draw the model’s attention to image foreground at both training and evaluation. Our method is only one possible solution, and future work may include exploring the potential of unsupervised segmentation or detection algorithms which may be a more reliable alternative of random cropping, or looking for a completely different but better algorithm customized for foreground extraction.
+
+# Acknowledgments and Disclosure of Funding
+
+Special thanks to Qi Yong, who gives indispensable support on the spirit of this paper. We also thank Junran Peng for his help and fruitful discussions. This paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204), and a key program of fundamental research from Shenzhen Science and Technology Innovation Commission (No. JCYJ20200109113403826).
+
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parse/train/N1i6BJzouX4/N1i6BJzouX4_content_list.json

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+        "text": "Rectifying the Shortcut Learning of Background for Few-Shot Learning ",
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+        "text": "Xu Luo1, Longhui Wei2, Liangjian Wen1, Jinrong Yang4, Lingxi Xie3, Zenglin $\\mathbf { \\bar { X } u } ^ { 6 , 7 * }$ , Qi $\\mathbf { T i a n ^ { 5 * } }$ ",
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+        "text": "1University of Electronic Science and Technology of China   \n2University of Science and Technology of China 3Tsinghua University   \n4Huazhong University of Science and Technology 5Xidian University   \n6Harbin Institute of Technology Shenzhen 7Pengcheng Laboratory   \nFrank.Luox@outlook.com,{weilh2568,zenglin}@gmail.com ",
+        "bbox": [
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+        "text": "Abstract ",
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+        "text": "The category gap between training and evaluation has been characterised as one of the main obstacles to the success of Few-Shot Learning (FSL). In this paper, we for the first time empirically identify image background, common in realistic images, as a shortcut knowledge helpful for in-class classification but ungeneralizable beyond training categories in FSL. A novel framework, COSOC, is designed to tackle this problem by extracting foreground objects in images at both training and evaluation without any extra supervision. Extensive experiments carried on inductive FSL tasks demonstrate the effectiveness of our approaches. ",
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+        "text": "1 Introduction ",
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+        "text": "Through observing a few samples at a glance, humans can accurately identify brand-new objects. This advantage comes from years of experiences accumulated by the human vision system. Inspired by such learning capabilities, Few-Shot Learning (FSL) is developed to tackle the problem of learning from limited data [24, 53]. At training, FSL models absorb knowledge from a large-scale dataset; later at evaluation, the learned knowledge is leveraged to solve a series of downstream classification tasks, each of which contains very few support (training) images from brand-new categories. ",
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+        "text": "The category gap between training and evaluation has been considered as one of the core issues in FSL [10]. Intuitively, the prior knowledge of old categories learned at training may not be applicable to novel ones. [62] consider solving this problem from a causal perspective. Their backdoor adjustment method, however, adjusts the prior knowledge in a black-box manner and cannot tell which specific prior knowledge is harmful and should be suppressed. ",
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+        "text": "In this paper, we identify image background as one specific harmful source knowledge for FSL. Empirical studies in [56] suggest that there exists spurious correlations between background and category of images (e.g., birds usually stand on branches, and shells often lie on the beaches; see Fig. 1), which serves as a shortcut knowledge for modern CNN-based vision systems to learn. It is further revealed that background knowledge has positive impact on the performance of in-class classification tasks. As illustrated in the simple example of Fig. 1, images from the same category are more likely to share similar background, making it possible for background knowledge to generalize from training to testing in common classification tasks. For FSL, however, the category gap produces brand-new foreground, background and their combinations at evaluation. The correlations learned at training thus may not be able to generalize and would probably mislead the predictions. We take empirical investigations on the role of image foreground and background in FSL, revealing how image background drastically affects the learning and evaluation of FSL in a negative way. ",
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+        "image_caption": [
+            "Figure 1: An illustrative example that demonstrates why background information is useful for regular classification but harmful for few-shot learning. "
+        ],
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+        "text": "Since the background is harmful, it would be good if we could force the model to concentrate on foreground objects at both training and evaluation, but this is not easy since we do not have any prior knowledge of the entity and position of the foreground objects in images. When humans are going to recognize foreground objects of images from the same class, they usually look for a shared local pattern that appears in the majority of images, and recognize patches with this pattern as foreground. This inspires us to design a novel framework, COSOC, to extract foreground of images for both training and evaluation of FSL by seeking shared patterns among images. The approach does not depend on any additional fine-grained supervisions such as bounding boxes or pixel-level labelings. ",
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+        "text": "The procedure of foreground extraction of images in the training set is implemented before training. The corresponding algorithm, named Clustering-based Object Seeker (COS), first pre-trains a feature extractor on the training set using contrative learning, which has an outstanding performance, shown empirically in a later section, on the task of discriminating between ground-truth foreground objects. The feature extractor then maps random crops of images—candidates of foreground objects—into a well-shaped feature space. This is followed by runing a clustering algorithm on all of the features of the same class, imitating the procedure of seeking shared local patterns inspired by human behavior. Each cropped patch is then assigned a foreground score according to its distance to the nearest cluster centroid, for determining a sampling probability of that patch in the later formal training of FSL models. For evaluation, we develop Shared Object Concentrator (SOC), an algorithm that applies iterative feature matching within the support set, looking for one crop per image at one time that is most likely to be foreground. The sorted averaging features of obtained crops are further leveraged to match crops of query images so that foreground crops have higher matching scores. A weighted sum of matching scores are finally calculated as classification logits of each query sample. Compared to other potential foreground extracting algorithms such as saliency-based methods, our COS and SOC algorithms have additional capability of capturing shared, inter-image information, performing better in complicated, multi-object scenery. Our methods also have flexibility of dynamically assigning beliefs (probabilities) to all candidate foreground objects, relieving the risk of overconfidence. ",
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+        "text": "Our contributions can be summarized as follows. i) By conducting empirical studies on the role of image foreground and background in FSL, we reveal that image background serves as a source of shortcut knowledge which harms the evaluation performance. ii) To solve this problem, we propose COSOC, a framework combining COS and SOC, which can draw the model’s attention to image foreground at both training and evaluation. iii) Extensive experiments for non-transductive FSL tasks demonstrate the effectiveness of our method. ",
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+    {
+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+        "text": "Few-shot Image Classification. Plenty of previous work tackled few-shot learning in meta-learning framework [18, 50], where a model learns experience about how to solve few-shot learning tasks by tackling pseudo few-shot classification tasks constructed from the training set. Existing methods that ultilize meta-learning can be generally divided into three groups: (1) Optimization-based methods learn the experience of how to optimize the model given few training samples. This kind of methods either meta-learn a good model initialization point [12, 45, 39, 70, 20] or the whole optimization process [40, 58, 34, 27] or both [3, 36]. (2) Hallucination-based methods [16, 54, 46, 67, 25, 9, 26, 37] learn to augment similar support samples in few-shot tasks, thus can greatly alleviate the low-shot problem. (3) Metric-based methods [53, 48, 49, 61, 59] learn to map images into a metric feature space and classify query images by computing feature distances to support images. Among them, several recent works [19, 63, 57, 10] intended to seek correspondence between images either by attention or meta-filter, in order to obtain a more reasonable similarity measure. Our SOC algorithm in one-shot setting is in spirit similar to these methods, in that we both apply pair-wise feature alignment between support and query images, implicitly removing backgrounds that are more likely to be dissimilar across images. SOC differs in multi-shot setting, where potentially useful shared inter-image information in support set exists and can be captured by our SOC algorithm. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Influence of Background. A body of prior work studied the impact of image background on learning-based vision systems from different perspectives. [52] showed initial evidence of the existence of background correlations and how it influences the predictions of vision models. [64, 43] analyzed background dependence for object detection. Another relevant work [4] utilized camera traps for investigating how performance drops when adapting classifiers to unseen cameras with novel backgrounds. They explore the effect of class-independent background (i.e., background changes from training to testing while categories remain the same) on classification performance. Although the problem is also concerned with image background, no shortcut learning of background exists under this setting. This is because under each training camera trap, the classifier must distinguish different categories with background fixed, causing the background knowledge being not useful for predictions of training images. Instead, the learning signal during training pushes the classifier towards ignoring each specific background. The difficulties under this setting lie in the domain shift challenge—the classifier is confident to handle previously existing backgrounds, but lost in novel backgrounds. More recently, [56] systematically explore the role of image background in modern deep-learning-based vision systems through well-designed experiments. The results give clear evidence on the existence of background correlations and identify it as a positive shortcut knowledge for models to learn. Our results, on the contrary, identify background correlations as a negative knowledge in the context of few-shot learning. ",
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+        "type": "text",
+        "text": "Contrastive Learning. Recent success on contrastive learning of visual representations has greatly promoted the development of unsupervised learning [6, 17, 15, 5]. The promising performance of contrastive learning relies on the instance-level discrimination loss which maximizes agreement between transformed views of the same image and minimizes agreement between transformed views of different images. Recently there have been some attempts [30, 13, 10, 33, 35, 31] at integrating contrastive learning into the framework of FSL. Although achieving good results, these work struggle to have an in-depth understanding of why contrastive learning has positive effects on FSL. Our work takes a step forward, revealing the advantages of contrastive learning over supervised FSL models in identifying core objects of images. ",
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+        "type": "text",
+        "text": "3 Empirical Investigation ",
+        "text_level": 1,
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+        "text": "Problem Definition. Few-shot learning consists of a training set $\\mathcal { D } _ { B }$ and an evaluation set $\\mathcal { D } _ { v }$ which share no overlapping classes. $\\mathcal { D } _ { B }$ contains a large amount of labeled data and is usually used at first to train a backbone network $f _ { \\theta } ( \\cdot )$ . After training, a set of $N$ -way $K$ -shot classification tasks $\\mathcal { T } = \\{ ( \\boldsymbol { S } _ { \\tau } , \\boldsymbol { \\mathcal { Q } } _ { \\tau } ) \\} _ { \\tau = 1 } ^ { N _ { T } }$ are constructed, each by first sampling $N$ classes in $D _ { v }$ and then sampling $K$ and $M$ images from each class to constitute $S _ { \\tau }$ and $\\mathcal { Q } _ { \\tau }$ , respectively. In each task $\\tau$ , given the learned backbone fθ(·) and a small support set Sτ = {(xτk,n, yτk,n)}K,Nk,n=1 consisting of $K$ images $x _ { k , n } ^ { \\tau }$ and corresponding labels $y _ { k , n } ^ { \\tau }$ from each of $N$ classes, a few-shot classification algorithm is designed to classify $M N$ images from the query set $\\mathcal { Q } _ { \\tau } = \\{ ( x _ { m n } ^ { \\tau } ) \\} _ { m , n = 1 } ^ { M , N }$ . ",
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+        "text": "Preparation. To investigate the role of background and foreground in FSL, we need ground-truth image foreground for comparison. However, it is time-consuming to label the whole dataset. Thus we select only a subset $\\mathcal { D } _ { \\mathrm { n e w } } = ( \\mathcal { D } _ { B } , \\mathcal { D } _ { v } )$ of miniImageNet [53] and crop each image manually according to the largest rectangular bounding box that contains the foreground object. We denote the uncropped version of the subset as ( $\\mathcal { D } _ { B }$ -Ori, $\\mathcal { D } _ { v }$ -Ori), and the cropped foreground version as $( \\mathcal { D } _ { B } \\mathrm { - F G } , \\mathbf { \\bar { \\mathcal { D } } } _ { v } \\mathbf { - F G }$ ). Two well-known FSL baselines are selected in our empirical studies: Cosine ",
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+        "img_path": "images/745128dbbb1ff084c514b47b51cba1da332833c2bb78aa51acaa8934edc52c59.jpg",
+        "image_caption": [
+            "Figure 2: 5-way 5-shot FSL performance on different variants of training and evaluation datasets detailed in Sec. 3. (a) Empirical exploration of image foreground and background in FSL using two models: PN and CC. (b) Comparison between CC and Exemplar trained on the full training set of miniImageNet and evaluated on $\\mathcal { D } _ { v }$ -Ori and $\\mathcal { D } _ { v }$ -FG. "
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+        "type": "text",
+        "text": "Classifier (CC) [14] and Prototypical Networks (PN) [48]. See Appendix A for details of constructing $\\mathcal { D } _ { \\mathrm { n e w } }$ and a formal introdcution of CC and PN. ",
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+        "type": "text",
+        "text": "3.1 The Role of Foreground and Background in Few-Shot Image Classification ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fig. 2(a) shows the average of 5-way 5-shot classification accuracy obtained by training CC and PN on $\\mathcal { D } _ { B }$ -Ori and $\\mathcal { D } _ { B }$ -FG, and evaluating on $\\mathcal { D } _ { v }$ -Ori and $\\mathcal { D } _ { v }$ -FG, respectively. See Appendix F for additional 5-way 1-shot experiments. ",
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+        "type": "text",
+        "text": "Category gap disables generalization of background knowledge. It can be first noticed that, under any condition, the performance is consistently and significantly improved if background is removed at the evaluation stage (switch from $\\mathcal { D } _ { v }$ -Ori to $\\mathcal { D } _ { v }$ -FG). The result implies that background at the evaluation stage in FSL is harmful. This is the opposite of that reported in [56] which shows background helps improve on the performance of traditional classification task, where no category gap exists between training and evaluation. Thus we can infer that the class/distribution gap in FSL disables generalization of background knowledge and degrades performance. ",
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+        "type": "text",
+        "text": "Removing background at training prevents shortcut learning. When only foreground is given at evaluation $\\mathcal { D } _ { v }$ -FG), the models trained with only foreground ( $\\mathcal { D } _ { B }$ -FG) perform much better than those trained with original images $( \\mathcal { D } _ { B } \\mathrm { - O r i } )$ . This indicates that models trained with original images may not pay enough attention to the foreground object that really matters for classification. Background information at training serves as a shortcut for models to learn and cannot generalize to brand-new classes. In contrast, models trained with only foreground \"learn to compare\" different objects—a desirable ability for reliable generalization to downstream few-shot learning tasks with out-of-domain classes. ",
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+        "type": "text",
+        "text": "Training with background helps models to handle complex scenes. When evaluating on $\\mathcal { D } _ { v }$ -Ori, the models trained with original dataset $\\mathcal { D } _ { B }$ -Ori are slightly better than those with foreground dataset $\\mathcal { D } _ { B }$ -FG. We attribute this to a sort of domain shift: models trained with $\\mathcal { D } _ { B }$ -FG never meet images with complex background and do not know how to handle it. In Appendix D.1 we further verify the assertion by showing evaluation accuracy of each class under the above two training situations. Note that since we apply random crop augmentation at training, domain shift does not exist if the models are instead trained on $\\mathcal { D } _ { B }$ -Ori and evaluated on $\\mathcal { D } _ { v }$ -FG. ",
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+        "type": "text",
+        "text": "Simple fusion sampling combines advantages of both sides. One may wish to cut off shortcut learning of background while maintaining adaptability of model to complex scenes. A simple solution may be fusion sampling: given an image as input, choose its foreground version with probability $p$ , and its original version with probability $1 - p$ . We simply set $p$ equal to 0.5. We denote the dataset using this sampling strategy as $\\mathcal { D } _ { B }$ -Fuse. As observed in Fig. 2(a), models trained this way indeed combine advantages of both sides: achieving relatively good performance on both $\\mathcal { D } _ { v }$ -Ori and $\\mathcal { D } _ { v }$ -FG. In Appendix C, we compare the training curves of PN trained on three versions of datasets to further investigate the effectiveness of fusion sampling. ",
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+        "type": "text",
+        "text": "The above analysis provides new inspiration for how to improve FSL further: (1) Fusion sampling of foreground and original images could be applied to training. (2) Since background information disturbs evaluation, it is needed to focus on foreground objects or assign image patches, that are more likely to be foreground, a larger weight for classification. Therefore, a foreground object identification mechanism is required at both training (for fusion sampling) and evaluation. ",
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+        "type": "text",
+        "text": "3.2 Contrastive Learning is Good at Identifying Objects ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this subsection, we reveal the potential of contrastive learning in identifying foreground objects, which we will use later for foreground extraction. Given one transformed view of one image, contrastive learning tends to distinguish another transformed view of that same image from thousands of views of other images. A more detailed introduction of contrastive learning is given in Appendix B. The two augmented views of the same image always cover the same object, but probably with different parts, sizes and color. To discriminate two augmented patches from thousands of other image patches, the model has to learn to identify the key discriminative information of the object under varying environment. In this manner, semantic relations among crops of images are explicitly modeled, thereby clustering semantically similar contents automatically. The features of different images are pushed away, while those of similar objects in different images are pulled closer. Thus it is reasonable to speculate that contrastive learning may enable models with better identification of centered foreground object. ",
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+        "type": "text",
+        "text": "To verify this, we train CC and contrastive learning models on the whole training set of miniImageNet $\\mathcal { D } _ { B }$ -Full) and compare their accuracy on $\\mathcal { D } _ { v }$ -Ori and $\\mathcal { D } _ { v }$ -FG. The contrastive learning method we use is Exemplar [68], a modified version of MoCo [17]. Fig 2(b) shows that, while the evaluation accuracy of Exemplar on $\\mathcal { D } _ { v }$ -Ori is slightly worse than that of CC, Exemplar performs much better when only foreground of images are given at evaluation, affirming that contrastive learning indeed has a better discriminative ability of single centered object. In Appendix D.2, we provide a more in-depth analysis of why contrastive learning has such properties and infer that the shape bias and viewpoint invariance may play an important role. ",
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+        "type": "text",
+        "text": "4 Rectifying the Shortcut Learning of Background ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given the analysis in the previous section, we wish to focus more on image foreground both at training and evaluation. Inspired by how humans recognise foreground objects, we propose COSOC, a framework ultilizing contrastive learning to draw the model’s attention to the foreground objects of images. ",
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+        "type": "text",
+        "text": "4.1 Clustering-based Object Seeker (COS) with Fusion Sampling for Training ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Since contrastive learning is good at discriminating foreground objects, we utilize it to extract foreground objects before training. The first step is to pre-train a backbone $f _ { \\theta } ( \\cdot )$ on the training set $\\mathcal { D } _ { B }$ using Exemplar [68]. Then a clustering-based algorithm is used to extract \"objects\" identified by the pre-trained model. The basic idea is that features of foreground objects in images within one class extracted by contrastive learalgorithm; see a simple example in ng models are similar, therig. 3. All images within the $i$ by can be i-th class in $\\mathcal { D } _ { B }$ ified via a form a set $\\{ \\mathbf { x } _ { n } ^ { i } \\} _ { n = 1 } ^ { N }$ $i$   \nobjects in one class is detailed as follows: ",
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+    {
+        "type": "text",
+        "text": "1) For each image ${ \\bf x } _ { n }$ , we randomly crop it $L$ times to obtain $L$ image patches $\\{ \\mathbf { p } _ { n , m } \\} _ { m = 1 } ^ { L }$ . Each image patch $\\mathbf { p } _ { n , m }$ is then passed through the pre-trained model $f _ { \\theta }$ and we get a normalized feature vector vn,m $\\begin{array} { r } { \\mathbf { v } _ { n , m } = \\frac { f _ { \\theta } ( \\mathbf { p } _ { n , m } ) } { | | f _ { \\theta } ( \\mathbf { p } _ { n , m } ) | | _ { 2 } } \\in \\mathbb { R } ^ { d } } \\end{array}$ . ",
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+        "type": "text",
+        "text": "2) We run a clustering algorithm $\\mathcal { A }$ on all features vectors of the class and obtain $H$ clusters $\\{ \\mathbf { z } _ { j } \\} _ { j = 1 } ^ { H } = \\mathcal { A } ( \\{ \\mathbf { v } _ { n , m } \\} _ { n , m = 1 } ^ { \\bar { N } , L ^ { - } } )$ , where $\\mathbf { z } _ { j }$ is the feature centroid of the $j$ -th cluster. ",
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+        "type": "text",
+        "text": "3) We say an image ${ \\bf x } _ { n } \\in { \\bf z } _ { j }$ , if there exists $k \\in [ L ]$ s.t. ${ \\bf v } _ { n , k } \\in { \\bf z } _ { j }$ , where $[ L ] = \\{ 1 , 2 , \\dots , L \\}$ . Let $\\begin{array} { r } { l ( \\mathbf { z } _ { j } ) = \\frac { \\# \\{ \\mathbf { x } | \\mathbf { x } \\in \\mathbf { z } _ { j } \\} } { N } } \\end{array}$ be the proportion of images in the class that belong to $\\mathbf { z } _ { j }$ . If $l ( \\mathbf { z } _ { j } )$ is small, then the cluster $\\mathbf { z } _ { j }$ is not representative for the whole class and is possibly background. Thus we remove all the cluremaining ers cl $\\mathbf { z }$ witters $l ( \\mathbf { z } ) < \\gamma$ , where represe $\\gamma$ is a threshold that controls the generality of clusters. The “objects” of the class that we are looking for. $h$ $\\{ \\mathbf { z } _ { j } \\} _ { j = \\alpha _ { 1 } } ^ { \\alpha _ { h } }$ ",
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+    {
+        "type": "image",
+        "img_path": "images/7920e4d487b562d01c46a9f315f77b58b065d7f7621741046f41a9f7bda24e5b.jpg",
+        "image_caption": [
+            "Figure 3: Simplified schematic illustration of COS algorithm. We show how we obtain foreground objects from three exemplified images. The value under each crop denotes its foreground score. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "4) The foreground score of image patch $p _ { n , m }$ is defined as $\\begin{array} { r } { s _ { n , m } = 1 - \\operatorname* { m i n } _ { j \\in [ h ] } | | \\mathbf { v } _ { n , m } - \\mathbf { z } _ { \\alpha _ { j } } | | _ { 2 } / \\eta } \\end{array}$ , where $\\eta \\ : = \\ : \\operatorname* { m a x } _ { n , m } \\operatorname* { m i n } _ { j \\in [ h _ { c } ] } | | \\mathbf { v } _ { n , m } - \\mathbf { z } _ { \\alpha _ { j } } | | _ { 2 }$ is used to normalize the score into $[ 0 , 1 ]$ . Then top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ scores of each image xn are obtained as {sn,m}βkm=β1 $\\{ s _ { n , m } \\} _ { m = \\beta _ { 1 } } ^ { \\beta _ { k } } = \\operatorname { T o p k } ( s _ { n , m } )$ . The corresponding patches {pn,m} km=β1 are seen as possible crops of the foreground object in image ${ \\bf x } _ { n }$ , and the foreground scores {sn,m}βkm=β as the confidence. We then use it as prior knowledge to rectify the shortcut learning of background for FSL models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The training strategy resembles fusion sampling introduced before. For an image ${ \\bf x } _ { n }$ , the probability that we choose the original version is $1 - \\operatorname* { m a x } _ { i \\in [ k ] } s _ { n , \\beta _ { i } }$ , and the probability of choosing $\\mathbf { p } _ { n , \\beta _ { j } }$ from top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ patches is $\\big ( s _ { n , \\beta _ { j } } / \\sum _ { i \\in [ k ] } s _ { n , \\beta _ { i } } \\big ) \\cdot \\operatorname* { m a x } _ { i \\in [ k ] } s _ { n , \\beta _ { i } }$ . Then we adjust the chosen image patch and make sure that the least area proportion to the original image keeps as a constant. We use this strategy to train a backbone $f _ { \\theta } ( \\cdot )$ using a FSL algorithm. ",
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+        "type": "text",
+        "text": "4.2 Few-shot Evaluation with Shared Object Concentrator (SOC) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As discussed before, if the foreground crop of the image is used at evaluation, the performance of FSL model will be boosted by a large margin, serving as an upper bound of the model performance. To approach this upper bound, we propose SOC algorithm to capture foreground objects by seeking shared contents among support images of the same class and query images. ",
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+        "type": "image",
+        "img_path": "images/80351e67f7854e28e64b79bed52f3ba918f8d433473dc457fce81a0209cc47e2.jpg",
+        "image_caption": [
+            "Figure 4: The overall pipeline of step 1 in SOC. Points in one color represent features of crops from one image. The red points are $\\omega _ { 1 } , \\omega _ { 2 }$ and $\\omega _ { 3 }$ . "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Step 1: Shared Content Searching within Each Class. For each image $\\mathbf { x } _ { k }$ within one class $c$ from support set $S _ { \\tau }$ , we randomly crop it $V$ times and obtain corresponding candidates $\\{ \\mathbf { p } _ { k , n } \\} _ { n = 1 , . . , V }$ . Each patch $\\mathbf { p } _ { k , n }$ is individually sent to the learned backbone $f _ { \\theta }$ to obtain a normalized feature vector ${ \\mathbf v } _ { k , n }$ . Thus we have totally $K \\times V$ feature vectors within a class $c$ Our goal is to obtain a feature vector $\\omega _ { 1 }$ that ",
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+    {
+        "type": "text",
+        "text": "contains maximal shared information of all images in class $c$ . Ideally, $\\omega _ { 1 }$ represents the centroid o the most similar $K$ image patches, each from one image, which can be formulated as ",
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+    {
+        "type": "equation",
+        "img_path": "images/4503c5d589129aee9fdf5f85b99eda20d383b98354fc9cb49a689592a299e646.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\boldsymbol { \\omega } _ { 1 } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbf { v } _ { k , \\lambda _ { o p t } ( k ) } , } \\\\ & { \\lambda _ { o p t } = \\arg \\operatorname* { m a x } _ { 1 \\leq i < j \\leq K } \\cos ( \\mathbf { v } _ { i , \\lambda ( i ) } , \\mathbf { v } _ { j , \\lambda ( j ) } ) , } \\\\ & { \\quad \\quad \\lambda { \\in } [ K ] ^ { [ V ] } \\mathbf { 1 } _ { 1 \\leq i < j \\leq K } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\cos ( \\cdot , \\cdot )$ denotes cosine similarity and $[ K ] ^ { [ V ] }$ denotes the set of functions that take $[ K ]$ as domain and $[ V ]$ as range. While $\\lambda _ { o p t }$ can be obtained by enumerating all possible combinations of image patches, the computation complexity of this brute-force method is $\\mathcal { O } ( V ^ { K } )$ , which is computation prohibitive when $V$ or $K$ is large. Thus when the computation is not affordable, we turn to use a simplified method that leverages iterative optimization. Instead of seeking for the closest image patches, we directly optimize $\\omega _ { 1 }$ so that the sum of minimum distance to patches of each image is minimized, i.e., ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/0090ea23f03d59094ce2b068bdbf0252bafdbc075b90d8ee77ab6d50fe8e30b4.jpg",
+        "text": "$$\n\\omega _ { 1 } = \\underset { \\omega \\in \\mathcal { R } ^ { d } } { \\arg \\operatorname* { m a x } } \\sum _ { k = 1 } ^ { K } \\underset { n } { \\operatorname* { m a x } } [ \\cos ( \\omega , \\mathbf { v } _ { k , n } ) ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which can be achieved by iterative optimization algorithms. We apply SGD in our experiments. After optimization, we remove the patch of each image that is most similar to $\\omega _ { 1 }$ , and obtain $K \\times ( V - 1 )$ feature vectors. Then we repeatedly implement the above optimization process until no features are left, as shown in Fig. 4. We eventually obtain $V$ sorted feature vectors $\\{ \\omega _ { n } \\} _ { n = 1 } ^ { V }$ , which we use to represent the class $c$ . As for the case where shot $K = 1$ , there is no shared inter-image information inside class, so similar to the handling in PN and DeepEMD [63] , we just skip step 1 and use the original $V$ feature vectors. ",
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+    {
+        "type": "text",
+        "text": "Step 2: Feature Matching for Concentrating on Foreground Object of Query Images. Once the foreground class representations are identified, the next step is to use them to implicitly concentrate on foreground of query images by feature matching. For each image $\\mathbf { x }$ in the query set $\\mathcal { Q } _ { \\tau }$ , we also randomly crop it for $V$ times and obtain $V$ candidate features $\\{ \\mu _ { n } \\} _ { n = 1 } ^ { V }$ . For each class $c$ , we have $V$ sorted representative feature vectors $\\{ \\omega _ { n } \\} _ { n = 1 } ^ { V }$ obtained in step 1. We then match the most similar patches between query features and class features, i.e., ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/fc5ad6e6085fe5c505fbfc6a355709985c416c98dd824bb51e2fe3cf3574780e.jpg",
+        "text": "$$\ns _ { 1 } = \\operatorname* { m a x } _ { 1 \\leq i , j \\leq V } [ \\alpha ^ { j - 1 } \\mathrm { c o s } ( \\pmb { \\mu } _ { i } , \\pmb { \\omega } _ { j } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha \\leq 1$ is an importance factor. Thus the weight $\\alpha ^ { j - 1 }$ decreases exponentially in index $n - 1$ , indicating a decreased belief of each vector representing foreground. Similarly, the two matched class features are removed and the above process repeats until no features left. Finally, the score of $c$ is obtained as a weighted sum of all similarities, i.e., $\\begin{array} { r } { S _ { c } = \\sum _ { n = 1 } ^ { V } \\beta ^ { n - 1 } s _ { n } } \\end{array}$ , where $\\beta \\leq 1$ $\\mathbf { x }$ w.r.t. is another importance factor controlling the belief of each crop being foreground objects. In this way, features matched earlier—thus more likely to be foreground—will have higher contributions to the score. The predicted class of $\\mathbf { x }$ is the one with the highest score. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 Experiment Setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset. We adopt two benchmark datasets which are the most representative in few-shot learning. The first is miniImageNet [53], a small subset of ILSVRC-12 [44] that contains 600 images within each of the 100 categories. The categories are split into 64, 16, 20 classes for training, validation and evaluation, respectively. The second dataset, tieredImageNet [41], is a much larger subset of ILSVRC12 and is more challenging. It is constructed by choosing 34 super-classes with 608 categories. The super-classes are split into 20, 6, 8 super-classes which ensures separation between training and evaluation categories. The final dataset contains 351, 97, 160 classes for training, validation and evaluation, respectively. On both datasets, the input image size is $8 4 \\times 8 4$ for fair comparison. ",
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+        "type": "text",
+        "text": "Evaluation Protocols. We follow the 5-way 5-shot (1-shot) FSL evaluation setting. Specifically, 2000 tasks, each contains 15 testing images and 5 (1) training images per class, are randomly sampled from the evaluation set $\\mathcal { D } _ { v }$ and the average classification accuracy is computed. This is repeated 5 times and the mean of the average accuracy with $9 5 \\%$ confidence intervals is reported. ",
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+        "text": "Implementation Details. The backbone we use throughout the article is ResNet-12, which is widely used in few-shot learning. We use Pytorch [38] to implement all our experiments on two NVIDIA 1080Ti GPUs. We train the model using SGD with cosine learning rate schedule without restart to reduce the number of hyperparameters (Which epochs to decay the learning rate). The initial learning rate for training Exemplar is 0.1, and for CC is 0.005. The batch size for Exemplar, CC are 256 and 128, respectively. For miniImageNet, we train Exemplar for 150k iterations, and train CC for $^ \\mathrm { 6 k }$ iterations. For tieredImageNet, we train Exemplar for approximately $9 0 0 \\mathrm { k }$ iterations, and train CC for $1 2 0 \\mathrm { k }$ iterations. We choose $\\mathbf { k }$ -means [32] as the clustering algorithm for COS. The threshold $\\gamma$ is set to 0.5, and top 3 out of 30 features are chosen per image at the training stage. At the evaluation stage, we crop each image 7 times. The importance factors $\\alpha$ and $\\beta$ are both set to 0.8. ",
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+    {
+        "type": "table",
+        "img_path": "images/30b7011ff597e47ac90d406b30e50291f7405a7a51d498bb7a4785fdc8cc9e9e.jpg",
+        "table_caption": [
+            "Table 1: Ablative study on miniImageNet. All models are trained on the full training set of miniImageNet. Since the aim of SOC algorithm is to find foreground objects, it is unnecessary to evaluate SOC on the foreground dataset $\\mathcal { D } _ { v }$ -FG. FT means finetuning from Exemplar used in COS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">CC</td><td rowspan=\"2\">FT</td><td rowspan=\"2\">COS</td><td rowspan=\"2\">SOC</td><td colspan=\"2\">Du-Ori</td><td colspan=\"2\">Du-FG</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>√</td><td></td><td></td><td></td><td>62.67 ± 0.32</td><td>80.22 ± 0.24</td><td>66.69 ± 0.32</td><td>82.86 ± 0.19</td></tr><tr><td>√</td><td></td><td>√</td><td></td><td>64.76 ± 0.13</td><td>81.18 ± 0.21</td><td>71.13 ± 0.36</td><td>86.21 ± 0.15</td></tr><tr><td>卜</td><td>√</td><td>√</td><td></td><td>65.05 ± 0.06</td><td>81.16 ± 0.17</td><td>71.36 ± 0.30</td><td>86.20 ± 0.14</td></tr><tr><td>厂</td><td></td><td></td><td>√</td><td>64.41 ± 0.22</td><td>81.54 ± 0.28</td><td></td><td>=</td></tr><tr><td></td><td></td><td>√</td><td>√</td><td>69.29 ± 0.12</td><td>84.94 ± 0.28</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7975cdad3521398109f74f8758855b86540ef9f0f95a4fcb76bdbd9e246f3fd0.jpg",
+        "table_caption": [
+            "Table 2: Comparisons with baselines of foreground extractors using saliency detection algorithms on miniImageNet. For fair comparison, all models in the right column at evaluation use multi-cropping. GT means evaluating with ground truth foreground. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Used for training</td><td colspan=\"3\">Used forevaluation</td></tr><tr><td>Method</td><td>1-shot</td><td>5-shot</td><td>Method</td><td>1-shot</td><td>5-shot</td></tr><tr><td>CC</td><td>62.67 ± 0.32</td><td>80.22 ± 0.24</td><td>COS</td><td>67.23 ± 0.35</td><td>82.79 ± 0.31</td></tr><tr><td>CC+RBD</td><td>63.24 ± 0.41</td><td>80.45 ± 0.37</td><td>COS+RBD</td><td>67.03 ± 0.52</td><td>82.57 ± 0.27</td></tr><tr><td>CC+MBD</td><td>61.50 ± 0.31</td><td>79.12 ± 0.32</td><td>COS+MBD</td><td>62.98 ± 0.45</td><td>79.56 ± 0.38</td></tr><tr><td>CC+FT</td><td>62.71 ± 0.11</td><td>80.06 ± 0.08</td><td>COS+FT</td><td>64.74 ± 0.28</td><td>80.74 ± 0.13</td></tr><tr><td>CC+COS</td><td>64.76 ± 0.13</td><td>81.18 ± 0.21</td><td>COSOC</td><td>69.28 ± 0.49</td><td>85.16 ± 0.42</td></tr><tr><td>1</td><td>1</td><td>=</td><td>COS+GT</td><td>72.71 ± 0.57</td><td>87.43 ± 0.36</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.2 Model Analysis ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this subsection, we show the effectiveness of each component of our method. Tab. 1 shows the ablation study conducted on miniImageNet. ",
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+    {
+        "type": "text",
+        "text": "On the effect of finetuning. Since a feature extractor is pre-trained using contrastive learning in COS, it may help accelerate convergence if we directly finetune from the pre-trained model instead of training from scratch. As shown in line 2-3 in Tab. 1, fintuning gives no improvement on the performance over training from scratch. Thus we adopt finetuning mainly for speeding up convergence $5 \\times$ faster). ",
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+    {
+        "type": "text",
+        "text": "Effectiveness of COS Algorithm. As observed in Tab. 1, When COS is applied on CC, the performance is improved on both versions of datasets. In Fig. 5, we show the curves of training and validation error of CC during training with and without COS. Both models are trained from scratch and validated on the full miniImageNet. We observe that CC sinks into overfitting: the training accuracy drops to zero, and validation accuracy stops improving before ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/71cf6cb98a6373acd448719b594eadf68dc2fbab7cf77278bf966fac3ce2ff3b.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of training and validation curves between CC with and without COS. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "the end of the training. Meanwhile, the COS algorithm helps slow down convergence and prevent training accuracy from reaching zero. This makes validation accuracy comparable at first but higher at the end. Our COS algorithm weakens the “background shortcut” for learning, draws model’s attention on foreground objects, and improves upon generalization. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Effectiveness of SOC Algorithm. The result in Tab. 1 shows that the SOC algorithm is the key to maximally exploit the potential of good object-discrimination ability. The performance even approaches the upper bound performance obtained by evaluating the model on the ground-truth foreground $\\mathcal { D } _ { v }$ -FG. One potential unfairness in our SOC algorithm may lie in the use of multicropping, which could possibly lead to performance improvement for other approaches as well. We ablate this concern in Appendix G, as well as in the comparisons to other methods in the later subsections. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ee5c7967ef79a227756257c56f308be55b1519b62174ad7276e40593991e3eba.jpg",
+        "table_caption": [
+            "Table 3: Comparisons with state-of-the-art models on miniImageNet and tieredImageNet. The average inductive 5-way few-shot classification accuracies with 95 confidence interval are reported. \\* indicates methods evaluated using multi-cropping. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">backbone</td><td colspan=\"2\">miniImageNet</td><td colspan=\"2\">tieredImageNet</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>MetaOptNet [22]</td><td>ResNet-12</td><td>62.64 ±0.82</td><td>78.63 ±0.46</td><td>65.99 ± 0.72</td><td>81.56± 0.53</td></tr><tr><td>DC[28]</td><td>ResNet-12</td><td>62.53 ± 0.19</td><td>79.77 ± 0.19</td><td>=</td><td>1</td></tr><tr><td>CTM[25]</td><td>ResNet-18</td><td>64.12 ± 0.82</td><td>80.51 ± 0.13</td><td>68.41 ± 0.39</td><td>84.28 ± 1.73</td></tr><tr><td>CAM[19]</td><td>ResNet-12</td><td>63.85 ± 0.48</td><td>79.44 ± 0.34</td><td>69.89 ± 0.51</td><td>84.23 ± 0.37</td></tr><tr><td>AFHN[26]</td><td>ResNet-18</td><td>62.38 ± 0.72</td><td>78.16 ± 0.56</td><td>=</td><td>1</td></tr><tr><td>DSN [47]</td><td>ResNet-12</td><td>62.64 ± 0.66</td><td>78.83 ± 0.45</td><td>66.22 ± 0.75</td><td>82.79 ± 0.48</td></tr><tr><td>AM3+TRAML[23]</td><td>ResNet-12</td><td>67.10 ± 0.52</td><td>79.54 ± 0.60</td><td></td><td>=</td></tr><tr><td>Net-Cosine [29]</td><td>ResNet-12</td><td>63.85 ± 0.81</td><td>81.57 ± 0.56</td><td></td><td>=</td></tr><tr><td>CA [2]</td><td>WRN-28-10</td><td>65.92 ± 0.60</td><td>82.85 ± 0.55</td><td>74.40 ± 0.68</td><td>86.61 ± 0.59</td></tr><tr><td>MABAS [21]</td><td>ResNet-12</td><td>65.08 ± 0.86</td><td>82.70 ± 0.54</td><td></td><td></td></tr><tr><td>ConsNet [59]</td><td>ResNet-12</td><td>64.89 ±0.23</td><td>79.95 ± 0.17</td><td></td><td>=</td></tr><tr><td>IEPT[66]</td><td>ResNet-12</td><td>67.05 ± 0.44</td><td>82.90 ±0.30</td><td>72.24 ± 0.50</td><td>86.73 ± 0.34</td></tr><tr><td>MELR[11]</td><td>ResNet-12</td><td>67.40 ± 0.43</td><td>83.40 ±0.28</td><td>72.14 ± 0.51</td><td>87.01 ±0.35</td></tr><tr><td>IER-Distill [42]</td><td>ResNet-12</td><td>67.28 ± 0.80</td><td>84.78 ± 0.52</td><td>72.21 ± 0.90</td><td>87.08 ± 0.58</td></tr><tr><td>LDAMF [57]</td><td>ResNet-12</td><td>67.76 ± 0.46</td><td>82.71 ± 0.31</td><td>71.89 ± 0.52</td><td>85.96 ± 0.35</td></tr><tr><td>FRN [55]</td><td>ResNet-12</td><td>66.45 ± 0.19</td><td>82.83 ± 0.13</td><td>72.06 ± 0.22</td><td>86.89 ± 0.14</td></tr><tr><td>Baseline*[7]</td><td>ResNet-12</td><td>63.83 ± 0.67</td><td>81.38 ± 0.41</td><td></td><td></td></tr><tr><td>DeepEMD* [63]</td><td>ResNet-12</td><td>67.63 ± 0.46</td><td>83.47 ± 0.61</td><td>74.29 ± 0.32</td><td>86.98 ± 0.60</td></tr><tr><td>RFS-Distill* [51]</td><td>ResNet-12</td><td>65.02 ± 0.44</td><td>82.04 ± 0.38</td><td>71.52 ± 0.69</td><td>86.03 ± 0.49</td></tr><tr><td>FEAT*[60]</td><td>ResNet-12</td><td>68.03 ±0.38</td><td>82.99 ± 0.31</td><td></td><td>=</td></tr><tr><td>Meta-baseline* [8]</td><td>ResNet-12</td><td>65.31 ± 0.51</td><td>81.26 ± 0.23</td><td>68.62 ± 0.27</td><td>83.74 ±0.18</td></tr><tr><td>COSOC* (ours)</td><td>ResNet-12</td><td>69.28 ± 0.49</td><td>85.16 ± 0.42</td><td>73.57 ± 0.43</td><td>87.57 ± 0.10</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that if we apply only the SOC algorithm on CC, the performance degrades. This indicates that COS and SOC are both necessary: COS provides the discrimination ability of foreground objects and SOC leverages it to maximally boost the performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 Comparison to Saliency-based Foreground Extractors ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "There could be other possible ways of extracting foreground objects. A simple yet possibly strong baseline could be running saliency detection to extract the most salient region in an image, followed by cropping to obtain patches without background. We consider comparing with three classical unsupervised saliency methods—RBD [69], FT [1] and MBD [65]. The cropping threshold is specially tuned. For training, fusion sampling with probability 0.5 is used for unsupervised saliency methods. For evaluation, We replace the original images with crops obtained by unsupervised saliency methods directly for classification. Tab. 2 displays the comparisons of performance using different foreground extraction methods applied at training or evaluation. For fair comparison, all methods are trained from scratch, and all compared baselines are evaluated with multi-cropping (i.e. using the average of features obtained from multiple crops for classification)and tested on the same backbone (COS trained). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The results show that: (1) Our method performs consistently much better than the listed unsupervised saliency methods. (2) The performance of different unsupervised saliency methods varies. While RBD gives a small improvement, MBD and FT have negative effect on the performance. The performance severely depends on the effectiveness of unsupervised saliency methods, and is very sensitive to the cropping threshold. Intuitively speaking, saliency detection methods focus on noticeable objects in the image, and might fail when there is another irrelevant salient object in the image (e.g., a man is walking a dog. Dog is the label, but the man is of high saliency). On the contrary, our method focuses on shared objects across images in the same class, thereby avoiding this problem. In addition, our COS algorithm has the ability to dynamically assign foreground scores to different patches, which reduces the risk of overconfidence. One of our main contributions is paving a new way towards improving FSL by rectifying shortcut learning of background, which can be implemented using any effective methods. Given the upper bound with ground truth foreground, we believe there is room to improve and there can be other more effective approaches in the future. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ab616a6195363bd563d3bf2e73da6c5f9695a968f315c522360602c0f3b52ac.jpg",
+        "image_caption": [
+            "Figure 6: Examples of objects obtained with COS from the training set of miniImageNet. The first row shows the original images;the second row shows the picked patch with the highest foreground score. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+            787,
+            169
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/7f65b76b5deebbac9324f58291087107217b0b2cc19a6557a4f2fa52a22bee6a.jpg",
+        "image_caption": [
+            "Figure 7: Visualization examples of the SOC algorithm. The first row displays 5 images that belong to dalmatian and guitar classes respectively from evaluation set of miniImageNet. The second row shows image patches that are picked up from the first round of SOC algorithm. Our method succesfully puts focus on the shared contents/foreground. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.4 Comparison to State-of-the-Arts ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            396,
+            439,
+            411
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tab. 3 presents 5-way 1-shot and 5-shot classification results on miniImageNet and tieredImageNet. We compare with state-of-the-art few-shot learning methods. For fair comparison, we reimplement some methods, and evaluate them with multi-cropping. See Appendix G for a detailed study on the influence of multi-cropping. Our method achieves state-of-the-art performance under all settings except for 1-shot task on tieredImageNet, on which the performance of our method is slightly worse than CA, which uses WRN-28-10, a deeper backbone, as the feature extractor. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.5 Visualization ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            529,
+            303,
+            542
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fig. 6 and 7 display visualization examples of the COS and SOC algorithms. See more examples in Appendix H. Thanks to the well-designed mechanism of capturing shared inter-image information, the COS and SOC algorithms are capable of locating foreground patches embodied in complicated, multi-object scenery. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            637,
+            299,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Few-shot image classification benefits from increasingly more complex network and algorithm design, but little attention has been focused on image itself. In this paper, we reveal that image background serves as a source of harmful knowledge that few-shot learning models easily absorb in. This problem is tackled by our COSOC framework that can draw the model’s attention to image foreground at both training and evaluation. Our method is only one possible solution, and future work may include exploring the potential of unsupervised segmentation or detection algorithms which may be a more reliable alternative of random cropping, or looking for a completely different but better algorithm customized for foreground extraction. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Special thanks to Qi Yong, who gives indispensable support on the spirit of this paper. We also thank Junran Peng for his help and fruitful discussions. This paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204), and a key program of fundamental research from Shenzhen Science and Technology Innovation Commission (No. JCYJ20200109113403826). ",
+        "bbox": [
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+        "text": "References ",
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parse/train/hu2aMLzOxC/hu2aMLzOxC.md

@@ -0,0 +1,270 @@
+# ASYMMETRIC SELF-PLAY FOR AUTOMATIC GOAL DIS-COVERY IN ROBOTIC MANIPULATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We train a single, goal-conditioned policy that can solve many robotic manipulation tasks, including tasks with previously unseen goals and objects. We rely on asymmetric self-play for goal discovery, where two agents, Alice and Bob, play a game. Alice is asked to propose challenging goals and Bob aims to solve them. We show that this method can discover highly diverse and complex goals without any human priors. Bob can be trained with only sparse rewards, because the interaction between Alice and Bob results in a natural curriculum and Bob can learn from Alice’s trajectory when relabeled as a goal-conditioned demonstration. Finally, our method scales, resulting in a single policy that can generalize to many unseen tasks such as setting a table, stacking blocks, and solving simple puzzles. Videos of a learned policy is available at https://robotics-self-play.github.io.
+
+# 1 INTRODUCTION
+
+We are motivated to train a single goal-conditioned policy $( \mathbf { | K a e l b l i n g right| } , \mathbf { | 9 9 3 | } )$ that can solve any robotic manipulation task that a human may request in a given environment. In this work, we make progress towards this goal by solving a robotic manipulation problem in a table-top setting where the robot’s task is to change the initial configuration of a variable number of objects on a table to match a given goal configuration. This problem is simple in its formulation but likely to challenge a wide variety of cognitive abilities of a robot as objects become diverse and goals become complex.
+
+Motivated by the recent success of deep reinforcement learning for robotics (Levine et al., 2016; Gu et al., 2017; Hwangbo et al., 2019; OpenAI et al., 2019a), we tackle this problem using deep reinforcement learning on a very large training distribution. An open question in this approach is how we can build a training distribution rich enough to achieve generalization to many unseen manipulation tasks. This involves defining both an environment’s initial state distribution and a goal distribution. The initial state distribution determines how we sample a set of objects and their configuration at the beginning of an episode, and the goal distribution defines how we sample target states given an initial state. In this work, we focus on a scalable way to define a rich goal distribution.
+
+The research community has started to explore automated ways of defining goal distributions. For example, previous works have explored learning a generative model of goal distributions (Florensa et al., 2018; Nair et al., 2018b; Racaniere et al., 2020) and collecting teleoperated robot trajectories to identify goals (Lynch et al., 2020; Gupta et al., 2020) In this paper, we extend an alternative approach called asymmetric self-play (Sukhbaatar et al., $\mathbf { \widehat { 2 0 1 8 b } } _ { \mathbf { \widehat { a } } }$ for automated goal generation. Asymmetric self-play trains two RL agents named Alice and Bob. Alice learns to propose goals that Bob is likely to fail at, and Bob, a goal-conditioned policy, learns to solve the proposed goals. Alice proposes a goal by manipulating objects and Bob has to solve the goal starting from the same initial state as Alice’s. By embodying these two agents into the same robotic hardware, this setup ensures that all proposed goals are provided with at least one solution: Alice’s trajectory.
+
+![](images/49d5111f82cae1b6b17a10d6278e05a1b75a98dd77f7bce7bcdaa1b8bb0a2ed9.jpg)  
+(a) Table-top setting with a robot arm (b) Example initial state for training (c) Example holdout tasks   
+Figure 1: (a) We train a policy that controls a robot arm operating in a table-top setting. (b) Randomly placed ShapeNet (Chang et al., 2015) objects constitute an initial state distribution for training. (c) We use multiple manually designed holdout tasks to evaluate the learned policy.
+
+![](images/e2247335901cc0511f696f18bd35da1a330b87b99954b68c194aa0ffbce10894.jpg)  
+Figure 2: (a) We train a goal-conditioned policy on a single training distribution and evaluate its performance on many unseen holdout tasks. (b) To construct a training distribution, we sample an initial state from a predefined distribution, and run a goal setting policy (Alice) to generate a goal. In one episode, Alice is asked to generate 5 goals and Bob solves them in sequence until it fails.
+
+There are two main reasons why we consider asymmetric self-play to be a promising goal generation and learning method. First, any proposed goal is achievable, meaning that there exists at least one solution trajectory that Bob can follow to achieve the goal. Because of this property, we can exploit Alice’s trajectory to provide additional learning signal to Bob via behavioral cloning. This additional learning signal alleviates the overhead of heuristically designing a curriculum or reward shaping for learning. Second, this approach does not require labor intensive data collection.
+
+In this paper, we show that asymmetric self-play can be used to train a goal-conditioned policy for complex object manipulation tasks, and the learned policy can zero-shot generalize to many manually designed holdout tasks, which consist of either previously unseen goals, previously unseen objects, or both. To the best of our knowledge, this is the first work that presents zero-shot generalization to many previously unseen tasks by training purely with asymmetric self-play.
+
+# 2 PROBLEM FORMULATION
+
+Our training environment for robotic manipulation consists of a robot arm with a gripper attached and a wide range of objects placed on a table surface (Figure $\boxed { 1 \mathrm { a } } \boxed { 1 \mathrm { b } }$ . The goal-conditioned policy learns to control the robot to rearrange randomly placed objects (the initial state) into a specified goal configuration (Figure 1c). We aim to train a policy on a single training distribution and to evaluate its performance over a suite of holdout tasks which are independently designed and not explicitly present during training $\mathbb { ( F i g u r e 2 a ) }$ . In this work, we construct the training distribution via asymmetric self-play (Figure 2b) to achieve generalization to many unseen holdout tasks (Figure 1c)
+
+Mathematical formulation Formally, we model the interaction between an environment and a goal-conditioned policy as a goal-augmented Markov decision process $\mathcal { M } = \langle \boldsymbol { S } , \mathcal { A } , \mathcal { P } , \mathcal { R } , \boldsymbol { \mathcal { G } } \rangle$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \mapsto \mathbb { R }$ denotes the transition probability, ${ \mathcal { G } } \subseteq S$ specifies the goal space and $\mathcal { R } : \mathcal { S } \times \mathcal { G } \mapsto \mathbb { R }$ is a goal-specific reward function. A goalaugmented trajectory sequence is $\left\{ \left( s _ { 0 } , g , a _ { 0 } , r _ { 0 } \right) , \ldots , \left( s _ { t } , g , a _ { t } , r _ { t } \right) \right\}$ , where the goal is provided to the policy as part of the observation at every step. We say a goal is achieved if $s _ { t }$ is sufficiently close to $g$ (Appendix $\mathbf { A } . 2 )$ . With a slightly overloaded notation, we define the goal distribution $\mathcal { G } ( \boldsymbol { g } | \boldsymbol { s } _ { 0 } )$ as the probability of a goal state $g \in { \mathcal { G } }$ conditioned on an initial state $s _ { 0 } \in S$ .
+
+Training goal distribution A naive design of the goal distribution $\mathcal { G } ( \boldsymbol { g } | \boldsymbol { s } _ { 0 } )$ is to randomly place objects uniformly on the table, but it is unlikely to generate interesting goals, such as an object picked up and held above the table surface by a robot gripper. Another possible approach, collecting tasks and goals manually, is expensive and hard to scale. We instead sidestep these issues and automatically generate goals via training based on asymmetric self-play (Sukhbaatar et al., $\textcircled { 2 0 1 8 6 } \textcircled { 2 }$ . Asymmetric self-play involves using a policy named Alice $\pi _ { A } ( a | s )$ to set goals and a goal-conditioned policy Bob $\pi _ { B } ( a | s , g )$ to solve goals proposed by Alice, as illustrated in ${ \overline { { \mathbb { F } \mathrm { i g u r e ~ } 2 \mathrm { b } } } } .$ We run $\pi _ { A }$ to generate a trajectory $\dot { \tau } _ { A } = \{ ( s _ { 0 } , \bar { a } _ { 0 } , \bar { r _ { 0 } } ) , \dot { \mathrm { ~ . ~ . ~ } } , ( s _ { T } , a _ { T } , r _ { T } ) \}$ and the last state is labelled as a goal $g$ for $\pi _ { B }$ to solve. The goal distribution $\mathcal { G } ( s _ { T } = g | s _ { 0 } )$ is fully determined by $\pi _ { A }$ and we train Bob only on this goal distribution. We therefore say zero-shot generalization when Bob generalizes to a holdout task which is not explicitly encoded into the training distribution.
+
+Evaluation on holdout tasks To assess zero-shot generalization of $\pi _ { B } ( a | s , g )$ from our training setup, we hand-designed a suite of holdout tasks with goals that are never directly incorporated into the training distribution. Some holdout tasks also feature previously unseen objects. The holdout tasks are designed to either test whether a specific skill has been learned, such as the ability to pick up objects $\overbrace { ( \mathrm { F i g u r e } 3 ) }$ , or represent a semantically interesting task, such as setting a table $( { \mathrm { f i g u r e } } \ { \mathrm { l c } } )$ Appendix $\overline { { \mathbb { B } . 6 } }$ describes the list of holdout tasks that we use in our experiments. Note that none of the holdout tasks are used for training $\pi _ { B } ( a | s , g )$ .
+
+# 3 ASYMMETRIC SELF-PLAY
+
+To train Alice policy $\pi _ { A } ( a | s )$ and Bob policy $\pi _ { B } ( a | s , g )$ , we run the following multi-goal game within one episode, as illustrated in Figure 2b:
+
+1. An initial state $s _ { 0 }$ is sampled from an initial state distribution. Alice and Bob are instantiated into their own copies of the environment. Alice and Bob alternate turns as follows.   
+2. Alice’s turn. Alice interacts with its environment for a fixed number of $T$ steps and may rearrange the objects. The state at the end of Alice’s turn $s _ { T }$ will be used as a goal $g$ for Bob. If the proposed goal is invalid (e.g. if Alice has not moved any objects, or if an object has fallen off the table), the episode terminates.   
+3. Bob’s turn. Bob receives reward if it successfully achieves the goal $g$ in its environment. Bob’s turn ends when it succeeds at achieving the goal or reaches a timeout. If Bob’s turn ends in a failure, its remaining turns are skipped and treated as failures, while we let Alice to keep generating goals.   
+4. Alice receives reward if Bob fails to solve the goal that Alice proposed. Steps 2–3 are repeated until 5 goals are set by Alice or Alice proposes an invalid goal, and then the episode terminates.
+
+The competition created by this game encourages Alice to propose goals that are increasingly challenging to Bob, while Bob is forced to solve increasingly complex goals. The multi-goal setup was chosen to allow Bob to take advantage of environmental information discovered earlier in the episode to solve its remaining goals, which OpenAI et al. (2019a) found to be important for transfer to physical systems. Note however that in this work we focus on solving goals in simulation only. To improve stability and avoid forgetting, we have Alice and Bob play against past versions of their respective opponent in $2 0 \%$ of games. More details about the game structure and pseudocode for training with asymmetric self-play are available in Appendix A.
+
+# 3.1 REWARD STRUCTURE
+
+For Bob, we assign sparse goal-conditioned rewards. We measure the positional and rotational distance between an object and its goal state as the Euclidean distance and the Euler angle rotational distance, respectively. Whenever both distance metrics are below a small error (the success threshold), this object is deemed to be placed close enough to the goal state and Bob receives 1 reward immediately. But if this object is moved away from the goal state that it has arrived at in past steps, Bob obtains -1 reward such that the sum of per-object reward is at most 1 during a given turn. When all of the objects are in their goal state, Bob receives 5 additional reward and its turn is over.
+
+For Alice, we assign a reward after Bob has attempted to solve the goal: 5 reward if Bob failed at solving the goal, and 0 if Bob succeeded. We shape Alice’s reward slightly by adding 1 reward if it has set a valid goal, defined to be when no object has fallen off the table and any object has been moved more than the success threshold. An additional penalty of $- 3$ reward is introduced when Alice sets a goal with objects outside of the placement area, defined to be a fixed 3D volume within the view of the robot’s camera. More details are discussed in Appendix A.2.
+
+# 3.2 ALICE BEHAVIORAL CLONING (ABC)
+
+One of the main benefits of using asymmetric self-play is that the generated goals come with at least one solution to achieve it: Alice’s trajectory. Similarly to Sukhbaatar et al. $\overline { { ( 2 0 1 8 \mathrm { a } ) } }$ , we exploit this property by training Bob with Behavioral Cloning (BC) from Alice’s trajectory, in addition to the reinforcement learning (RL) objective. We call this learning mechanism Alice Behavioral Cloning (ABC). We propose several improvements over the original formulation in Sukhbaatar et al. (2018a).
+
+Demonstration trajectory filtering Compared to BC from expert demonstrations, using Alice’s trajectory needs extra care. Alice’s trajectory is likely to be suboptimal for solving the goal, as Alice might arrive at the final state merely by accident. Therefore, we only consider trajectories with goals that Bob failed to solve as demonstrations, to avoid distracting Bob with suboptimal examples. Whenever Bob fails, we relabel Alice’s trajectory $\tau _ { A }$ to be a goal-augmented version $\tau _ { \mathrm { B C } } = \{ ( s _ { 0 } , s _ { T } , a _ { 0 } , r _ { 0 } ) , \dots , ( s _ { T } , s _ { T } , a _ { T } , r _ { T } ) \}$ as a demonstration for BC, where $s _ { T }$ is the goal.
+
+PPO-style BC loss clipping The objective for training Bob is $\mathcal { L } = \mathcal { L } _ { \mathrm { R L } } + \beta \mathcal { L } _ { \mathrm { a b c } }$ , where ${ \mathcal { L } } _ { \mathrm { R L } }$ is an RL objective and $\mathcal { L } _ { \mathrm { a b c } }$ is the ABC loss. $\beta$ is a hyperparameter controlling the relative importance of the BC loss. We set $\beta = 0 . 5$ throughout the whole experiment. A naive BC loss is to minimize the negative log-likelihood of demonstrated actions, $- \mathbb { E } _ { ( s _ { t } , g _ { t } , a _ { t } ) \in \mathcal { D } _ { \mathrm { B C } } } \left[ \log \pi _ { B } ( a _ { t } | s _ { t } , g _ { t } ; \theta ) \right]$ where $\mathcal { D } _ { \mathrm { B C } }$ is a mini-batch of demonstration data and $\pi _ { B }$ is parameterized by $\theta$ . We found that overly-aggressive policy changes triggered by BC sometimes led to learning instabilities. To prevent the policy from changing too drastically, we introduce PPO-style loss clipping (Schulman et al., $\bar { 2 0 1 7 } \dot { ) }$ on the BC loss by setting the advantage $\hat { A } = 1$ in the clipped surrogate objective:
+
+$$
+\mathcal { L } _ { \mathrm { a b c } } = - \mathbb { E } _ { ( s _ { t } , g _ { t } , a _ { t } ) \in \mathcal { D } _ { \mathrm { B C } } } \left[ \mathrm { c l i p } \Big ( \frac { \pi _ { B } \big ( a _ { t } \big | s _ { t } , g _ { t } ; \theta \big ) } { \pi _ { B } \big ( a _ { t } \big | s _ { t } , g _ { t } ; \theta _ { \mathrm { o l d } } \big ) } , 1 - \epsilon , 1 + \epsilon \Big ) \right]
+$$
+
+where $\pi _ { B } ( a _ { t } | s _ { t } , g _ { t } ; \theta )$ is Bob’s likelihood on a demonstration based on the parameters that we are optimizing, and $\pi _ { B } \big ( \dot { a } _ { t } | s _ { t } , g _ { t } ; \theta _ { \mathrm { o l d } } \big )$ is the likelihood based on Bob’s behavior policy (at the time of demonstration collection) evaluated on a demonstration. This behavior policy is identical to the policy that we use to collect RL trajectories. By setting $\hat { A } = 1$ , this objective optimizes the naive BC loss, but clips the loss whenever $\frac { \pi _ { B } \left( a _ { t } | s _ { t } , g _ { t } ; \theta \right) } { \pi _ { B } \left( a _ { t } | s _ { t } , g _ { t } ; \theta _ { \mathrm { o l d } } \right) }$ is bigger than $1 + \epsilon$ , to prevent the policy from changing too much. $\epsilon$ is a clipping threshold and we use $\epsilon = 0 . 2$ in all the experiments.
+
+# 4 RELATED WORK
+
+Training distribution for RL In the context of multi-task RL (Beattie et al., 2016; Hausman et al., 2018; Yu et al., 2020), multi-goal RL (Kaelbling, 1993; Andrychowicz et al., $\overline { { \mathbb { Z } 0 1 7 } }$ , and meta RL (Wang et al., 2016; Duan et al., 2016), previous works manually designed a distribution of tasks or goals to see better generalization of a policy to a new task or goal. Domain randomization (Sadeghi & Levine, 2017b; Tobin et al., 2017; OpenAI et al., 2020) manually defines a distribution of simulated environments, but in service of generalizing to the same task in the real world.
+
+There are approaches to grow the training distribution automatically (Srivastava et al., 2013). Selfplay (Tesauro, 1995; Silver et al., 2016; 2017; Bansal et al., 2018; OpenAI et al., 2019b; Vinyals et al., 2019) constructs an ever-growing training distribution where multiple agents learn by competing with each other, so that the resulting agent shows strong performance on a single game. OpenAI et al. $\textcircled { 1 2 0 1 9 2 }$ automatically grew a distribution of domain randomization parameters to accomplish better generalization in the task of solving a Rubik’s cube on the physical robot. Wang et al. (2019;
+
+$\boxed { 2 0 2 0 }$ studied an automated way to keep discovering challenging 2D terrains and locomotion policies that can solve them in a 2D bipedal walking environment.
+
+We employ asymmetric self-play to construct a training distribution for learning a goal-conditioned policy and to achieve generalization to unseen tasks. Florensa et al. (2018); Nair et al. (2018b); Racaniere et al. $\underline { { ( 2 0 2 0 ) } }$ had the same motivation as ours, but trained a generative model instead of a goal setting policy. Thus, the difficulties of training a generative model were inherited by these methods: difficulty of modeling a high dimensional space and generation of unrealistic samples. Lynch et al. $\underline { { \widehat { ( 2 0 2 0 ) } } }$ ; Gupta et al. $\underline { { \widehat { ( 2 0 2 0 ) } } }$ used teleoperation to collect arbitrary robot trajectories, and defined a goal distribution from the states in the collected trajectories. This approach likely requires a large number of robot trajectories for each environment configuration (e.g. various types of objects on a table), and randomization of objects was not studied in this context.
+
+Asymmetric self-play Asymmetric self-play was proposed by Sukhbaatar et al. (2018b) as a way to supplement RL training. Sukhbaatar et al. $\underline { { \dot { \left( 2 0 1 8 \dot { 6 } \right) } } }$ mixed asymmetric self-play training with standard RL training on the target task and measured the performance on the target task. Sukhbaatar et al. $\textcircled { 2 0 1 8 \mathrm { a } }$ used asymmetric self-play to pre-train a hierarchical policy and evaluated the policy after fine-tuning it on a target task. $\boxed { \mathrm { L i u ~ e t ~ a l . } } \dot { \textcircled { 1 2 0 1 9 } }$ adopted self-play to encourage efficient learning with sparse reward in the context of an exploration competition between a pair of agents. As far as we know, no previous work has trained a goal-conditioned policy purely based on asymmetric selfplay and evaluated generalization to unseen holdout tasks.
+
+Curriculum learning Many previous works showed the difficulty of RL and proposed an automated curriculum (Andrychowicz et al., 2017; Florensa et al., 2017; Salimans & Chen, 2018; Matiisen et al., 2019; Zhang et al., 2020) or auxiliary exploration objectives (Oudeyer et al., 2007; Baranes & Oudeyer, 2013; Pathak et al., 2017; Burda et al., 2019; Ecoffet et al., 2019; 2020) to learn predefined tasks. When training goal-conditioned policies, relabeling or reversing trajectories (Andrychowicz et al., 2017; Florensa et al., 2017; Salimans & Chen, 2018) or imitating successful demonstrations (Oh et al., 2018; Ecoffet et al., 2019; 2020) naturally reduces the task complexity. Our work shares a similarity in that asymmetric self-play alleviates the difficulty of learning a goal-conditioned policy via an intrinsic curriculum and imitation from the goal setter’s trajectory, but our work does not assume any predefined task or goal distribution.
+
+Hierarchical reinforcement learning (HRL) Some HRL methods jointly trained a goal setting policy (high-level or manager policy) and a goal solving policy (low-level or worker policy) $\ " { \mathbb { V } } ^ { \mathrm { e z h - } }$ nevets et al., 2017; Levy et al., 2019; Nachum et al., 2018). However, the motivation for learning a goal setting policy in HRL is not to challenge the goal solving policy, but to cooperate to tackle a task that can be decomposed into a sequence of sub-goals. Hence, this goal setting policy is trained to optimize task reward for the target task, unlike asymmetric self-play where the goal setter is rewarded upon the other agent’s failure.
+
+Robot learning for object manipulation. It has been reported that training a policy for multiobject manipulation is very challenging with sparse rewards (Riedmiller et al., 2018; Vecerik et al., $\dot { \textcircled { 2 0 1 8 } }$ One example is block stacking, which has been studied for a long time in robotics as it involves complex contact reasoning and long horizon motion planning (Deisenroth et al., $\textcircled { 2 0 1 1 }$ . Learning block stacking often requires a hand-designed curriculum (Li et al., 2019), meticulous reward shaping $( \mathrm { \overline { { P o p o v \ e t { a l . } } } } , \mathrm { \overline { { 2 0 1 7 } } } )$ , fine-tuning (Rusu et al., 2017), or human demonstrations (Nair et al., $\underline { { 2 0 1 8 \mathrm { a } } } ,$ Duan et al., 2017). In this work, we use block stacking as one of the holdout tasks to test zero-shot generalization, but without training on it.
+
+# 5 EXPERIMENTS
+
+In this section, we first show that asymmetric self-play generates an effective training curriculum that enables generalization to unseen hold-out tasks. Then, the experiment is scaled up to train in an environment containing multiple random complex objects and evaluate it with a set of holdout tasks containing unseen objects and unseen goal configurations. Finally, we demonstrate how critical ABC is for Bob to make progress in a set of ablation studies.
+
+![](images/51c53dff2b7856da23701a4470f29bc6a9d53435ef286d47195a05e326f7f6e0.jpg)  
+Figure 3: Holdout tasks in the environment using 1 or 2 blocks. The transparent blocks denote the desired goal state, while opaque blocks are the current state. (a) push: The blocks must be moved to their goal locations and orientations. There is no differentiation between the six block faces. (b) flip: Each side of the block is labelled with a unique letter. The blocks must be moved to make every face correctly positioned as what the goal specifies. (c) pick-and-place: One goal block is in the air. (d) stack: Two blocks must be stacked in the right order at the right location.
+
+![](images/69cf0d99128f68ae3433a8dfcf2136da5e5821da3aaf989f43570a34e612f2e0.jpg)  
+Figure 4: Generalization to unseen holdout tasks for blocks. Baselines are trained over a mixture of all holdout tasks. The solid lines represent 2-blocks, while the dashed lines are for 1-block. The $\mathbf { X }$ -axis denotes the number of training steps via asymmetric self-play. The y-axis is the zero-shot generalization performance of Bob policy at corresponding training checkpoints. Note that success rate curves of completely failed baselines are occluded by others.
+
+# 5.1 EXPERIMENTAL SETUP
+
+We implement the training environment2 described in Sec. 2 with randomly placed ShapeNet objects (Chang et al., 2015) as an initial state distribution. In addition, we set up another simpler environment using one or two blocks of fixed size, used for small-scale comparisons and ablation studies. Figure 3 visualizes four holdout tasks for this environment. Each task is designed to evaluate whether the robot has acquired certain manipulation skills: pushing, flipping, picking up and stacking blocks. Experiments in Sec. 5.2, 5.3 and 5.5 focus on blocks and experimental results based on ShapeNet objects are present on Sec. 5.4. More details on our training setups are in Appendix B.
+
+We implement Alice and Bob as two independent policies of the same network architecture with memory (Appendix B.4), except that Alice has no observation on goal state. The policies take state observations (“state policy”) for experiments with blocks (Sec. $\underline { { \breve { 5 . 2 } } } \mathrm { , } \underline { { \breve { 5 . 3 } } } \mathrm { , }$ and $\underline { { \vec { \left. 5 . 5 \right. } } }$ , and take both vision and state observations (“hybrid policy”) for experiments with ShapeNet objects (Sec. 5.4). Both policies are trained with Proximal Policy Optimization (PPO) (Schulman et al., 2017).
+
+# 5.2 GENERALIZATION TO UNSEEN GOALS WITHOUT MANUAL CURRICULA
+
+One way to train a single policy to acquire all the skills in Figure 3 is to train a goal-conditioned policy directly over a mixture of these tasks. However, training directly over these tasks without a curriculum turns out to be very challenging, as the policy completely fails to make any progress.3 In contrast, Bob is able to solve all these holdout tasks quickly when learning via asymmetric self-play, without explicitly encoding any prior knowledge of the holdout tasks into the training distribution.
+
+To gauge the effect of an intrinsic curriculum introduced by self-play, we carefully designed a set of non-self-play baselines using explicit curricula controlled by Automatic Domain Randomization (OpenAI et al., 2019a). All baselines are trained over a mixture of block holdout tasks as the goal distribution. We measure the effectiveness of a training setup by tracking the success rate for each holdout task, as shown in Figure 4. The no curriculum baseline fails drastically. The curriculum:distance baseline expands the distance between the initial and goal states gradually as training progresses, but only learns to push and flip a single block. The curriculum:distribution baseline, which slowly increases the proportion of pick-and-place and stacking goals in the training distribution, fails to acquire any skill. The curriculum:full baseline incorporates all hand-designed curricula yet still cannot learn how to pick up or stack blocks. We have spent a decent amount of time iterating and improving these baselines but found it especially difficult to develop a scheme good enough to compete with asymmetric self-play. See Appendix C.1 for more details of our baselines.
+
+![](images/096c95bff00f179a267aac7fec859b99c410a06426f2f4bda4f25b664fa1bd98.jpg)  
+Figure 5: Goals discovered by asymmetric self-play. Alice discovers many goals that are not covered by our manually designed holdout tasks on blocks.
+
+![](images/bcdd9d1c76011b061b14d9b15fb5d87525c007e7a7c5b33ed7c3c1cd811758b7.jpg)  
+Figure 6: The empirical payoff matrix between Alice and Bob. Average success rate over multiple self-play episode is visualized. Alice with more training steps generates more challenging goals that Bob cannot solve yet. Bob with more training steps can achieve more goals against the same Alice.
+
+# 5.3 DISCOVERY OF NOVEL GOALS AND SOLUTIONS
+
+Asymmetric self-play discovers novel goals and solutions that are not covered by our holdout tasks. As illustrated in Figure 5, Alice can lift multiple blocks at the same time, build a tower and then keep it balanced using an arm joint. Although it is a tricky strategy for Bob to learn on its own, with ABC, Bob eventually acquires the skills for solving such complex tasks proposed by Alice. Videos are available at https://robotics-self-play.github.io.
+
+Figure 6 summarizes Alice and Bob’s learning progress against each other. For every pair of Alice and Bob, we ran multiple self-play episodes and measured the success rate. We observe an interesting trend with 2 blocks. As training proceeds, Alice tends to generate more challenging goals, where Bob shows lower success rate. With past sampling, Bob continues to make progress against versions of Alices from earlier optimization steps. This visualization suggests a desired dynamic of asymmetric self-play that could potentially lead to unbounded complexity: Alice continuously generates goals to challenge Bob, and Bob keeps making progress on learning to solve new goals.
+
+# 5.4 GENERALIZATION TO UNSEEN OBJECTS AND GOALS
+
+The experiments above show strong evidence that efficient curricula and novel goals can autonomously emerge in asymmetric self-play. To further challenge our approach, we scale it up to work with many more complex objects using more computational resources for training. We train a hybrid policy in an environment containing up to 10 random ShapeNet $\mathtt { ( F h a n g ~ e t ~ a l . ) } \overline { { \mathbb { P } \mathrm { 2 0 1 5 } } } \mathrm { ) }$ objects. During training, we randomize the number of objects and the object sizes via Automatic Domain Randomization (OpenAI et al., $\textcircled { 2 0 1 9 9 }$ . The hybrid policy uses vision observations to extract information about object geometry and size. We evaluate the Bob policy on a more diverse set of manipulation tasks, including semantically interesting ones. Many tasks contain unseen objects and complex goals, as illustrated in Figure 7.
+
+The learned Bob policy achieves decent zero-shot generalization performance for many tasks. Success rates are reported in Figure 8. Several tasks are still challenging. For example, ball-capture requires delicate handling of rolling objects and lifting skills. The rainbow tasks call for an understanding of concave shapes. Understanding the ordering of placement actions is crucial for stacking more than 3 blocks in the desired order. The Bob policy learns such an ordering to some degree, but fails to fully generalize to an arbitrary number of stacked blocks.
+
+![](images/a8ccd1da66404feecc3078c235632509a3fedb5ab7c9d28036f03e83b2135b8e.jpg)  
+Figure 7: Example holdout tasks involving unseen objects and complex goal states. The goal states are illustrated here, and the initial states have randomly placed objects.
+
+![](images/1872ae926f0c2ae5662bf7f23df92b965843d67db8850cec53ceda8aa91ec729.jpg)  
+Figure 8: Success rates of a single goal-conditioned policy solving a variety of holdout tasks, averaged over 100 trials. The error bars indicate the $9 9 \%$ confidence intervals. Yellow, orange and blue bars correspond to success rates of manipulation tasks with blocks, YCB4objects and other uniquely built objects, respectively. Videos are available at https://robotics-self-play.github.io.
+
+# 5.5 ABLATION STUDIES
+
+We present a series of ablation studies designed for measuring the importance of each component in our asymmetric self-play framework, including Alice behavioral cloning (ABC), BC loss clipping, demonstration filtering, and the multi-goal game setup. We disable a single ingredient in each ablation run and compare with the complete self-play baseline in Figure 9.
+
+![](images/34b1337c62e618872a4347e82eeffa8875f8ebc03dac7c9443a335b759d03220.jpg)  
+Figure 9: The ablation studies compare four ablation runs each with one component disabled with the full baseline. Solid lines are for 2-blocks, dashed lines are for 1-block. The $\mathbf { X }$ -axis denotes the number of training steps via asymmetric self-play. The y-axis is the zero-shot generalization performance of Bob policy at corresponding training steps.
+
+The no ABC baseline shows that Bob completely fails to solve any holdout task without ABC, indicating that ABC is a critical mechanism in asymmetric self-play. The no BC loss clipping baseline shows slightly slower learning on pick-and-place and stack, as well as some instabilities in the middle of training. Clipping in the BC loss is expected to help alleviate this instability by controlling the rate of policy change per optimizer iteration. The no demonstration filter baseline shows noticeable instability on flip, suggesting the importance of excluding suboptimal demonstrations from behavioral cloning. Finally, the single-goal baseline uses a single goal instead of 5 goals per episode during training. The evaluation tasks are also updated to require a single success per episode. Generalization of this baseline to holdout tasks turns out to be much slower and less stable. It signifies some advantages of using multiple goals per episode, perhaps due to the policy memory internalizing environmental information during multiple trials of goal solving.
+
+The results of the ablation studies suggest that ABC with proper configuration and multi-goal gameplay are critical components of asymmetric self-play, alleviating the importance of manual curricula and facilitating efficient learning.
+
+# 6 CONCLUSION
+
+One limitation of our asymmetric self-play approach is that it depends on a resettable simulation environment as Bob needs to start from the same initial state as Alice’s. Therefore asymmetric self-play training has to happen in a simulator which can be easily updated to a desired state. In order to run the goal-solving policy on physical robots, we plan to adopt sim-to-real techniques in future work. Sim-to-real has been shown to achieve great performance on many robotic tasks in the real world (Sadeghi & Levine, 2017a; Tobin et al., 2017; James et al., 2019; OpenAI et al., 2020). One potential approach is to pre-train two agents via asymmetric self-play in simulation, and then fine-tune the Bob policy with domain randomization or data collected on physical robots.
+
+In conclusion, we studied asymmetric self-play as a framework for defining a single training distribution to learn many arbitrary object manipulation tasks. Even without any prior knowledge about the target tasks, asymmetric self-play is able to train a strong goal-conditioned policy that can generalize to many unseen holdout tasks. We found that asymmetric self-play not only generates a wide range of interesting goals but also alleviates the necessity of designing manual curricula for learning such goals. We provided evidence that using the goal setting trajectory as a demonstration for training a goal solving policy is essential to enable efficient learning. We further scaled up our approach to work with various complex objects using more computation, and achieved zero-shot generalization to a collection of challenging manipulation tasks involving unseen objects and unseen goals.
+
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parse/train/rh0vIXw6i33/rh0vIXw6i33.md

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+# Multiple Descent: Design Your Own Generalization Curve
+
+Lin Chen Simons Institute for the Theory of Computing University of California, Berkeley CA 94720 lin.chen@berkeley.edu
+
+Yifei Min Department of Statistics and Data Science Yale University CT 06511 yifei.min@yale.edu
+
+# Mikhail Belkin
+
+Halıcıoglu Data Science Institute˘   
+University of California, San Diego CA 92093 mbelkin@ucsd.edu
+
+Amin Karbasi School of Engineering and Applied Science Yale University CT 06511 amin.karbasi@yale.edu
+
+# Abstract
+
+This paper explores the generalization loss of linear regression in variably parameterized families of models, both under-parameterized and over-parameterized. We show that the generalization curve can have an arbitrary number of peaks, and moreover, locations of those peaks can be explicitly controlled. Our results highlight the fact that both classical U-shaped generalization curve and the recently observed double descent curve are not intrinsic properties of the model family. Instead, their emergence is due to the interaction between the properties of the data and the inductive biases of learning algorithms.
+
+# 1 Introduction
+
+The main goal of machine learning methods is to provide an accurate out-of-sample prediction, known as generalization. For a fixed family of models, a common way to select a model from this family is through empirical risk minimization, i.e., algorithmically selecting models that minimize the risk on the training dataset. Given a variably parameterized family of models, the statistical learning theory aims to identify the dependence between model complexity and model performance. The empirical risk usually decreases monotonically as the model complexity increases, and achieves its minimum when the model is rich enough to interpolate the training data, resulting in zero (or near-zero) training error. In contrast, the behaviour of the test error as a function of model complexity is far more complicated. Indeed, in this paper we show how to construct a model family for which the generalization curve can be fully controlled (away from the interpolation threshold) in both under-parameterized and over-parameterized regimes. Classical statistical learning theory supports a U-shaped curve of generalization versus model complexity [31, 33]. Under such a framework, the best model is found at the bottom of the U-shaped curve, which corresponds to appropriately balancing under-fitting and over-fitting the training data. From the view of the bias-variance trade-off, a higher model complexity increases the variance while decreasing the bias. A model with an appropriate level of complexity achieves a relatively low bias while still keeping the variance under control. On the other hand, a model that interpolates the training data is deemed to over-fit and tends to worsen the generalization performance due to the soaring variance.
+
+Although classical statistical theory suggests a pattern of behavior for the generalization curve up to the interpolation threshold, it does not describe what happens beyond the interpolation threshold, commonly referred to as the over-parameterized regime. This is the exact regime where many modern machine learning models, especially deep neural networks, achieved remarkable success. Indeed, neural networks generalize well even when the models are so complex that they have the potential to interpolate all the training data points [61, 10, 32, 34].
+
+Modern practitioners commonly deploy deep neural networks with hundreds of millions or even billions of parameters. It has become widely accepted that large models achieve performance superior to small models that may be suggested by the classical U-shaped generalization curve [13, 38, 55, 35, 36]. This indicates that the test error decreases again once model complexity grows beyond the interpolation threshold, resulting in the so called double-descent phenomenon described in [9], which has been broadly supported by empirical evidence [49, 48, 29, 30] and confirmed empirically on modern neural architectures by Nakkiran et al. [46]. On the theoretical side, this phenomenon has been recently addressed by several works on various model settings. In particular, Belkin et al. [11] proved the existence of double-descent phenomenon for linear regression with random feature selection and analyzed the random Fourier feature model [50]. Mei and Montanari [44] also studied the Fourier model and computed the asymptotic test error which captures the double-descent phenomenon. Bartlett et al. [8], Tsigler and Bartlett [56] analyzed and gave explicit conditions for “benign overfitting” in linear and ridge regression, respectively. Caron and Chretien [16] provided a finite sample analysis of the nonlinear function estimation and showed that the parameter learned through empirical risk minimization converges to the true parameter with high probability as the model complexity tends to infinity, implying the existence of double descent. Liu et al. [42] studied the high dimensional kernel ridge regression in the under- and over-parameterized regimes and showed that the risk curve can be double descent, bell-shaped, and monotonically decreasing.
+
+Among all the aforementioned efforts, one particularly interesting question is whether one can observe more than two descents in the generalization curve. d’Ascoli et al. [21] empirically showed a samplewise triple-descent phenomenon under the random Fourier feature model. Similar triple-descent was also observed for linear regression [47]. More rigorously, Liang et al. [41] presented an upper bound on the risk of the minimum-norm interpolation versus the data dimension in Reproducing Kernel Hilbert Spaces (RKHS), which exhibits multiple descent. However, a multiple-descent upper bound without a properly matching lower bound does not imply the existence of a multiple-descent generalization curve. In this work, we study the multiple descent phenomenon by addressing the following questions:
+
+• Can the existence of a multiple descent generalization curve be rigorously proven? • Can an arbitrary number of descents occur? • Can the generalization curve and the locations of descents be designed?
+
+In this paper, we show that the answer to all three of these questions is yes. Further related work is presented in Section 2.
+
+Our Contribution. We consider the linear regression model and analyze how the risk changes as the dimension of the data grows. In the linear regression setting, the data dimension is equal to the dimension of the parameter space, which reflects the model complexity. We rigorously show that the multiple descent generalization curve exists under this setting. To our best knowledge, this is the first work proving a multiple descent phenomenon.
+
+Our analysis considers both the underparametrized and overparametrized regimes. In the overparametrized regime, we show that one can control where a descent or an ascent occurs in the generalization curve. This is realized through our algorithmic construction of a feature-revealing process. To be more specific, we assume that the data is in $\mathbb { R } ^ { D }$ , where $D$ can be arbitrarily large or even essentially infinite. We view each dimension of the data as a feature. We consider a linear regression problem restricted on the first $d$ features, where $d < D$ . New features are revealed by increasing the dimension of the data. We then show that by specifying the distribution of the newly revealed feature to be either a standard Gaussian or a Gaussian mixture, one can determine where an ascent or a descent occurs. In order to create an ascent when a new feature is revealed, it is sufficient that the feature follows a Gaussian mixture distribution. In order to have a descent, it is sufficient that the new feature follows a standard Gaussian distribution. Therefore, in the overparametrized regime, we can fully control the occurrence of a descent and an ascent. As a comparison, in the underparametrized regime, the generalization loss always increases regardless of the feature distribution. Generally speaking, we show that we are able to design the generalization curve.
+
+On the one hand, we show theoretically that the generalization curve is malleable and can be constructed in an arbitrary fashion. On the other hand, we rarely observe complex generalization curves in practice, besides carefully curated constructions. Putting these facts together, we arrive at the conclusion that realistic generalization curves arise from specific interactions between properties of typical data and the inductive biases of algorithms. We should highlight that the nature of these interactions is far from being understood and should be an area of further investigations.
+
+# 2 Related Work
+
+Our work is directly related to the recent line of research in the theoretical understanding of the double descent [11, 34, 60, 44] and the multiple descent phenomenon [41, 39]. Here we briefly discuss some other work that is closely related to this paper.
+
+Least Square Regression. In this paper we focus on the least square linear regression with no regularization. For the regularized least square regression, De Vito et al. [22] proposed a selection procedure for the regularization parameter. Advani and Saxe [1] analyzed the generalization of neural networks with mean squared error under the asymptotic regime where both the sample size and model complexity tend to infinity. Richards et al. [52] proved for least square regression in the asymptotic regime that as the dimension-to-sample-size ratio $d / n$ grows, an additional peak can occur in both the variance and bias due to the covariance structure of the features. As a comparison, in this paper the sample size is fixed and the model complexity increases. Rudi and Rosasco [53] studied kernel ridge regression and gave an upper bound on the number of the random features to reach certain risk level. Our result shows that there exists a natural setting where by manipulating the random features one can control the risk curve.
+
+Over-Parameterization and Interpolation. The double descent occurs when the model complexity reaches and increases beyond the interpolation threshold. Most previous works focused on proving an upper bound or optimal rate for the risk. Caponnetto and De Vito [15] gave the optimal rate for least square ridge regression via careful selection of the regularization parameter. Belkin et al. [12] showed that the optimal rate for risk can be achieved by a model that interpolates the training data. In a series of work on kernel regression with regularization parameter tending to zero (a.k.a. kernel ridgeless regression), Rakhlin and Zhai [51] showed that the risk is bounded away from zero when the data dimension is fixed with respect to the sample size. Liang and Rakhlin [40] then considered the case when $d \asymp n$ , showed empirically the multiple descent phenomenon and proved a risk upper bound that can be small given favorable data and kernel assumptions. Instead of giving a bound, our paper presents an exact computation of risk in the cases of underparametrized and overparametrized linear regression, and proves the existence of the multiple descent phenomenon. Wyner et al. [59] analyzed AdaBoost and Random Forest from the perspective of interpolation. There has also been a line of work on wide neural networks [4–6, 23, 3, 58, 14, 2, 18, 62, 54].
+
+Sample-wise Double Descent and Non-monotonicity. There has also been recent development beyond the model-complexity double-descent phenomenon. For example, regarding sample-wise non-monotonicity, Nakkiran et al. [46] empirically observed the epoch-wise double-descent and sample-wise non-monotonicity for neural networks. Chen et al. [19] and Min et al. [45] identified and proved the sample-wise double descent under the adversarial training setting, and Javanmard et al. [37] discovered double-descent under adversarially robust linear regression. Loog et al. [43] showed that empirical risk minimization can lead to sample-wise non-monotonicity in the standard linear model setting under various loss functions including the absolute loss and the squared loss, which covers the range from classification to regression. We also refer the reader to their discussion of the earlier work on non-monotonicity of generalization curves. Dar et al. [20] demonstrated the double descent curve of the generalization errors of subspace fitting problems. Fei et al. [28] studied the risk-sample tradeoff in reinforcement learning.
+
+# 3 Preliminaries and Problem Formulation
+
+Notation. For $x \in \mathbb { R } ^ { D }$ and $d \leq D$ , we let $x [ 1 : d ] \in \mathbb { R } ^ { d }$ denote a $d$ -dimensional vector with $x [ 1 : d ] _ { i } = x _ { i }$ for all $1 \ \leq \ i \ \leq \ d$ . For a matrix $A \ \in \ \mathbb { R } ^ { n \times d }$ , we denote its Moore-Penrose pseudoinverse by $A ^ { + } \in \mathbb { R } ^ { d \times n }$ and denote its spectral norm by $\textstyle \| A \| \triangleq \operatorname* { s u p } _ { x \neq 0 } { \frac { \| A x \| _ { 2 } } { \| x \| _ { 2 } } }$ kAxk2 , where k · k2 is the Euclidean norm for vectors. If $v$ is a vector, its spectral norm $\lVert v \rVert$ agrees with the Euclidean norm $\lVert \boldsymbol { v } \rVert _ { 2 }$ . Therefore, we write $\lVert v \rVert$ for $\lVert \boldsymbol { v } \rVert _ { 2 }$ to simplify the notation. We use the big $\mathrm { o }$ notation $\mathcal { O }$ and write variables in the subscript of $\mathcal { O }$ if the implicit constant depends on them. For example, ${ \mathcal { O } } _ { n , d , \sigma } ( 1 )$ is a constant that only depends on $n , d ,$ , and $\sigma$ . If $f ( \sigma )$ and $g ( \sigma )$ are functions of $\sigma$ , write $f ( \sigma ) \sim g ( \sigma )$ if $\begin{array} { r } { \operatorname* { l i m } \frac { f ( \sigma ) } { g ( \sigma ) } = 1 } \end{array}$ . It will be given in the context how we take the limit.
+
+Distributions. Let ${ \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ $( \mu , \sigma \in \mathbb { R } )$ and $\mathcal { N } ( \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ $\mathbf { \mathcal { \mu } } _ { \mathbf { \lambda } } ( \mathbf { \mathcal { \mu } } _ { \mathbf { \lambda } } \mathbf { \mathbb { R } } ^ { n }$ , $\Sigma \in \mathbb { R } ^ { n \times n }$ ) denote the univariate and multivariate Gaussian distributions, respectively, where $\boldsymbol { \mu } \in \mathbb { R } ^ { n }$ and $\boldsymbol { \Sigma } \in \mathbb { R } ^ { n \times n }$ is a positive semi-definite matrix. We define a family of trimodal Gaussian mixture distributions as follows
+
+$$
+\mathcal { N } _ { \sigma , \mu } ^ { \mathrm { m i x } } \triangleq \frac 1 3  { N ( 0 , \sigma ^ { 2 } ) + \frac { 1 } { 3 } }  { N ( - \mu , \sigma ^ { 2 } ) + \frac { 1 } { 3 } }  { N ( \mu , \sigma ^ { 2 } ) } .
+$$
+
+For an illustration, please see Fig. 1.
+
+![](images/34bccdb4d4a2022179ac5125e4c54a675031218de867a057d0a8c1b92194f060.jpg)  
+Figure 1: Density functions of the $\mathcal { N } ( 0 , 1 )$ and $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ feature. A new entry is independently sampled from the 1-dimensional distribution being either a standard Gaussian or trimodal Gaussian mixture. Smaller $\sigma$ leads to higher concentration around each modes.
+
+Let $\chi ^ { 2 } ( k , \lambda )$ denote the noncentral chi-squared distribution with $k$ degrees of freedom and the non-centrality parameter $\lambda$ . For example, if $X _ { i } \sim \mathcal { N } ( \mu _ { i } , 1 )$ (for $i = 1 , 2 , \ldots , k )$ are independent Gaussian random variables, we have ${ \textstyle \sum _ { i = 1 } ^ { k } X _ { i } ^ { 2 } \sim \chi ^ { 2 } ( k , \lambda ) }$ , where $\begin{array} { r } { \lambda = \sum _ { i = 1 } ^ { k } \mu _ { i } ^ { 2 } } \end{array}$ . We also denote by $\chi ^ { 2 } ( k )$ the (central) chi-squared distribution with $k$ degrees and the $F$ -distribution by $F ( d _ { 1 } , d _ { 2 } )$ where $d _ { 1 }$ and $d _ { 2 }$ are the degrees of freedom.
+
+Problem Setup. Let $x _ { 1 } , \ldots , x _ { n } \in \mathbb { R } ^ { D }$ be column vectors that represent the training data of size $n$ and let $\boldsymbol { x } _ { \mathrm { t e s t } } \boldsymbol { \bar { \in } } \mathbb { R } ^ { D }$ be a column vector that represents the test data. We assume that they are all independently drawn from a distribution
+
+$$
+x _ { 1 } , \ldots , x _ { n } , x _ { \mathrm { t e s t } } \overset { i i d } { \sim } \mathcal { D } .
+$$
+
+Let us consider a linear regression problem on the first $d$ features, where $d \leq D$ for some arbitrary large $D$ . Here, $d$ can be viewed as the number of features revealed. Then the feature vectors are $\tilde { x } _ { 1 } , \ldots , \tilde { x } _ { n }$ , where $\widetilde { x } _ { i } = x _ { i } [ 1 : d ] \in \mathbb { R } ^ { d }$ denotes the first $d$ entries of $x _ { i }$ . The corresponding response variable $y _ { i }$ satisfies
+
+$$
+y _ { i } = \tilde { x } _ { i } ^ { \top } \beta + \varepsilon _ { i } , \quad i = 1 , \ldots , n ,
+$$
+
+where the noise $\varepsilon _ { i } \sim \mathcal { N } ( 0 , \eta ^ { 2 } )$ . We use the same setup as in [34] (see Equations (1) and (2) in [34]). Moreover, in another closely related work [41], if the kernel is set to the linear kernel, it is equivalent to our setup.
+
+Next, we introduce the estimate $\hat { \beta }$ of $\beta$ and its excess generalization loss. Let $\varepsilon = ( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } ) ^ { \top } \in \mathbb { R } ^ { n }$ denote the noise vector. The design matrix $A$ equals $[ \tilde { x } _ { 1 } , \ldots , \tilde { x } _ { n } ] ^ { \top } \in \mathbb { R } ^ { n \times d }$ . Let $x = x _ { \mathrm { t e s t } } [ 1 : d ]$ denote the first $d$ features of the test data. For the underparametrized regime where $d < n$ , the least square solution on the training data is $A ^ { + } ( A \beta + \varepsilon )$ . For the overparametrized regime where $d > n$ , $A ^ { + } ( A \beta + \varepsilon )$ is the minimum-norm solution. In both regimes we consider the solution ${ \hat { \boldsymbol { \beta } } } \triangleq A ^ { + } ( A \beta + \varepsilon )$ . The excess generalization loss on the test data is then given by
+
+$$
+\begin{array} { r l } & { L _ { d } \triangleq \mathbb { E } [ ( y - x ^ { \top } \hat { \beta } ) ^ { 2 } - ( y - x ^ { \top } \beta ) ^ { 2 } ] } \\ & { \quad = \mathbb { E } [ ( x ^ { \top } ( \hat { \beta } - \beta ) ) ^ { 2 } ] } \\ & { \quad = \mathbb { E } [ ( x ^ { \top } ( ( A ^ { + } A - I ) \beta + A ^ { + } \varepsilon ) ) ^ { 2 } ] } \\ & { \quad = \mathbb { E } [ ( x ^ { \top } ( A ^ { + } A - I ) \beta ) ^ { 2 } ] + \mathbb { E } [ ( x ^ { \top } A ^ { + } \varepsilon ) ^ { 2 } ] } \\ & { \quad = \mathbb { E } [ ( x ^ { \top } ( A ^ { + } A - I ) \beta ) ^ { 2 } ] + \eta ^ { 2 } \mathbb { E }  ( A ^ { \top } ) ^ { + } x  ^ { 2 } , } \end{array}
+$$
+
+where $y = x ^ { \top } \beta + \varepsilon _ { \mathrm { t e s t } }$ and $\varepsilon _ { \mathrm { t e s t } } \sim \mathcal { N } ( 0 , \eta ^ { 2 } )$ . We call the term $\mathbb { E } \left[ ( x ^ { \top } ( A ^ { + } A - I ) \beta ) ^ { 2 } \right]$ the bias and call the term $\eta ^ { 2 } \mathbb { E } \left\| ( A ^ { \top } ) ^ { + } x \right\| ^ { 2 }$ the variance.
+
+The next remark shows that in the underparametrized regime, the bias vanishes. The vanishing bias in the underparametrized regime is also observed by Hastie et al. [34] and shown in their Proposition 2.
+
+Remark 1. In the underparametrized regime, if $\mathcal { D }$ is a continous distribution (our construction presented later satisfies this condition), the matrix $A$ has independent column almost surely. In this case, we have $A ^ { + } A = I$ and therefore the bias $\mathbb { E } \left[ ( x ^ { \top } ( A ^ { + } A - I ) \beta ) ^ { 2 } \right]$ vanishes irrespective of $\beta$ . In other words, in the underparametrized regime, $L _ { d }$ equals $\eta ^ { 2 } \mathbb { E } \| ( A ^ { \top } ) ^ { + } x \| ^ { 2 }$ .
+
+According to Remark 1, we have $L _ { d } = \eta ^ { 2 } \mathbb { E } \| ( A ^ { \top } ) ^ { + } x \| ^ { 2 }$ in the underparametrized regime. It also holds in the overparametrized regime when $\beta = 0$ . Without loss of generality, we assume $\eta = 1$ in the underparametrized regime (for all $\beta$ ). In the overparametrized regime, we also assume $\eta = 1$ for the $\beta = 0$ case. In this case, we have
+
+$$
+L _ { d } = \mathbb { E } \| ( A ^ { \top } ) ^ { + } x \| ^ { 2 } .
+$$
+
+We assume a general $\eta$ (i.e., not necessarily being 1) in the overparametrized regime when $\beta$ is non-zero.
+
+We would like to study the change in the loss caused by the growth in the number of features revealed. Recall $L _ { d } = \mathbb { E } \| ( A ^ { \top } ) ^ { + } x \| ^ { 2 }$ . Once we reveal a new feature, which adds a new row $b ^ { \top }$ to $A ^ { \top }$ and a new component $a _ { 1 }$ to $x$ , we have $L _ { d + 1 } = \mathbb { E } { \left. \left[ \binom { A ^ { \top } } { b ^ { \top } } \right] ^ { + } \left[ \frac { x } { a _ { 1 } } \right] \right. } ^ { 2 } .$
+
+Local Maximum and Multiple Descent. Throughout the paper, we say that a local maximum occurs at a dimension $d \geq 1$ if $L _ { d - 1 } < L _ { d }$ and $L _ { d } > L _ { d + 1 }$ . Intuitively, a local maximum occurs if there is an increasing stage of the generalization loss, followed by a decreasing stage, as the dimension $d$ grows. Additionally, we define $L _ { 0 } \triangleq - \infty$ . If the generalization loss exhibits a single descent, based on our definition, a unique local maximum occurs at $d = 1$ . For a double-descent generalization curve, a local maximum occurs at two different dimensions. In general, if we observe local maxima at multiple dimensions, we say there is a multiple descent.
+
+# 4 Underparametrized Regime
+
+First, we present our main theorem for the underparametrized regime below, whose proof is deferred to the end of Section 4. It states that the generalization loss $L _ { d }$ is always non-decreasing as $d$ grows. Moreover, it is possible to have an arbitrarily large ascent, i.e., $L _ { d + 1 } - L _ { d } > C$ for any $C > 0$ .
+
+Theorem 1 (Proof in Section 4.1). If $d < n$ , we have $L _ { d + 1 } \ge L _ { d }$ irrespective of the data distribution Moreover, for any $C > 0$ , there exists a distribution $\mathcal { D }$ such that $L _ { d + 1 } - L _ { d } > C$ .
+
+Remark 2 ( $\mathcal { D }$ can be a product distribution). The first part of Theorem 1 holds irrespective of the data distribution. For the second part of the theorem ( i.e., for any $C > 0$ there exists a distribution such that $L _ { d + 1 } - L _ { d } > C )$ to hold, one extremely simple and elegant choice of the distribution $\mathcal { D }$ is a product distribution $\mathcal { D } = \mathcal { D } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { D } _ { D }$ such that $x _ { i , j } \stackrel { i i d } { \sim } \mathcal { D } _ { j }$ for all $1 \leq i \leq n$ , where $\mathcal { D } _ { j }$ is a Gaussian mixture $\mathcal { N } _ { \sigma _ { j } , 1 } ^ { \mathrm { m i x } }$ for some $\sigma _ { j } > 0$ . Since the second part of Theorem 1 is of independent interest, the result is summarized by Theorem 4.
+
+Remark 3 (Kernel regression on Gaussian data). In light of Remark 2, $\mathcal { D }$ can be chosen to be a product distribution that consists $\mathcal { N } _ { \sigma _ { j } } ^ { \mathrm { m i x } }$ . Note that one can simulate $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ with $\mathcal { N } ( 0 , 1 )$ through the inverse transform sampling. To see this, let $F _ { \mathcal { N } ( 0 , 1 ) }$ and $F _ { \mathrm { \mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } } } }$ be the cdf of $\mathcal { N } ( 0 , 1 )$ and $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ , respectively. If $X \sim \mathcal { N } ( 0 , 1 )$ , we have $F _ { \mathcal { N } ( 0 , 1 ) } ( X ) \sim \mathrm { U n i f } ( ( 0 , 1 ) )$ and therefore $\varphi _ { \sigma } ( X ) \triangleq$ $F _ { \mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } } } ^ { - 1 } ( F _ { \mathcal { N } ( 0 , 1 ) } ( X ) ) \sim \mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ . In fact, we can use a multivariate Gaussian $\mathcal { D } ^ { \prime } = \mathcal { N } ( 0 , I _ { D \times D } )$ and a sequence of non-linear kernels $k ^ { [ 1 : d ] } ( x , x ^ { \prime } )$ , $\langle \phi ^ { [ 1 : d ] } ( x ) , \phi ^ { [ 1 : d ] } ( x ^ { \prime } ) \rangle$ , where the feature map is $\phi ^ { [ 1 : d ] } ( x ) \ \triangleq \ [ \phi _ { 1 } ( x _ { 1 } ) , \phi _ { 2 } ( x _ { 2 } ) , \ldots , \phi _ { d } ( x _ { d } ) ] ^ { \top } \ \in \ \mathbb { R } ^ { d }$ . Here is a simple rule for defining $\phi _ { j }$ : if $\mathcal { D } _ { j } = \mathcal { N } _ { \sigma _ { j } } ^ { \mathrm { m i x } }$ , we set $\phi _ { j }$ to $\varphi _ { \sigma _ { j } }$ . Thus, the problem becomes a kernel regression problem on the standard Gaussian data.
+
+The first part of Theorem 1, which says that $L _ { d }$ is increasing (or more precisely, non-decreasing), agrees with Figure 1 of [11] and Proposition 2 of [34]. In [34], they proved that the risk increases with $\gamma = d / n$ . Note that, at first glance, Theorem 1 may look counterintuitive since it does not obey the classical U-shaped generalization curve. However, we would like to emphasize that the U-shaped curve does not always occur. In Figure 1 and Proposition 2 of these two papers respectively, there is no U-shaped curve. The intuition behind Theorem 1 is that in the underparametrized setting, the bias is always zero and as $d$ approaches $n$ , the variance keeps increasing.
+
+Coming to the second part of Theorem 1, we now discuss how we will construct such a distribution $\mathcal { D }$ inductively to satisfy $L _ { d + 1 } - L _ { d } > C$ . We fix $d$ . Again, denote the first $d$ features of $x _ { \mathrm { t e s t } }$ by $x \triangleq x _ { \mathrm { t e s t } } [ 1 : d ]$ . Let us add an additional component to the training data $x _ { 1 } [ 1 : d ] , \dotsc , x _ { n } [ 1 : d ]$ and test data $x$ so that the dimension $d$ is incremented by 1. Let $b _ { i } \in \mathbb { R }$ denote the additional component that we add to the vector $x _ { i }$ (so that the new vector is given as $[ x _ { i } [ 1 : d ] ^ { \top } , b _ { i } ] ^ { \top }$ . Similarly, let $a _ { 1 } \in \mathbb { R }$ denote the additional component that we add to the test vector $x$ . We form the column vector $b = [ b _ { 1 } , \ldots , b _ { n } ] ^ { \top } \in \mathbb { R } ^ { n }$ that collects all additional components that we add to the training data.
+
+We consider the change in the generalization loss as follows
+
+$$
+L _ { d + 1 } - L _ { d } = \mathbb { E } \left[ \left. \left[ \mathbf { \Sigma } _ { b } ^ { A } \right] ^ { + } \left[ \mathbf { \Sigma } _ { a _ { 1 } } ^ { x } \right] \right. ^ { 2 } - \left. ( A ^ { + } ) ^ { \top } x \right. ^ { 2 } \right] .
+$$
+
+Note that the components $b _ { 1 } , \ldots , b _ { n } , a _ { 1 }$ are i.i.d. The proof of Theorem 1 starts with Lemma 2 which relates the pseudo-inverse of $[ A , b ] ^ { \top }$ to that of $A ^ { \top }$ . In this way, we can decompose $\left\| \left[ \binom { A ^ { \top } } { b ^ { \top } } \right] ^ { + } \left[ \binom { x } { a _ { 1 } } \right] \right\| ^ { 2 }$ into multiple terms for further careful analysis in the proofs hereinafter.
+
+Lemma 2 (Proof in Appendix B.1). Let $A \in \mathbb { R } ^ { n \times d }$ and $0 \neq b \in \mathbb { R } ^ { n \times 1 }$ , where $n \geq d + 1$ Additionally, let $P = A A ^ { + }$ and $\begin{array} { r } { Q = b b ^ { + } = \frac { b b ^ { \top } } { \| b \| ^ { 2 } } } \end{array}$ bb>2 , and define z , b>(I−P )b2 . If $z \neq 0$ and the columnwise partitioned matrix $[ A , b ]$ has linearly independent columns, we have
+
+$$
+\begin{array} { r l } & { \left[ \boldsymbol { A } ^ { \top } \right] ^ { + } = \left[ \left( I - \frac { b b ^ { \top } } { \| b \| ^ { 2 } } \right) \left( I + \frac { \boldsymbol { A } \boldsymbol { A } ^ { + } b \boldsymbol { b } ^ { \top } } { \| b \| ^ { 2 } - b ^ { \top } \boldsymbol { A } \boldsymbol { A } ^ { + } b } \right) ( \boldsymbol { A } ^ { + } ) ^ { \top } , \frac { ( I - \boldsymbol { A } \boldsymbol { A } ^ { + } ) \boldsymbol { b } } { \| b \| ^ { 2 } - b ^ { \top } \boldsymbol { A } \boldsymbol { A } ^ { + } \boldsymbol { b } } \right] } \\ & { = \left[ ( I - \boldsymbol { Q } ) ( I + \frac { P Q } { 1 - \mathrm { t r } ( P Q ) } ) ( \boldsymbol { A } ^ { + } ) ^ { \top } , \frac { ( I - P ) \boldsymbol { b } } { b ^ { \top } ( I - P ) \boldsymbol { b } } \right] } \\ & { = \left[ ( I - Q ) ( I + \frac { P Q } { z } ) ( \boldsymbol { A } ^ { + } ) ^ { \top } , \frac { ( I - P ) \boldsymbol { b } } { b ^ { \top } ( I - P ) \boldsymbol { b } } \right] . } \end{array}
+$$
+
+In our construction of $\mathcal { D }$ , the components $\mathcal { D } _ { j }$ are all continuous distributions. The matrix $I - P$ is an orthogonal projection matrix and therefore $\operatorname { i r a n k } ( I - P ) = n - d .$ . As a result, it holds almost surely that $b \neq 0$ , $z \neq 0$ , and $[ A , b ]$ has linearly independent columns. Thus the assumptions of Lemma 2 are satisfied almost surely. In the sequel, we assume that these assumptions are always fulfilled.
+
+Theorem 3 guarantees that if $L _ { d } = \mathbb { E } \left\| ( A ^ { + } ) ^ { \top } x \right\| ^ { 2 }$ is finite and the $( d + 1 )$ -th features $b _ { 1 } , \ldots , b _ { n } , a _ { 1 }$ are i.i.d. sampled from $\mathcal { N } ( 0 , 1 )$ or $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ , $L _ { d + 1 } = \mathbb { E } \| [ { \binom { A ^ { \top } } { b ^ { \top } } } ^ { + } [ { \binom { x } { a _ { 1 } } } ] \| ^ { 2 }$ is also finite.
+
+Theorem 3 (Proof in Appendix B.2). Let $z$ be as defined in Lemma 2. If $b _ { 1 } , \ldots , b _ { n } , a _ { 1 }$ are i.i.d. and follow a distribution with mean zero, conditioned on $A$ and $x$ , we have
+
+$$
+\mathbb { E } _ { b , a _ { 1 } } \left[ \left. \left[ \boldsymbol { \mathsf { A } } ^ { \top } \right] ^ { + } \left[ \boldsymbol { \mathsf { x } } _ { 1 } \right] \right. ^ { 2 } \right] \leq \mathbb { E } _ { b , a _ { 1 } } \left[ \frac { 1 } { z } \left. ( \boldsymbol { A } ^ { + } ) ^ { \top } \boldsymbol { \mathsf { x } } \right. ^ { 2 } + \frac { a _ { 1 } ^ { 2 } } { b ^ { \top } ( I - P ) b } \right] .
+$$
+
+In particular, if $d + 2 < n$ and $b _ { 1 } , \dots , b _ { n } , a _ { 1 } \overset { i i d } { \sim } \mathcal { N } ( 0 , 1 )$ , conditioned on $A$ and $x$ , we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { b , a _ { 1 } } \left[ \left. \left[ \boldsymbol { \mathsf { A } } ^ { \top } \right] ^ { + } \left[ \boldsymbol { \mathsf { x } } _ { 1 } \right] \right. ^ { 2 } \right] \leq \frac { ( n - 2 ) \left. ( \boldsymbol { \mathsf { A } } ^ { + } ) ^ { \top } \boldsymbol { \mathsf { x } } \right. ^ { 2 } + 1 } { n - d - 2 } . } \end{array}
+$$
+
+$d + 2 < n$ and $b _ { 1 } , \dots , b _ { n } , a _ { 1 } \overset { i i d } { \sim } \mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ , conditioned on $A$ and $x$ , we have
+
+$$
+\mathbb { E } _ { b , a _ { 1 } } \| [ \binom { A ^ { \top } } { b ^ { \top } } ^ { + } [ \frac { x } { a _ { 1 } } ] \| ^ { 2 } \leq \frac { ( n - 2 + \sqrt { d } ) \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } + 2 / ( 3 \sigma ^ { 2 } ) + 1 } { n - d - 2 } .
+$$
+
+Using Theorem 3, we can show inductively (on $d$ ) that $L _ { d }$ is finite for every $d$ . Provided that we are able to guarantee finite $L _ { 1 }$ , Theorem 3 implies that $L _ { d }$ is finite for every $d$ if the components are always sampled from $\mathcal { N } ( 0 , 1 )$ or $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ .
+
+Making a large $L _ { d }$ can be achieved by adding an entry sampled from $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ when the data dimension increases from $d - 1$ to $d$ in the previous step. Theorem 4 shows that adding a $\mathcal { N } _ { \sigma , 1 } ^ { \mathrm { m i x } }$ feature can increase the loss by arbitrary amount, which in turn implies the second part of Theorem 1.
+
+Theorem 4 (Proof in Appendix B.4). For any $\sigma > 0$ such that if $\mathbf { \dot { \cdot } } b _ { 1 } , \dots , b _ { n } , a _ { 1 } \overset { i i d } { \sim } \mathcal { N } _ { \sigma , 1 } ^ { \operatorname* { m i x } }$ , we have $C > 0$ and $\mathbb { E } \left\| ( A ^ { + } ) ^ { \top } x \right\| ^ { 2 } < + \infty$ , there exists $a$
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \left. \left[ \boldsymbol { \mathsf { A } } ^ { \top } \right] ^ { + } \left[ \boldsymbol { \mathsf { x } } _ { 1 } \right] \right. ^ { 2 } - \left. ( \boldsymbol { \mathsf { A } } ^ { + } ) ^ { \top } \boldsymbol { \mathsf { x } } \right. ^ { 2 } \right] > C . } \end{array}
+$$
+
+We are now ready to prove Theorem 1.
+
+# 4.1 Proof of Theorem 1
+
+Proof. We follow the notation convention in (3):
+
+$$
+\begin{array} { r } { L _ { d + 1 } - L _ { d } = \mathbb { E } \left[ \left. \left[ \boldsymbol { \mathsf { A } } ^ { \top } \right] ^ { + } \left[ \boldsymbol { \mathsf { a } } _ { 1 } \right] \right. ^ { 2 } - \left. ( \boldsymbol { A } ^ { \top } ) ^ { + } \boldsymbol { \mathsf { x } } \right. ^ { 2 } \right] . } \end{array}
+$$
+
+Recall d < n and the matrix B0 , $B ^ { \prime } \triangleq { \left[ \begin{array} { l } { A ^ { \top } } \\ { b ^ { \top } } \end{array} \right] }$ is of size $( d + 1 ) \times n$ . Both matrices $B ^ { \prime }$ and $B \triangleq A ^ { \intercal }$ are fat matrices. As a result, if $x ^ { \prime } \triangleq { \left[ \begin{array} { l } { x } \\ { a _ { 1 } } \end{array} \right] }$ , we have
+
+$$
+\| B ^ { \prime + } x ^ { \prime } \| ^ { 2 } = \operatorname* { m i n } _ { z : B ^ { \prime } z = x ^ { \prime } } \| z \| ^ { 2 } , \quad \| B ^ { + } x \| ^ { 2 } = \operatorname* { m i n } _ { z : B z = x } \| z \| ^ { 2 } .
+$$
+
+Since $\{ z \mid B ^ { \prime } z = x ^ { \prime } \} \subseteq \{ z \mid B z = x \}$ , we get $\| B ^ { \prime + } x ^ { \prime } \| ^ { 2 } \geq \| B ^ { + } x \| ^ { 2 }$ . Therefore, we obtain $L _ { d + 1 } \ge L _ { d }$ . The second part follows from Theorem 4.
+
+Remark 4. Remark 2 and the proof of Theorem 4 indicate that $\mathcal { D } = \mathcal { D } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { D } _ { D }$ is a product distribution. The construction in the proof also shows that the generalization curve is determined by the specific choice of the $\mathcal { D } _ { i }$ ’s. Note that permuting the order of $\mathcal { D } _ { i }$ ’s is equivalent to changing the order by which the features are being revealed (i.e., permuting the entries of the data $x _ { i }$ ’s). Therefore, given the same data points $x _ { 1 } , \cdot \cdot \cdot , x _ { n } \in \mathbb { R } ^ { D }$ , one can create different generalization curves simply by changing the order of the feature-revealing process.
+
+![](images/3ae6272ce8f7e69e181cc3091b02f047c831fd44bf49439b06b86812553f0333.jpg)  
+Figure 2: Illustration of the multiple descent phenomenon for the generalization loss $L _ { d }$ versus the dimension of data $d$ in the overparametrized regime starting from $d = n { + } 8$ . One can fully control the generalization curve to increase or decrease as specified by the sequence $\Delta = \{ \downarrow , \uparrow , \downarrow , \downarrow , \uparrow , \downarrow , . . . \}$ . Adding a new feature with Gaussian mixture distribution increases the loss, while adding one with Gaussian distribution decreases the loss.
+
+# 5 Overparametrized Regime
+
+In this section, we study the multiple decent phenomenon in the overparametrized regime. Note that as stated in Section 3, we consider the minimum-norm solution here. We first consider the case where the model $\beta = 0$ and $L _ { d }$ is as defined in (2). Then we discuss the setting $\beta \neq 0$ .
+
+As stated in the following theorem, we require $d \ge n + 8$ . This is merely a technical requirement and we can still say that $d$ starts at roughly the same order as $n$ . In other words, the result covers almost the entire spectrum of the overparametrized regime.
+
+Theorem 5 (Overparametrized regime, $\beta ~ = ~ 0 ,$ . Let $\begin{array} { l l l } { n } & { < } & { D ~ - ~ 9 } \end{array}$ . Given any sequence $\Delta _ { n + 8 } , \Delta _ { n + 9 } , . . . , \Delta _ { D - 1 }$ where $\Delta _ { d } \in \{ \uparrow , \downarrow \}$ , there exists a distribution $\mathcal { D }$ such that for every $n + 8 \leq d \leq D - 1$ , we have
+
+$$
+L _ { d + 1 } \left\{ \stackrel { > } { _ { < } } L _ { d } , \quad i f \Delta _ { d } = \uparrow \right.
+$$
+
+In Theorem 5, the sequence $\Delta _ { n + 8 } , \Delta _ { n + 9 } , \cdot \cdot \cdot , \Delta _ { D - 1 }$ is just used to specify the increasing/decreasing behavior of the $L _ { d }$ sequence for $d > n + 8$ . Compared to Theorem 1 for the underparametrized regime, where $L _ { d }$ always increases, Theorem 5 indicates that one is able to fully control both ascents and descents in the overparametrized regime. Fig. 2 is an illustration.
+
+We now present tools for proving Theorem 5. Lemma 6 gives the pseudo-inverse of $A$ when $d > n$ . Lemma 6 (Proof in Appendix C.1). Let $A \in \mathbb { R } ^ { n \times d }$ and $b \in \mathbb { R } ^ { n \times 1 }$ , where $n \leq d$ . Assume that matrix $A$ and the columnwise partitioned matrix $B \triangleq [ A , b ]$ have linearly independent rows. Let $G \triangleq ( A A ^ { \top } ) ^ { - 1 } \in \mathbb { R } ^ { n \times n }$ and $\begin{array} { r } { u \triangleq \frac { b ^ { \intercal } G } { 1 + b ^ { \intercal } G b } \in \mathbb { R } ^ { 1 \times n } } \end{array}$ . We have
+
+$$
+\left[ \begin{array} { l }  \boldsymbol { A } ^ { \top } \right] ^ { + } = \left[ ( \boldsymbol { I } - b \boldsymbol { u } ) ^ { \top } ( \boldsymbol { A } ^ { + } ) ^ { \top } , \boldsymbol { u } ^ { \top } \right] . \end{array}
+$$
+
+Lemma 7 establishes finite expectation for several random variables. These finite expectation results are necessary for Theorem 8 and Theorem 9 to hold. Technically, they are the dominating random variables needed in Lebesgue’s dominated convergence theorem. Lemma 7 indicates that to guarantee these finite expectations, it suffices to set the first $n + 8$ distributions to the standard normal distribution and then set $\mathcal { D } _ { n + 8 } , \ldots , \mathcal { D } _ { D }$ to either a Gaussian or a Gaussian mixture distribution. In fact, in Theorem 8 and Theorem 9, we always add a Gaussian distribution or a Gaussian mixture.
+
+Lemma 7 (Proof in Appendix C.2). Let $\mathcal { D } = \mathcal { D } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { D } _ { D }$ be a product distribution where
+
+Let $\mathcal { D } _ { [ 1 : d ] }$ denote $\mathcal { D } _ { 1 } \times \cdots \times \mathcal { D } _ { d }$ . Assume that every row of $A \in \mathbb { R } ^ { n \times d }$ and $x \in \mathbb { R } ^ { d \times 1 }$ are i.i.d. and follow $\mathcal { D } _ { [ 1 : d ] }$ . For any $d$ such that $n + 8 \leq d \leq D$ , all of the followings hold:
+
+$$
+\begin{array} { r l r } & { \mathbb E [ \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } ] < + \infty , } & { \mathbb E [ \lambda _ { \operatorname* { m a x } } ^ { 2 } ( ( A A ^ { \top } ) ^ { - 1 } ) ] < + \infty , } \\ & { \mathbb E [ \lambda _ { \operatorname* { m a x } } ( ( A A ^ { \top } ) ^ { - 1 } ) \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } ] < + \infty , } & { \mathbb E [ \lambda _ { \operatorname* { m a x } } ^ { 2 } ( ( A A ^ { \top } ) ^ { - 1 } ) \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } ] < + \infty . } \end{array}
+$$
+
+Theorems 8 and 9 are the key technical results for constructing multiple descent in the overparametrized regime. One can create a descent $( L _ { d + 1 } < L _ { d } )$ by adding a Gaussian feature (Theorem 8) and create an ascent $( L _ { d + 1 } > L _ { d } )$ by adding a Gaussian mixture feature (Theorem 9).
+
+Theorem 8 (Proof in Appendix C.3). If E $\bar { \mathsf { x } } [ \| ( A ^ { \top } A ) ^ { + } x \| ^ { 2 } ] > 0$ and all equations in (4) hold, there exists $\sigma > 0$ such that i $f a _ { 1 } , b _ { 1 } , \ldots , b _ { n } \stackrel { i i d } { \sim } { \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ , we have
+
+$$
+L _ { d + 1 } - L _ { d } = \mathbb { E } \| [ { \binom { A ^ { \top } } { b ^ { \top } } } ^ { + } [ { \binom { x } { a _ { 1 } } } ] \| ^ { 2 } - \mathbb { E } \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } < 0 .
+$$
+
+Theorem 9 shows that adding a Gaussian mixture feature can make $L _ { d + 1 } > L _ { d }$ .
+
+Theorem 9 (Proof in Appendix C.4). Assume $\sigma > 0$ such that $i f a _ { 1 } , b _ { 1 } , \ldots , b _ { n } \stackrel { i i d } { \sim } \mathcal { N } _ { \sigma , \mu } ^ { \mathrm { m i x } }$ , we have $\mathbb { E } \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } < + \infty$ . For any $C > 0$ , there exist $\mu$
+
+$$
+L _ { d + 1 } - L _ { d } = \mathbb { E } \| [ { \binom { A ^ { \top } } { b ^ { \top } } } ^ { + } [ { \binom { x } { a _ { 1 } } } ] \| ^ { 2 } - \mathbb { E } \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } > C .
+$$
+
+The proof of Theorem 5 immediately follows from Theorem 8 and Theorem 9.
+
+Proof of Theorem 5. We construct the product distribution $\begin{array} { r } { \mathcal { D } = \prod _ { d = 1 } ^ { D } \mathcal { D } _ { d } } \end{array}$ . We set $\mathcal { D } _ { d } = \mathcal { N } ( 0 , 1 )$ for $d = 1 , \dotsc , n + 8$ . For $n + 8 < d \leq D$ , $\mathcal { D } _ { d }$ is either $\textstyle \mathcal { N } ( 0 , \sigma _ { d } ^ { 2 } )$ dor $\sqrt { \operatorname * { m i x } _ { \sigma _ { d } , \mu _ { d } } }$ depending on $\Delta _ { d }$ being either $\downarrow \mathrm { o r } \uparrow$ .
+
+First we show that for each step $d$ , the assumption $\mathbb { E } [ \| ( A ^ { \top } A ) ^ { + } x \| ^ { 2 } ] > 0$ of Theorem 8 is satisfied. If $\mathbb { E } [ \| ( A ^ { \top } A ) ^ { + } x \| ^ { 2 } ] = 0$ , we know that $( A ^ { \top } A ) ^ { + } x = 0$ almost surely. Since $\mathcal { D }$ is a continuous distribution, the matrix $A$ has full row rank almost surely. Therefore, $\operatorname { r a n k } ( ( A ^ { \top } A ) ^ { + } ) = \operatorname { r a n k } ( A ^ { \top } A ) = n$ almost surely. Thus $\dim \ker ( A ^ { \top } A ) ^ { + } = d - n \leq d - 1$ almost surely, which implies $x \not \in \ker ( A ^ { \top } A ) ^ { + }$ . In other words, $( A ^ { \top } A ) ^ { + } x \neq 0$ almost surely. We reach a contradiction. Moreover, by Lemma 7, the assumption $\mathbb { E } \| ( A ^ { + } ) ^ { \top } x \| ^ { 2 } < + \infty$ of Theorem 9 is also satisfied.
+
+If $\Delta _ { d - 1 } = \downarrow$ , by Theorem 8, there exists $\sigma _ { d } > 0$ such that if $\mathcal { D } _ { d } = \mathcal { N } ( 0 , \sigma _ { d } ^ { 2 } )$ , then $L _ { d } < L _ { d - 1 }$ . Similarly if $\Delta _ { d - 1 } = \uparrow$ , by Theorem 9, there exists $\sigma _ { d }$ and $\mu _ { d }$ such that $\mathcal { D } _ { d } = \mathcal { N } _ { \sigma _ { d } , \mu _ { d } } ^ { \mathrm { m i x } }$ N mixσd,µd guarantees $L _ { d } > L _ { d - 1 }$ .
+
+Gaussian $\beta$ setting. In what follows, we study the case where the model $\beta$ is non-zero. In particular, we consider a setting where each entry of $\beta$ is i.i.d. $\mathcal { N } ( 0 , \rho ^ { 2 } )$ . Recalling (1), define the biases
+
+$$
+\begin{array} { r } { \mathcal { E } _ { d } \triangleq ( x ^ { \top } ( A ^ { + } A - I ) \beta ) ^ { 2 } , \quad \mathcal { E } _ { d + 1 } \triangleq \left( [ x ^ { \top } , a _ { 1 } ] ( [ A , b ] ^ { + } [ A , b ] - I ) \left[ \beta \right] \right) ^ { 2 } , } \end{array}
+$$
+
+and the expected risks
+
+$$
+\begin{array} { r } { L _ { d } ^ { \mathrm { e x p } } \triangleq \mathbb E [ \mathcal { E } _ { d } ] + \eta ^ { 2 } \mathbb E \| ( A ^ { \top } ) ^ { + } x \| ^ { 2 } , \quad L _ { d + 1 } ^ { \mathrm { e x p } } \triangleq \mathbb E [ \mathcal { E } _ { d + 1 } ] + \eta ^ { 2 } \mathbb E \| [ [ A ^ { \top } ] ^ { + } [ a _ { 1 } ] ] ^ { 2 } , } \end{array}
+$$
+
+where $\beta \sim \mathcal { N } ( 0 , \rho ^ { 2 } I _ { d } )$ and $\beta _ { 1 } \sim \mathcal { N } ( 0 , \rho ^ { 2 } )$ . The second term in $\boldsymbol { L } _ { d } ^ { \mathrm { e x p } }$ and $L _ { d + 1 } ^ { \mathrm { e x p } }$ is the variance term. Note that ${ \cal L } _ { d } ^ { \mathrm { e x p } }$ is the expected value of $L _ { d }$ in (1) and averages over $\beta$ . Theorem 10 shows that one d can add a Gaussian mixture feature in order to make $L _ { d + 1 } ^ { \mathrm { e x p } } > L _ { d } ^ { \mathrm { e x p } }$ , and add a Gaussian feature in order to make Lexpd+1 $L _ { d + 1 } ^ { \exp } < L _ { d } ^ { \exp }$ .
+
+Theorem 10 (Proof in Appendix C.5). Let $a _ { 1 } , \beta _ { 1 } \in \mathbb { R } ,$ , $x \in \mathbb { R } ^ { d \times 1 }$ , $\beta \in \mathbb { R } ^ { d \times 1 }$ , $A \in \mathbb { R } ^ { n \times d }$ and $b \in \mathbb { R } ^ { n \times 1 }$ , where $n \leq d$ . Assume that $x , a _ { 1 } , \beta _ { 1 } , \beta , A , b$ are jointly independent, $[ \beta ^ { \top } , \beta _ { 1 } ] ^ { \top } \sim$ $\mathcal { N } ( 0 , \rho ^ { 2 } I _ { d + 1 } )$ . Moreover, assume that the matrix $[ A , b ]$ has linearly independent rows almost surely. The following statements hold:
+
+$f a _ { 1 } , b _ { 1 } , \ldots , b _ { n } \stackrel { i i d } { \sim } \mathcal { N } _ { \sigma , \mu } ^ { \mathrm { m i x } } ,$ , for any $C > 0$ , there exist $\mu , \sigma$ such that $L _ { d + 1 } ^ { \mathrm { e x p } } - L _ { d } ^ { \mathrm { e x p } } > C$ (b) If $\cdot _ { a _ { 1 } , b _ { 1 } , . . . , b _ { n } } \stackrel { i i d } { \sim } \mathcal { N } ( 0 , \sigma ^ { 2 } )$ , there exists $\sigma > 0$ such that for all
+
+$$
+\rho \leq \eta \sqrt { \frac { \mathbb { E } [ \| ( A ^ { \top } A ) ^ { + } x \| ^ { 2 } ] } { \mathbb { E } \| A ^ { + \top } x \| ^ { 2 } + 1 } } ,
+$$
+
+we hav e $L _ { d + 1 } ^ { \exp } < L _ { d } ^ { \exp }$
+
+Theorem 10 indicates that for $\beta$ obeying a normal distribution, one can still construct a generalization curve as desired by adding a Gaussian or Gaussian mixture feature properly. We make this construction explicit for any desired generalization curve in (the proof of) Theorem 11. Similar to the construction in the underparametrized regime (for all $\beta$ ) and overparametrization regime (for $\beta = 0$ ), the distribution $\mathcal { D }$ can be made a product distribution.
+
+Theorem 11 (Overparametrized regime, $\beta$ being Gaussian). Let $n < D - 9$ . Given any sequence $\Delta _ { n + 8 } , \Delta _ { n + 9 } , . . . , \Delta _ { D - 1 }$ where $\Delta _ { d } \in \{ \uparrow , \downarrow \}$ , there exists $\rho > 0$ and a distribution $\mathcal { D }$ such that for $\beta \sim \mathcal { N } ( 0 , \rho ^ { 2 } )$ and every $n + 8 \leq d \leq D - 1$ , we have
+
+$$
+\begin{array} { r } { L _ { d + 1 } ^ { \mathrm { e x p } } \left\{ { \stackrel { > } { \sim } } L _ { d } ^ { \mathrm { e x p } } , \quad i f \Delta _ { d } = \uparrow \right. } \\ { \left. < L _ { d } ^ { \mathrm { e x p } } , \quad i f \Delta _ { d } = \downarrow . \right. } \end{array}
+$$
+
+Proof of Theorem $I I$ . Define the design matrix $A _ { d } \triangleq [ x _ { 1 } [ 1 : d ] , \dots , x _ { n } [ 1 : d ] ] ^ { \intercal } \in \mathbb { R } ^ { n \times d }$ . Similar to the proof of Theorem 5, we construct the product distribution $\begin{array} { r } { \mathcal { D } = \prod _ { d = 1 } ^ { D } \mathcal { D } _ { d } } \end{array}$ . We set $\mathcal { D } _ { d } = \mathcal { N } ( 0 , 1 )$ for $d = 1 , \ldots , n + 8$ . For $n + 8 < d \leq D$ , $\mathcal { D } _ { d }$ is either $\textstyle \mathcal { N } ( 0 , \sigma _ { d } ^ { 2 } )$ or $\sqrt { \operatorname* { m i x } _ { \sigma _ { d } , \mu _ { d } } }$ depending on $\Delta _ { d }$ being either $\downarrow$ or $\uparrow$ .
+
+If $\Delta _ { d - 1 } = \uparrow$ , by Theorem 10, there exists $\sigma _ { d }$ and $\mu _ { d }$ such that $\mathcal { D } _ { d } = \mathcal { N } _ { \sigma _ { d } , \mu _ { d } } ^ { \mathrm { m i x } }$ guarantees $L _ { d } ^ { \exp } > L _ { d - 1 } ^ { \exp }$ $\Delta _ { d - 1 } = \downarrow$
+
+$$
+\rho _ { d } \triangleq \eta \sqrt { \frac { \mathbb { E } [ \| ( A _ { d - 1 } ^ { \top } A _ { d - 1 } ) ^ { + } x _ { \mathrm { t e s t } } [ 1 : d - 1 ] \| ^ { 2 } ] } { \mathbb { E } \| A _ { d - 1 } ^ { + \top } x _ { \mathrm { t e s t } } [ 1 : d - 1 ] \| ^ { 2 } + 1 } } .
+$$
+
+By Theorem 10, there exists $\sigma _ { d } > 0$ such that if $\rho \leq \rho _ { d }$ and $\mathcal { D } _ { d } = \mathcal { N } ( 0 , \sigma _ { d } ^ { 2 } )$ , then $L _ { d } ^ { \exp } < L _ { d - 1 } ^ { \exp }$ . We
+
+$$
+\rho = \operatorname* { m i n } _ { \substack { d : \Delta _ { d - 1 } = \downarrow } } \rho _ { d } .
+$$
+
+# 6 Conclusion
+
+Our work proves that the expected risk of linear regression can manifest multiple descents when the number of features increases and sample size is fixed. This is carried out through an algorithmic construction of a feature-revealing process where the newly revealed feature follows either a Gaussian distribution or a Gaussian mixture distribution. Notably, the construction also enables us to control local maxima in the underparametrized regime and control ascents/descents freely in the overparametrized regime. Overall, this allows us to design the generalization curve away from the interpolation threshold.
+
+We believe that our analysis of linear regression in this paper is a good starting point for explaining non-monotonic generalization curves observed in machine learning studies. Extending these results to more complex problem setups would be a meaningful future direction.
+
+# Funding Transparency Statement
+
+LC: Funding in direct support of this work: postdoctoral research fellowship by the Simons Institute for the Theory of Computing, University of California, Berkeley, and Google PhD Fellowship by Google. Additional revenues related to this work: internships at Google.
+
+MB acknowledges support from NSF IIS-1815697, and the support of the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and #814639.
+
+AK: Funding in direct support of this work: NSF (IIS-1845032) and ONR (N00014-19-1-2406).
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(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] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(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)? [N/A]
+
+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? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/rh0vIXw6i33/rh0vIXw6i33_content_list.json

@@ -0,0 +1,1655 @@
+[
+    {
+        "type": "text",
+        "text": "Multiple Descent: Design Your Own Generalization Curve ",
+        "text_level": 1,
+        "bbox": [
+            186,
+            122,
+            812,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lin Chen Simons Institute for the Theory of Computing University of California, Berkeley CA 94720 lin.chen@berkeley.edu ",
+        "bbox": [
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+            222,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yifei Min Department of Statistics and Data Science Yale University CT 06511 yifei.min@yale.edu ",
+        "bbox": [
+            524,
+            222,
+            803,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mikhail Belkin ",
+        "text_level": 1,
+        "bbox": [
+            277,
+            313,
+            383,
+            325
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Halıcıoglu Data Science Institute˘   \nUniversity of California, San Diego CA 92093 mbelkin@ucsd.edu ",
+        "bbox": [
+            214,
+            328,
+            449,
+            382
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Amin Karbasi School of Engineering and Applied Science Yale University CT 06511 amin.karbasi@yale.edu ",
+        "bbox": [
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+            313,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            417,
+            535,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper explores the generalization loss of linear regression in variably parameterized families of models, both under-parameterized and over-parameterized. We show that the generalization curve can have an arbitrary number of peaks, and moreover, locations of those peaks can be explicitly controlled. Our results highlight the fact that both classical U-shaped generalization curve and the recently observed double descent curve are not intrinsic properties of the model family. Instead, their emergence is due to the interaction between the properties of the data and the inductive biases of learning algorithms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The main goal of machine learning methods is to provide an accurate out-of-sample prediction, known as generalization. For a fixed family of models, a common way to select a model from this family is through empirical risk minimization, i.e., algorithmically selecting models that minimize the risk on the training dataset. Given a variably parameterized family of models, the statistical learning theory aims to identify the dependence between model complexity and model performance. The empirical risk usually decreases monotonically as the model complexity increases, and achieves its minimum when the model is rich enough to interpolate the training data, resulting in zero (or near-zero) training error. In contrast, the behaviour of the test error as a function of model complexity is far more complicated. Indeed, in this paper we show how to construct a model family for which the generalization curve can be fully controlled (away from the interpolation threshold) in both under-parameterized and over-parameterized regimes. Classical statistical learning theory supports a U-shaped curve of generalization versus model complexity [31, 33]. Under such a framework, the best model is found at the bottom of the U-shaped curve, which corresponds to appropriately balancing under-fitting and over-fitting the training data. From the view of the bias-variance trade-off, a higher model complexity increases the variance while decreasing the bias. A model with an appropriate level of complexity achieves a relatively low bias while still keeping the variance under control. On the other hand, a model that interpolates the training data is deemed to over-fit and tends to worsen the generalization performance due to the soaring variance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although classical statistical theory suggests a pattern of behavior for the generalization curve up to the interpolation threshold, it does not describe what happens beyond the interpolation threshold, commonly referred to as the over-parameterized regime. This is the exact regime where many modern machine learning models, especially deep neural networks, achieved remarkable success. Indeed, neural networks generalize well even when the models are so complex that they have the potential to interpolate all the training data points [61, 10, 32, 34]. ",
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+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Modern practitioners commonly deploy deep neural networks with hundreds of millions or even billions of parameters. It has become widely accepted that large models achieve performance superior to small models that may be suggested by the classical U-shaped generalization curve [13, 38, 55, 35, 36]. This indicates that the test error decreases again once model complexity grows beyond the interpolation threshold, resulting in the so called double-descent phenomenon described in [9], which has been broadly supported by empirical evidence [49, 48, 29, 30] and confirmed empirically on modern neural architectures by Nakkiran et al. [46]. On the theoretical side, this phenomenon has been recently addressed by several works on various model settings. In particular, Belkin et al. [11] proved the existence of double-descent phenomenon for linear regression with random feature selection and analyzed the random Fourier feature model [50]. Mei and Montanari [44] also studied the Fourier model and computed the asymptotic test error which captures the double-descent phenomenon. Bartlett et al. [8], Tsigler and Bartlett [56] analyzed and gave explicit conditions for “benign overfitting” in linear and ridge regression, respectively. Caron and Chretien [16] provided a finite sample analysis of the nonlinear function estimation and showed that the parameter learned through empirical risk minimization converges to the true parameter with high probability as the model complexity tends to infinity, implying the existence of double descent. Liu et al. [42] studied the high dimensional kernel ridge regression in the under- and over-parameterized regimes and showed that the risk curve can be double descent, bell-shaped, and monotonically decreasing. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Among all the aforementioned efforts, one particularly interesting question is whether one can observe more than two descents in the generalization curve. d’Ascoli et al. [21] empirically showed a samplewise triple-descent phenomenon under the random Fourier feature model. Similar triple-descent was also observed for linear regression [47]. More rigorously, Liang et al. [41] presented an upper bound on the risk of the minimum-norm interpolation versus the data dimension in Reproducing Kernel Hilbert Spaces (RKHS), which exhibits multiple descent. However, a multiple-descent upper bound without a properly matching lower bound does not imply the existence of a multiple-descent generalization curve. In this work, we study the multiple descent phenomenon by addressing the following questions: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Can the existence of a multiple descent generalization curve be rigorously proven? • Can an arbitrary number of descents occur? • Can the generalization curve and the locations of descents be designed? ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we show that the answer to all three of these questions is yes. Further related work is presented in Section 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our Contribution. We consider the linear regression model and analyze how the risk changes as the dimension of the data grows. In the linear regression setting, the data dimension is equal to the dimension of the parameter space, which reflects the model complexity. We rigorously show that the multiple descent generalization curve exists under this setting. To our best knowledge, this is the first work proving a multiple descent phenomenon. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our analysis considers both the underparametrized and overparametrized regimes. In the overparametrized regime, we show that one can control where a descent or an ascent occurs in the generalization curve. This is realized through our algorithmic construction of a feature-revealing process. To be more specific, we assume that the data is in $\\mathbb { R } ^ { D }$ , where $D$ can be arbitrarily large or even essentially infinite. We view each dimension of the data as a feature. We consider a linear regression problem restricted on the first $d$ features, where $d < D$ . New features are revealed by increasing the dimension of the data. We then show that by specifying the distribution of the newly revealed feature to be either a standard Gaussian or a Gaussian mixture, one can determine where an ascent or a descent occurs. In order to create an ascent when a new feature is revealed, it is sufficient that the feature follows a Gaussian mixture distribution. In order to have a descent, it is sufficient that the new feature follows a standard Gaussian distribution. Therefore, in the overparametrized regime, we can fully control the occurrence of a descent and an ascent. As a comparison, in the underparametrized regime, the generalization loss always increases regardless of the feature distribution. Generally speaking, we show that we are able to design the generalization curve. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "On the one hand, we show theoretically that the generalization curve is malleable and can be constructed in an arbitrary fashion. On the other hand, we rarely observe complex generalization curves in practice, besides carefully curated constructions. Putting these facts together, we arrive at the conclusion that realistic generalization curves arise from specific interactions between properties of typical data and the inductive biases of algorithms. We should highlight that the nature of these interactions is far from being understood and should be an area of further investigations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            195,
+            321,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our work is directly related to the recent line of research in the theoretical understanding of the double descent [11, 34, 60, 44] and the multiple descent phenomenon [41, 39]. Here we briefly discuss some other work that is closely related to this paper. ",
+        "bbox": [
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+            270
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Least Square Regression. In this paper we focus on the least square linear regression with no regularization. For the regularized least square regression, De Vito et al. [22] proposed a selection procedure for the regularization parameter. Advani and Saxe [1] analyzed the generalization of neural networks with mean squared error under the asymptotic regime where both the sample size and model complexity tend to infinity. Richards et al. [52] proved for least square regression in the asymptotic regime that as the dimension-to-sample-size ratio $d / n$ grows, an additional peak can occur in both the variance and bias due to the covariance structure of the features. As a comparison, in this paper the sample size is fixed and the model complexity increases. Rudi and Rosasco [53] studied kernel ridge regression and gave an upper bound on the number of the random features to reach certain risk level. Our result shows that there exists a natural setting where by manipulating the random features one can control the risk curve. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            436
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Over-Parameterization and Interpolation. The double descent occurs when the model complexity reaches and increases beyond the interpolation threshold. Most previous works focused on proving an upper bound or optimal rate for the risk. Caponnetto and De Vito [15] gave the optimal rate for least square ridge regression via careful selection of the regularization parameter. Belkin et al. [12] showed that the optimal rate for risk can be achieved by a model that interpolates the training data. In a series of work on kernel regression with regularization parameter tending to zero (a.k.a. kernel ridgeless regression), Rakhlin and Zhai [51] showed that the risk is bounded away from zero when the data dimension is fixed with respect to the sample size. Liang and Rakhlin [40] then considered the case when $d \\asymp n$ , showed empirically the multiple descent phenomenon and proved a risk upper bound that can be small given favorable data and kernel assumptions. Instead of giving a bound, our paper presents an exact computation of risk in the cases of underparametrized and overparametrized linear regression, and proves the existence of the multiple descent phenomenon. Wyner et al. [59] analyzed AdaBoost and Random Forest from the perspective of interpolation. There has also been a line of work on wide neural networks [4–6, 23, 3, 58, 14, 2, 18, 62, 54]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Sample-wise Double Descent and Non-monotonicity. There has also been recent development beyond the model-complexity double-descent phenomenon. For example, regarding sample-wise non-monotonicity, Nakkiran et al. [46] empirically observed the epoch-wise double-descent and sample-wise non-monotonicity for neural networks. Chen et al. [19] and Min et al. [45] identified and proved the sample-wise double descent under the adversarial training setting, and Javanmard et al. [37] discovered double-descent under adversarially robust linear regression. Loog et al. [43] showed that empirical risk minimization can lead to sample-wise non-monotonicity in the standard linear model setting under various loss functions including the absolute loss and the squared loss, which covers the range from classification to regression. We also refer the reader to their discussion of the earlier work on non-monotonicity of generalization curves. Dar et al. [20] demonstrated the double descent curve of the generalization errors of subspace fitting problems. Fei et al. [28] studied the risk-sample tradeoff in reinforcement learning. ",
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+    {
+        "type": "text",
+        "text": "3 Preliminaries and Problem Formulation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Notation. For $x \\in \\mathbb { R } ^ { D }$ and $d \\leq D$ , we let $x [ 1 : d ] \\in \\mathbb { R } ^ { d }$ denote a $d$ -dimensional vector with $x [ 1 : d ] _ { i } = x _ { i }$ for all $1 \\ \\leq \\ i \\ \\leq \\ d$ . For a matrix $A \\ \\in \\ \\mathbb { R } ^ { n \\times d }$ , we denote its Moore-Penrose pseudoinverse by $A ^ { + } \\in \\mathbb { R } ^ { d \\times n }$ and denote its spectral norm by $\\textstyle \\| A \\| \\triangleq \\operatorname* { s u p } _ { x \\neq 0 } { \\frac { \\| A x \\| _ { 2 } } { \\| x \\| _ { 2 } } }$ kAxk2 , where k · k2 is the Euclidean norm for vectors. If $v$ is a vector, its spectral norm $\\lVert v \\rVert$ agrees with the Euclidean norm $\\lVert \\boldsymbol { v } \\rVert _ { 2 }$ . Therefore, we write $\\lVert v \\rVert$ for $\\lVert \\boldsymbol { v } \\rVert _ { 2 }$ to simplify the notation. We use the big $\\mathrm { o }$ notation $\\mathcal { O }$ and write variables in the subscript of $\\mathcal { O }$ if the implicit constant depends on them. For example, ${ \\mathcal { O } } _ { n , d , \\sigma } ( 1 )$ is a constant that only depends on $n , d ,$ , and $\\sigma$ . If $f ( \\sigma )$ and $g ( \\sigma )$ are functions of $\\sigma$ , write $f ( \\sigma ) \\sim g ( \\sigma )$ if $\\begin{array} { r } { \\operatorname* { l i m } \\frac { f ( \\sigma ) } { g ( \\sigma ) } = 1 } \\end{array}$ . It will be given in the context how we take the limit. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Distributions. Let ${ \\mathcal { N } } ( \\mu , \\sigma ^ { 2 } )$ $( \\mu , \\sigma \\in \\mathbb { R } )$ and $\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )$ $\\mathbf { \\mathcal { \\mu } } _ { \\mathbf { \\lambda } } ( \\mathbf { \\mathcal { \\mu } } _ { \\mathbf { \\lambda } } \\mathbf { \\mathbb { R } } ^ { n }$ , $\\Sigma \\in \\mathbb { R } ^ { n \\times n }$ ) denote the univariate and multivariate Gaussian distributions, respectively, where $\\boldsymbol { \\mu } \\in \\mathbb { R } ^ { n }$ and $\\boldsymbol { \\Sigma } \\in \\mathbb { R } ^ { n \\times n }$ is a positive semi-definite matrix. We define a family of trimodal Gaussian mixture distributions as follows ",
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+    {
+        "type": "equation",
+        "img_path": "images/57cc300f58c1dd11885fedd07b3011525eec8a87ec445e5a0d2bf7014ec5ab8c.jpg",
+        "text": "$$\n\\mathcal { N } _ { \\sigma , \\mu } ^ { \\mathrm { m i x } } \\triangleq \\frac 1 3  { N ( 0 , \\sigma ^ { 2 } ) + \\frac { 1 } { 3 } }  { N ( - \\mu , \\sigma ^ { 2 } ) + \\frac { 1 } { 3 } }  { N ( \\mu , \\sigma ^ { 2 } ) } .\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "For an illustration, please see Fig. 1. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/34bccdb4d4a2022179ac5125e4c54a675031218de867a057d0a8c1b92194f060.jpg",
+        "image_caption": [
+            "Figure 1: Density functions of the $\\mathcal { N } ( 0 , 1 )$ and $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ feature. A new entry is independently sampled from the 1-dimensional distribution being either a standard Gaussian or trimodal Gaussian mixture. Smaller $\\sigma$ leads to higher concentration around each modes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let $\\chi ^ { 2 } ( k , \\lambda )$ denote the noncentral chi-squared distribution with $k$ degrees of freedom and the non-centrality parameter $\\lambda$ . For example, if $X _ { i } \\sim \\mathcal { N } ( \\mu _ { i } , 1 )$ (for $i = 1 , 2 , \\ldots , k )$ are independent Gaussian random variables, we have ${ \\textstyle \\sum _ { i = 1 } ^ { k } X _ { i } ^ { 2 } \\sim \\chi ^ { 2 } ( k , \\lambda ) }$ , where $\\begin{array} { r } { \\lambda = \\sum _ { i = 1 } ^ { k } \\mu _ { i } ^ { 2 } } \\end{array}$ . We also denote by $\\chi ^ { 2 } ( k )$ the (central) chi-squared distribution with $k$ degrees and the $F$ -distribution by $F ( d _ { 1 } , d _ { 2 } )$ where $d _ { 1 }$ and $d _ { 2 }$ are the degrees of freedom. ",
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+    },
+    {
+        "type": "text",
+        "text": "Problem Setup. Let $x _ { 1 } , \\ldots , x _ { n } \\in \\mathbb { R } ^ { D }$ be column vectors that represent the training data of size $n$ and let $\\boldsymbol { x } _ { \\mathrm { t e s t } } \\boldsymbol { \\bar { \\in } } \\mathbb { R } ^ { D }$ be a column vector that represents the test data. We assume that they are all independently drawn from a distribution ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c0f679e0126bf47de6049e53644f2e4b9a1866ed173bd008e502687f1da003d0.jpg",
+        "text": "$$\nx _ { 1 } , \\ldots , x _ { n } , x _ { \\mathrm { t e s t } } \\overset { i i d } { \\sim } \\mathcal { D } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let us consider a linear regression problem on the first $d$ features, where $d \\leq D$ for some arbitrary large $D$ . Here, $d$ can be viewed as the number of features revealed. Then the feature vectors are $\\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { n }$ , where $\\widetilde { x } _ { i } = x _ { i } [ 1 : d ] \\in \\mathbb { R } ^ { d }$ denotes the first $d$ entries of $x _ { i }$ . The corresponding response variable $y _ { i }$ satisfies ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49b235e3dfe13087c2d03cecadb1a05dd2e43e08e3d55b46d34739e0b9678d5d.jpg",
+        "text": "$$\ny _ { i } = \\tilde { x } _ { i } ^ { \\top } \\beta + \\varepsilon _ { i } , \\quad i = 1 , \\ldots , n ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where the noise $\\varepsilon _ { i } \\sim \\mathcal { N } ( 0 , \\eta ^ { 2 } )$ . We use the same setup as in [34] (see Equations (1) and (2) in [34]). Moreover, in another closely related work [41], if the kernel is set to the linear kernel, it is equivalent to our setup. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we introduce the estimate $\\hat { \\beta }$ of $\\beta$ and its excess generalization loss. Let $\\varepsilon = ( \\varepsilon _ { 1 } , \\ldots , \\varepsilon _ { n } ) ^ { \\top } \\in \\mathbb { R } ^ { n }$ denote the noise vector. The design matrix $A$ equals $[ \\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { n } ] ^ { \\top } \\in \\mathbb { R } ^ { n \\times d }$ . Let $x = x _ { \\mathrm { t e s t } } [ 1 : d ]$ denote the first $d$ features of the test data. For the underparametrized regime where $d < n$ , the least square solution on the training data is $A ^ { + } ( A \\beta + \\varepsilon )$ . For the overparametrized regime where $d > n$ , $A ^ { + } ( A \\beta + \\varepsilon )$ is the minimum-norm solution. In both regimes we consider the solution ${ \\hat { \\boldsymbol { \\beta } } } \\triangleq A ^ { + } ( A \\beta + \\varepsilon )$ . The excess generalization loss on the test data is then given by ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f493daeb59206f8fb83616613fb1e589ee4ecb40c040ea2377378ac22add92f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { d } \\triangleq \\mathbb { E } [ ( y - x ^ { \\top } \\hat { \\beta } ) ^ { 2 } - ( y - x ^ { \\top } \\beta ) ^ { 2 } ] } \\\\ & { \\quad = \\mathbb { E } [ ( x ^ { \\top } ( \\hat { \\beta } - \\beta ) ) ^ { 2 } ] } \\\\ & { \\quad = \\mathbb { E } [ ( x ^ { \\top } ( ( A ^ { + } A - I ) \\beta + A ^ { + } \\varepsilon ) ) ^ { 2 } ] } \\\\ & { \\quad = \\mathbb { E } [ ( x ^ { \\top } ( A ^ { + } A - I ) \\beta ) ^ { 2 } ] + \\mathbb { E } [ ( x ^ { \\top } A ^ { + } \\varepsilon ) ^ { 2 } ] } \\\\ & { \\quad = \\mathbb { E } [ ( x ^ { \\top } ( A ^ { + } A - I ) \\beta ) ^ { 2 } ] + \\eta ^ { 2 } \\mathbb { E }  ( A ^ { \\top } ) ^ { + } x  ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $y = x ^ { \\top } \\beta + \\varepsilon _ { \\mathrm { t e s t } }$ and $\\varepsilon _ { \\mathrm { t e s t } } \\sim \\mathcal { N } ( 0 , \\eta ^ { 2 } )$ . We call the term $\\mathbb { E } \\left[ ( x ^ { \\top } ( A ^ { + } A - I ) \\beta ) ^ { 2 } \\right]$ the bias and call the term $\\eta ^ { 2 } \\mathbb { E } \\left\\| ( A ^ { \\top } ) ^ { + } x \\right\\| ^ { 2 }$ the variance. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The next remark shows that in the underparametrized regime, the bias vanishes. The vanishing bias in the underparametrized regime is also observed by Hastie et al. [34] and shown in their Proposition 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark 1. In the underparametrized regime, if $\\mathcal { D }$ is a continous distribution (our construction presented later satisfies this condition), the matrix $A$ has independent column almost surely. In this case, we have $A ^ { + } A = I$ and therefore the bias $\\mathbb { E } \\left[ ( x ^ { \\top } ( A ^ { + } A - I ) \\beta ) ^ { 2 } \\right]$ vanishes irrespective of $\\beta$ . In other words, in the underparametrized regime, $L _ { d }$ equals $\\eta ^ { 2 } \\mathbb { E } \\| ( A ^ { \\top } ) ^ { + } x \\| ^ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "According to Remark 1, we have $L _ { d } = \\eta ^ { 2 } \\mathbb { E } \\| ( A ^ { \\top } ) ^ { + } x \\| ^ { 2 }$ in the underparametrized regime. It also holds in the overparametrized regime when $\\beta = 0$ . Without loss of generality, we assume $\\eta = 1$ in the underparametrized regime (for all $\\beta$ ). In the overparametrized regime, we also assume $\\eta = 1$ for the $\\beta = 0$ case. In this case, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a476d10f0893f26f74a94bf7d5a5871f6ce8326909f3e32fe7a2aa63af9e422c.jpg",
+        "text": "$$\nL _ { d } = \\mathbb { E } \\| ( A ^ { \\top } ) ^ { + } x \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We assume a general $\\eta$ (i.e., not necessarily being 1) in the overparametrized regime when $\\beta$ is non-zero. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We would like to study the change in the loss caused by the growth in the number of features revealed. Recall $L _ { d } = \\mathbb { E } \\| ( A ^ { \\top } ) ^ { + } x \\| ^ { 2 }$ . Once we reveal a new feature, which adds a new row $b ^ { \\top }$ to $A ^ { \\top }$ and a new component $a _ { 1 }$ to $x$ , we have $L _ { d + 1 } = \\mathbb { E } { \\left. \\left[ \\binom { A ^ { \\top } } { b ^ { \\top } } \\right] ^ { + } \\left[ \\frac { x } { a _ { 1 } } \\right] \\right. } ^ { 2 } .$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Local Maximum and Multiple Descent. Throughout the paper, we say that a local maximum occurs at a dimension $d \\geq 1$ if $L _ { d - 1 } < L _ { d }$ and $L _ { d } > L _ { d + 1 }$ . Intuitively, a local maximum occurs if there is an increasing stage of the generalization loss, followed by a decreasing stage, as the dimension $d$ grows. Additionally, we define $L _ { 0 } \\triangleq - \\infty$ . If the generalization loss exhibits a single descent, based on our definition, a unique local maximum occurs at $d = 1$ . For a double-descent generalization curve, a local maximum occurs at two different dimensions. In general, if we observe local maxima at multiple dimensions, we say there is a multiple descent. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Underparametrized Regime ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "First, we present our main theorem for the underparametrized regime below, whose proof is deferred to the end of Section 4. It states that the generalization loss $L _ { d }$ is always non-decreasing as $d$ grows. Moreover, it is possible to have an arbitrarily large ascent, i.e., $L _ { d + 1 } - L _ { d } > C$ for any $C > 0$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Proof in Section 4.1). If $d < n$ , we have $L _ { d + 1 } \\ge L _ { d }$ irrespective of the data distribution Moreover, for any $C > 0$ , there exists a distribution $\\mathcal { D }$ such that $L _ { d + 1 } - L _ { d } > C$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 2 ( $\\mathcal { D }$ can be a product distribution). The first part of Theorem 1 holds irrespective of the data distribution. For the second part of the theorem ( i.e., for any $C > 0$ there exists a distribution such that $L _ { d + 1 } - L _ { d } > C )$ to hold, one extremely simple and elegant choice of the distribution $\\mathcal { D }$ is a product distribution $\\mathcal { D } = \\mathcal { D } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { D } _ { D }$ such that $x _ { i , j } \\stackrel { i i d } { \\sim } \\mathcal { D } _ { j }$ for all $1 \\leq i \\leq n$ , where $\\mathcal { D } _ { j }$ is a Gaussian mixture $\\mathcal { N } _ { \\sigma _ { j } , 1 } ^ { \\mathrm { m i x } }$ for some $\\sigma _ { j } > 0$ . Since the second part of Theorem 1 is of independent interest, the result is summarized by Theorem 4. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 3 (Kernel regression on Gaussian data). In light of Remark 2, $\\mathcal { D }$ can be chosen to be a product distribution that consists $\\mathcal { N } _ { \\sigma _ { j } } ^ { \\mathrm { m i x } }$ . Note that one can simulate $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ with $\\mathcal { N } ( 0 , 1 )$ through the inverse transform sampling. To see this, let $F _ { \\mathcal { N } ( 0 , 1 ) }$ and $F _ { \\mathrm { \\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } } } }$ be the cdf of $\\mathcal { N } ( 0 , 1 )$ and $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ , respectively. If $X \\sim \\mathcal { N } ( 0 , 1 )$ , we have $F _ { \\mathcal { N } ( 0 , 1 ) } ( X ) \\sim \\mathrm { U n i f } ( ( 0 , 1 ) )$ and therefore $\\varphi _ { \\sigma } ( X ) \\triangleq$ $F _ { \\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } } } ^ { - 1 } ( F _ { \\mathcal { N } ( 0 , 1 ) } ( X ) ) \\sim \\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ . In fact, we can use a multivariate Gaussian $\\mathcal { D } ^ { \\prime } = \\mathcal { N } ( 0 , I _ { D \\times D } )$ and a sequence of non-linear kernels $k ^ { [ 1 : d ] } ( x , x ^ { \\prime } )$ , $\\langle \\phi ^ { [ 1 : d ] } ( x ) , \\phi ^ { [ 1 : d ] } ( x ^ { \\prime } ) \\rangle$ , where the feature map is $\\phi ^ { [ 1 : d ] } ( x ) \\ \\triangleq \\ [ \\phi _ { 1 } ( x _ { 1 } ) , \\phi _ { 2 } ( x _ { 2 } ) , \\ldots , \\phi _ { d } ( x _ { d } ) ] ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { d }$ . Here is a simple rule for defining $\\phi _ { j }$ : if $\\mathcal { D } _ { j } = \\mathcal { N } _ { \\sigma _ { j } } ^ { \\mathrm { m i x } }$ , we set $\\phi _ { j }$ to $\\varphi _ { \\sigma _ { j } }$ . Thus, the problem becomes a kernel regression problem on the standard Gaussian data. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first part of Theorem 1, which says that $L _ { d }$ is increasing (or more precisely, non-decreasing), agrees with Figure 1 of [11] and Proposition 2 of [34]. In [34], they proved that the risk increases with $\\gamma = d / n$ . Note that, at first glance, Theorem 1 may look counterintuitive since it does not obey the classical U-shaped generalization curve. However, we would like to emphasize that the U-shaped curve does not always occur. In Figure 1 and Proposition 2 of these two papers respectively, there is no U-shaped curve. The intuition behind Theorem 1 is that in the underparametrized setting, the bias is always zero and as $d$ approaches $n$ , the variance keeps increasing. ",
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+    {
+        "type": "text",
+        "text": "Coming to the second part of Theorem 1, we now discuss how we will construct such a distribution $\\mathcal { D }$ inductively to satisfy $L _ { d + 1 } - L _ { d } > C$ . We fix $d$ . Again, denote the first $d$ features of $x _ { \\mathrm { t e s t } }$ by $x \\triangleq x _ { \\mathrm { t e s t } } [ 1 : d ]$ . Let us add an additional component to the training data $x _ { 1 } [ 1 : d ] , \\dotsc , x _ { n } [ 1 : d ]$ and test data $x$ so that the dimension $d$ is incremented by 1. Let $b _ { i } \\in \\mathbb { R }$ denote the additional component that we add to the vector $x _ { i }$ (so that the new vector is given as $[ x _ { i } [ 1 : d ] ^ { \\top } , b _ { i } ] ^ { \\top }$ . Similarly, let $a _ { 1 } \\in \\mathbb { R }$ denote the additional component that we add to the test vector $x$ . We form the column vector $b = [ b _ { 1 } , \\ldots , b _ { n } ] ^ { \\top } \\in \\mathbb { R } ^ { n }$ that collects all additional components that we add to the training data. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We consider the change in the generalization loss as follows ",
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+        "type": "equation",
+        "img_path": "images/736d88a85092415446c8734e8319ba5396fea6efc3291241455e63109e7d4907.jpg",
+        "text": "$$\nL _ { d + 1 } - L _ { d } = \\mathbb { E } \\left[ \\left. \\left[ \\mathbf { \\Sigma } _ { b } ^ { A } \\right] ^ { + } \\left[ \\mathbf { \\Sigma } _ { a _ { 1 } } ^ { x } \\right] \\right. ^ { 2 } - \\left. ( A ^ { + } ) ^ { \\top } x \\right. ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            483,
+            678,
+            535
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the components $b _ { 1 } , \\ldots , b _ { n } , a _ { 1 }$ are i.i.d. The proof of Theorem 1 starts with Lemma 2 which relates the pseudo-inverse of $[ A , b ] ^ { \\top }$ to that of $A ^ { \\top }$ . In this way, we can decompose $\\left\\| \\left[ \\binom { A ^ { \\top } } { b ^ { \\top } } \\right] ^ { + } \\left[ \\binom { x } { a _ { 1 } } \\right] \\right\\| ^ { 2 }$ into multiple terms for further careful analysis in the proofs hereinafter. ",
+        "bbox": [
+            173,
+            546,
+            825,
+            617
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2 (Proof in Appendix B.1). Let $A \\in \\mathbb { R } ^ { n \\times d }$ and $0 \\neq b \\in \\mathbb { R } ^ { n \\times 1 }$ , where $n \\geq d + 1$ Additionally, let $P = A A ^ { + }$ and $\\begin{array} { r } { Q = b b ^ { + } = \\frac { b b ^ { \\top } } { \\| b \\| ^ { 2 } } } \\end{array}$ bb>2 , and define z , b>(I−P )b2 . If $z \\neq 0$ and the columnwise partitioned matrix $[ A , b ]$ has linearly independent columns, we have ",
+        "bbox": [
+            173,
+            621,
+            825,
+            672
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07545593772a5be926a3de5faf0e6743266719cbc28a8b923d2b6037e80bc33d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\left[ \\boldsymbol { A } ^ { \\top } \\right] ^ { + } = \\left[ \\left( I - \\frac { b b ^ { \\top } } { \\| b \\| ^ { 2 } } \\right) \\left( I + \\frac { \\boldsymbol { A } \\boldsymbol { A } ^ { + } b \\boldsymbol { b } ^ { \\top } } { \\| b \\| ^ { 2 } - b ^ { \\top } \\boldsymbol { A } \\boldsymbol { A } ^ { + } b } \\right) ( \\boldsymbol { A } ^ { + } ) ^ { \\top } , \\frac { ( I - \\boldsymbol { A } \\boldsymbol { A } ^ { + } ) \\boldsymbol { b } } { \\| b \\| ^ { 2 } - b ^ { \\top } \\boldsymbol { A } \\boldsymbol { A } ^ { + } \\boldsymbol { b } } \\right] } \\\\ & { = \\left[ ( I - \\boldsymbol { Q } ) ( I + \\frac { P Q } { 1 - \\mathrm { t r } ( P Q ) } ) ( \\boldsymbol { A } ^ { + } ) ^ { \\top } , \\frac { ( I - P ) \\boldsymbol { b } } { b ^ { \\top } ( I - P ) \\boldsymbol { b } } \\right] } \\\\ & { = \\left[ ( I - Q ) ( I + \\frac { P Q } { z } ) ( \\boldsymbol { A } ^ { + } ) ^ { \\top } , \\frac { ( I - P ) \\boldsymbol { b } } { b ^ { \\top } ( I - P ) \\boldsymbol { b } } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            680,
+            728,
+            773
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our construction of $\\mathcal { D }$ , the components $\\mathcal { D } _ { j }$ are all continuous distributions. The matrix $I - P$ is an orthogonal projection matrix and therefore $\\operatorname { i r a n k } ( I - P ) = n - d .$ . As a result, it holds almost surely that $b \\neq 0$ , $z \\neq 0$ , and $[ A , b ]$ has linearly independent columns. Thus the assumptions of Lemma 2 are satisfied almost surely. In the sequel, we assume that these assumptions are always fulfilled. ",
+        "bbox": [
+            176,
+            785,
+            823,
+            843
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 guarantees that if $L _ { d } = \\mathbb { E } \\left\\| ( A ^ { + } ) ^ { \\top } x \\right\\| ^ { 2 }$ is finite and the $( d + 1 )$ -th features $b _ { 1 } , \\ldots , b _ { n } , a _ { 1 }$ are i.i.d. sampled from $\\mathcal { N } ( 0 , 1 )$ or $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ , $L _ { d + 1 } = \\mathbb { E } \\| [ { \\binom { A ^ { \\top } } { b ^ { \\top } } } ^ { + } [ { \\binom { x } { a _ { 1 } } } ] \\| ^ { 2 }$ is also finite. ",
+        "bbox": [
+            173,
+            848,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (Proof in Appendix B.2). Let $z$ be as defined in Lemma 2. If $b _ { 1 } , \\ldots , b _ { n } , a _ { 1 }$ are i.i.d. and follow a distribution with mean zero, conditioned on $A$ and $x$ , we have ",
+        "bbox": [
+            169,
+            90,
+            826,
+            121
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b533c9e79d65e1ba3799b5ac474aa82d414370f9fe0389b60ba42a59f3c6d06.jpg",
+        "text": "$$\n\\mathbb { E } _ { b , a _ { 1 } } \\left[ \\left. \\left[ \\boldsymbol { \\mathsf { A } } ^ { \\top } \\right] ^ { + } \\left[ \\boldsymbol { \\mathsf { x } } _ { 1 } \\right] \\right. ^ { 2 } \\right] \\leq \\mathbb { E } _ { b , a _ { 1 } } \\left[ \\frac { 1 } { z } \\left. ( \\boldsymbol { A } ^ { + } ) ^ { \\top } \\boldsymbol { \\mathsf { x } } \\right. ^ { 2 } + \\frac { a _ { 1 } ^ { 2 } } { b ^ { \\top } ( I - P ) b } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            125,
+            728,
+            176
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In particular, if $d + 2 < n$ and $b _ { 1 } , \\dots , b _ { n } , a _ { 1 } \\overset { i i d } { \\sim } \\mathcal { N } ( 0 , 1 )$ , conditioned on $A$ and $x$ , we have ",
+        "bbox": [
+            171,
+            183,
+            771,
+            202
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b6d598af4f89ed38955604f80b592ce6d4cafe36931325d8a08ae819e52b7e33.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { b , a _ { 1 } } \\left[ \\left. \\left[ \\boldsymbol { \\mathsf { A } } ^ { \\top } \\right] ^ { + } \\left[ \\boldsymbol { \\mathsf { x } } _ { 1 } \\right] \\right. ^ { 2 } \\right] \\leq \\frac { ( n - 2 ) \\left. ( \\boldsymbol { \\mathsf { A } } ^ { + } ) ^ { \\top } \\boldsymbol { \\mathsf { x } } \\right. ^ { 2 } + 1 } { n - d - 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            207,
+            684,
+            257
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "$d + 2 < n$ and $b _ { 1 } , \\dots , b _ { n } , a _ { 1 } \\overset { i i d } { \\sim } \\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ , conditioned on $A$ and $x$ , we have ",
+        "bbox": [
+            174,
+            265,
+            665,
+            285
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/373979f6f26f89130635fcf71f23d1adc467e2bf7aad3b000ecc85d3c6740e4b.jpg",
+        "text": "$$\n\\mathbb { E } _ { b , a _ { 1 } } \\| [ \\binom { A ^ { \\top } } { b ^ { \\top } } ^ { + } [ \\frac { x } { a _ { 1 } } ] \\| ^ { 2 } \\leq \\frac { ( n - 2 + \\sqrt { d } ) \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } + 2 / ( 3 \\sigma ^ { 2 } ) + 1 } { n - d - 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            291,
+            732,
+            337
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using Theorem 3, we can show inductively (on $d$ ) that $L _ { d }$ is finite for every $d$ . Provided that we are able to guarantee finite $L _ { 1 }$ , Theorem 3 implies that $L _ { d }$ is finite for every $d$ if the components are always sampled from $\\mathcal { N } ( 0 , 1 )$ or $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ . ",
+        "bbox": [
+            174,
+            348,
+            825,
+            392
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Making a large $L _ { d }$ can be achieved by adding an entry sampled from $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ when the data dimension increases from $d - 1$ to $d$ in the previous step. Theorem 4 shows that adding a $\\mathcal { N } _ { \\sigma , 1 } ^ { \\mathrm { m i x } }$ feature can increase the loss by arbitrary amount, which in turn implies the second part of Theorem 1. ",
+        "bbox": [
+            173,
+            398,
+            825,
+            445
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4 (Proof in Appendix B.4). For any $\\sigma > 0$ such that if $\\mathbf { \\dot { \\cdot } } b _ { 1 } , \\dots , b _ { n } , a _ { 1 } \\overset { i i d } { \\sim } \\mathcal { N } _ { \\sigma , 1 } ^ { \\operatorname* { m i x } }$ , we have $C > 0$ and $\\mathbb { E } \\left\\| ( A ^ { + } ) ^ { \\top } x \\right\\| ^ { 2 } < + \\infty$ , there exists $a$ ",
+        "bbox": [
+            173,
+            449,
+            825,
+            488
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1977ccb28cb73ac1cc18f1cfee61a29c4b17519e6f423f6603cc389cc3dbf0ad.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\left. \\left[ \\boldsymbol { \\mathsf { A } } ^ { \\top } \\right] ^ { + } \\left[ \\boldsymbol { \\mathsf { x } } _ { 1 } \\right] \\right. ^ { 2 } - \\left. ( \\boldsymbol { \\mathsf { A } } ^ { + } ) ^ { \\top } \\boldsymbol { \\mathsf { x } } \\right. ^ { 2 } \\right] > C . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            494,
+            645,
+            545
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to prove Theorem 1. ",
+        "bbox": [
+            174,
+            556,
+            429,
+            571
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Proof of Theorem 1 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            587,
+            348,
+            603
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof. We follow the notation convention in (3): ",
+        "bbox": [
+            173,
+            613,
+            495,
+            628
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3bb8f11668637f710d3c179994c54557d750678adcff362035f9f2ec6b418013.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { d + 1 } - L _ { d } = \\mathbb { E } \\left[ \\left. \\left[ \\boldsymbol { \\mathsf { A } } ^ { \\top } \\right] ^ { + } \\left[ \\boldsymbol { \\mathsf { a } } _ { 1 } \\right] \\right. ^ { 2 } - \\left. ( \\boldsymbol { A } ^ { \\top } ) ^ { + } \\boldsymbol { \\mathsf { x } } \\right. ^ { 2 } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            632,
+            678,
+            684
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Recall d < n and the matrix B0 , $B ^ { \\prime } \\triangleq { \\left[ \\begin{array} { l } { A ^ { \\top } } \\\\ { b ^ { \\top } } \\end{array} \\right] }$ is of size $( d + 1 ) \\times n$ . Both matrices $B ^ { \\prime }$ and $B \\triangleq A ^ { \\intercal }$ are fat matrices. As a result, if $x ^ { \\prime } \\triangleq { \\left[ \\begin{array} { l } { x } \\\\ { a _ { 1 } } \\end{array} \\right] }$ , we have ",
+        "bbox": [
+            173,
+            689,
+            826,
+            755
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/387d6cf733b428adfcf37546909c7757c5f0671699813b6d720551a664de3dc4.jpg",
+        "text": "$$\n\\| B ^ { \\prime + } x ^ { \\prime } \\| ^ { 2 } = \\operatorname* { m i n } _ { z : B ^ { \\prime } z = x ^ { \\prime } } \\| z \\| ^ { 2 } , \\quad \\| B ^ { + } x \\| ^ { 2 } = \\operatorname* { m i n } _ { z : B z = x } \\| z \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            760,
+            684,
+            784
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Since $\\{ z \\mid B ^ { \\prime } z = x ^ { \\prime } \\} \\subseteq \\{ z \\mid B z = x \\}$ , we get $\\| B ^ { \\prime + } x ^ { \\prime } \\| ^ { 2 } \\geq \\| B ^ { + } x \\| ^ { 2 }$ . Therefore, we obtain $L _ { d + 1 } \\ge L _ { d }$ . The second part follows from Theorem 4. ",
+        "bbox": [
+            173,
+            790,
+            825,
+            820
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark 4. Remark 2 and the proof of Theorem 4 indicate that $\\mathcal { D } = \\mathcal { D } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { D } _ { D }$ is a product distribution. The construction in the proof also shows that the generalization curve is determined by the specific choice of the $\\mathcal { D } _ { i }$ ’s. Note that permuting the order of $\\mathcal { D } _ { i }$ ’s is equivalent to changing the order by which the features are being revealed (i.e., permuting the entries of the data $x _ { i }$ ’s). Therefore, given the same data points $x _ { 1 } , \\cdot \\cdot \\cdot , x _ { n } \\in \\mathbb { R } ^ { D }$ , one can create different generalization curves simply by changing the order of the feature-revealing process. ",
+        "bbox": [
+            173,
+            827,
+            825,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ae6272ce8f7e69e181cc3091b02f047c831fd44bf49439b06b86812553f0333.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the multiple descent phenomenon for the generalization loss $L _ { d }$ versus the dimension of data $d$ in the overparametrized regime starting from $d = n { + } 8$ . One can fully control the generalization curve to increase or decrease as specified by the sequence $\\Delta = \\{ \\downarrow , \\uparrow , \\downarrow , \\downarrow , \\uparrow , \\downarrow , . . . \\}$ . Adding a new feature with Gaussian mixture distribution increases the loss, while adding one with Gaussian distribution decreases the loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            323,
+            93,
+            671,
+            229
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Overparametrized Regime ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            337,
+            428,
+            354
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study the multiple decent phenomenon in the overparametrized regime. Note that as stated in Section 3, we consider the minimum-norm solution here. We first consider the case where the model $\\beta = 0$ and $L _ { d }$ is as defined in (2). Then we discuss the setting $\\beta \\neq 0$ . ",
+        "bbox": [
+            173,
+            368,
+            825,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As stated in the following theorem, we require $d \\ge n + 8$ . This is merely a technical requirement and we can still say that $d$ starts at roughly the same order as $n$ . In other words, the result covers almost the entire spectrum of the overparametrized regime. ",
+        "bbox": [
+            173,
+            416,
+            823,
+            459
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (Overparametrized regime, $\\beta ~ = ~ 0 ,$ . Let $\\begin{array} { l l l } { n } & { < } & { D ~ - ~ 9 } \\end{array}$ . Given any sequence $\\Delta _ { n + 8 } , \\Delta _ { n + 9 } , . . . , \\Delta _ { D - 1 }$ where $\\Delta _ { d } \\in \\{ \\uparrow , \\downarrow \\}$ , there exists a distribution $\\mathcal { D }$ such that for every $n + 8 \\leq d \\leq D - 1$ , we have ",
+        "bbox": [
+            173,
+            463,
+            825,
+            507
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/93c048186b3cf238bce7b8e498922367633ee7ada863077240ea65bda52533c9.jpg",
+        "text": "$$\nL _ { d + 1 } \\left\\{ \\stackrel { > } { _ { < } } L _ { d } , \\quad i f \\Delta _ { d } = \\uparrow \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            515,
+            583,
+            550
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Theorem 5, the sequence $\\Delta _ { n + 8 } , \\Delta _ { n + 9 } , \\cdot \\cdot \\cdot , \\Delta _ { D - 1 }$ is just used to specify the increasing/decreasing behavior of the $L _ { d }$ sequence for $d > n + 8$ . Compared to Theorem 1 for the underparametrized regime, where $L _ { d }$ always increases, Theorem 5 indicates that one is able to fully control both ascents and descents in the overparametrized regime. Fig. 2 is an illustration. ",
+        "bbox": [
+            173,
+            565,
+            826,
+            622
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now present tools for proving Theorem 5. Lemma 6 gives the pseudo-inverse of $A$ when $d > n$ . Lemma 6 (Proof in Appendix C.1). Let $A \\in \\mathbb { R } ^ { n \\times d }$ and $b \\in \\mathbb { R } ^ { n \\times 1 }$ , where $n \\leq d$ . Assume that matrix $A$ and the columnwise partitioned matrix $B \\triangleq [ A , b ]$ have linearly independent rows. Let $G \\triangleq ( A A ^ { \\top } ) ^ { - 1 } \\in \\mathbb { R } ^ { n \\times n }$ and $\\begin{array} { r } { u \\triangleq \\frac { b ^ { \\intercal } G } { 1 + b ^ { \\intercal } G b } \\in \\mathbb { R } ^ { 1 \\times n } } \\end{array}$ . We have ",
+        "bbox": [
+            171,
+            627,
+            825,
+            698
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f0220a941ba7b87cef7f93533316bc31f11203b20f8dbfe46b7ec2562155c94e.jpg",
+        "text": "$$\n\\left[ \\begin{array} { l }  \\boldsymbol { A } ^ { \\top } \\right] ^ { + } = \\left[ ( \\boldsymbol { I } - b \\boldsymbol { u } ) ^ { \\top } ( \\boldsymbol { A } ^ { + } ) ^ { \\top } , \\boldsymbol { u } ^ { \\top } \\right] . \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            707,
+            620,
+            743
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 7 establishes finite expectation for several random variables. These finite expectation results are necessary for Theorem 8 and Theorem 9 to hold. Technically, they are the dominating random variables needed in Lebesgue’s dominated convergence theorem. Lemma 7 indicates that to guarantee these finite expectations, it suffices to set the first $n + 8$ distributions to the standard normal distribution and then set $\\mathcal { D } _ { n + 8 } , \\ldots , \\mathcal { D } _ { D }$ to either a Gaussian or a Gaussian mixture distribution. In fact, in Theorem 8 and Theorem 9, we always add a Gaussian distribution or a Gaussian mixture. ",
+        "bbox": [
+            173,
+            756,
+            825,
+            842
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 7 (Proof in Appendix C.2). Let $\\mathcal { D } = \\mathcal { D } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { D } _ { D }$ be a product distribution where ",
+        "bbox": [
+            173,
+            845,
+            790,
+            861
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { D } _ { [ 1 : d ] }$ denote $\\mathcal { D } _ { 1 } \\times \\cdots \\times \\mathcal { D } _ { d }$ . Assume that every row of $A \\in \\mathbb { R } ^ { n \\times d }$ and $x \\in \\mathbb { R } ^ { d \\times 1 }$ are i.i.d. and follow $\\mathcal { D } _ { [ 1 : d ] }$ . For any $d$ such that $n + 8 \\leq d \\leq D$ , all of the followings hold: ",
+        "bbox": [
+            168,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a696906f67177230dcb64d4ef0c1471edbcc37f24a59353c8888ebb8078803ae.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { \\mathbb E [ \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } ] < + \\infty , } & { \\mathbb E [ \\lambda _ { \\operatorname* { m a x } } ^ { 2 } ( ( A A ^ { \\top } ) ^ { - 1 } ) ] < + \\infty , } \\\\ & { \\mathbb E [ \\lambda _ { \\operatorname* { m a x } } ( ( A A ^ { \\top } ) ^ { - 1 } ) \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } ] < + \\infty , } & { \\mathbb E [ \\lambda _ { \\operatorname* { m a x } } ^ { 2 } ( ( A A ^ { \\top } ) ^ { - 1 } ) \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } ] < + \\infty . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            126,
+            782,
+            170
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorems 8 and 9 are the key technical results for constructing multiple descent in the overparametrized regime. One can create a descent $( L _ { d + 1 } < L _ { d } )$ by adding a Gaussian feature (Theorem 8) and create an ascent $( L _ { d + 1 } > L _ { d } )$ by adding a Gaussian mixture feature (Theorem 9). ",
+        "bbox": [
+            174,
+            179,
+            825,
+            223
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 8 (Proof in Appendix C.3). If E $\\bar { \\mathsf { x } } [ \\| ( A ^ { \\top } A ) ^ { + } x \\| ^ { 2 } ] > 0$ and all equations in (4) hold, there exists $\\sigma > 0$ such that i $f a _ { 1 } , b _ { 1 } , \\ldots , b _ { n } \\stackrel { i i d } { \\sim } { \\mathcal { N } } ( 0 , \\sigma ^ { 2 } )$ , we have ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97bf275af72a9086d50c3b6d41b83592fef8868904bce270b8d1817becdce140.jpg",
+        "text": "$$\nL _ { d + 1 } - L _ { d } = \\mathbb { E } \\| [ { \\binom { A ^ { \\top } } { b ^ { \\top } } } ^ { + } [ { \\binom { x } { a _ { 1 } } } ] \\| ^ { 2 } - \\mathbb { E } \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } < 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            266,
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+            311
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 9 shows that adding a Gaussian mixture feature can make $L _ { d + 1 } > L _ { d }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 9 (Proof in Appendix C.4). Assume $\\sigma > 0$ such that $i f a _ { 1 } , b _ { 1 } , \\ldots , b _ { n } \\stackrel { i i d } { \\sim } \\mathcal { N } _ { \\sigma , \\mu } ^ { \\mathrm { m i x } }$ , we have $\\mathbb { E } \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } < + \\infty$ . For any $C > 0$ , there exist $\\mu$ ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3aa389b5c547e615add27c9dce5ddf0bb32d87e188fd6371534478fe4847913b.jpg",
+        "text": "$$\nL _ { d + 1 } - L _ { d } = \\mathbb { E } \\| [ { \\binom { A ^ { \\top } } { b ^ { \\top } } } ^ { + } [ { \\binom { x } { a _ { 1 } } } ] \\| ^ { 2 } - \\mathbb { E } \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } > C .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            383,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The proof of Theorem 5 immediately follows from Theorem 8 and Theorem 9. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 5. We construct the product distribution $\\begin{array} { r } { \\mathcal { D } = \\prod _ { d = 1 } ^ { D } \\mathcal { D } _ { d } } \\end{array}$ . We set $\\mathcal { D } _ { d } = \\mathcal { N } ( 0 , 1 )$ for $d = 1 , \\dotsc , n + 8$ . For $n + 8 < d \\leq D$ , $\\mathcal { D } _ { d }$ is either $\\textstyle \\mathcal { N } ( 0 , \\sigma _ { d } ^ { 2 } )$ dor $\\sqrt { \\operatorname * { m i x } _ { \\sigma _ { d } , \\mu _ { d } } }$ depending on $\\Delta _ { d }$ being either $\\downarrow \\mathrm { o r } \\uparrow$ . ",
+        "bbox": [
+            173,
+            470,
+            825,
+            517
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "First we show that for each step $d$ , the assumption $\\mathbb { E } [ \\| ( A ^ { \\top } A ) ^ { + } x \\| ^ { 2 } ] > 0$ of Theorem 8 is satisfied. If $\\mathbb { E } [ \\| ( A ^ { \\top } A ) ^ { + } x \\| ^ { 2 } ] = 0$ , we know that $( A ^ { \\top } A ) ^ { + } x = 0$ almost surely. Since $\\mathcal { D }$ is a continuous distribution, the matrix $A$ has full row rank almost surely. Therefore, $\\operatorname { r a n k } ( ( A ^ { \\top } A ) ^ { + } ) = \\operatorname { r a n k } ( A ^ { \\top } A ) = n$ almost surely. Thus $\\dim \\ker ( A ^ { \\top } A ) ^ { + } = d - n \\leq d - 1$ almost surely, which implies $x \\not \\in \\ker ( A ^ { \\top } A ) ^ { + }$ . In other words, $( A ^ { \\top } A ) ^ { + } x \\neq 0$ almost surely. We reach a contradiction. Moreover, by Lemma 7, the assumption $\\mathbb { E } \\| ( A ^ { + } ) ^ { \\top } x \\| ^ { 2 } < + \\infty$ of Theorem 9 is also satisfied. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "If $\\Delta _ { d - 1 } = \\downarrow$ , by Theorem 8, there exists $\\sigma _ { d } > 0$ such that if $\\mathcal { D } _ { d } = \\mathcal { N } ( 0 , \\sigma _ { d } ^ { 2 } )$ , then $L _ { d } < L _ { d - 1 }$ . Similarly if $\\Delta _ { d - 1 } = \\uparrow$ , by Theorem 9, there exists $\\sigma _ { d }$ and $\\mu _ { d }$ such that $\\mathcal { D } _ { d } = \\mathcal { N } _ { \\sigma _ { d } , \\mu _ { d } } ^ { \\mathrm { m i x } }$ N mixσd,µd guarantees $L _ { d } > L _ { d - 1 }$ . ",
+        "bbox": [
+            173,
+            618,
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+            664
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Gaussian $\\beta$ setting. In what follows, we study the case where the model $\\beta$ is non-zero. In particular, we consider a setting where each entry of $\\beta$ is i.i.d. $\\mathcal { N } ( 0 , \\rho ^ { 2 } )$ . Recalling (1), define the biases ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/763f0cfeefbccd51465a18c3321dac866a2ddac5c0c834ddd645ebae71418c00.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { E } _ { d } \\triangleq ( x ^ { \\top } ( A ^ { + } A - I ) \\beta ) ^ { 2 } , \\quad \\mathcal { E } _ { d + 1 } \\triangleq \\left( [ x ^ { \\top } , a _ { 1 } ] ( [ A , b ] ^ { + } [ A , b ] - I ) \\left[ \\beta \\right] \\right) ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "and the expected risks ",
+        "bbox": [
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+            776,
+            321,
+            790
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a28a7100cda91abd38adcc828c9460e0ecc68226ed3891724dc9fe9b564360c.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { d } ^ { \\mathrm { e x p } } \\triangleq \\mathbb E [ \\mathcal { E } _ { d } ] + \\eta ^ { 2 } \\mathbb E \\| ( A ^ { \\top } ) ^ { + } x \\| ^ { 2 } , \\quad L _ { d + 1 } ^ { \\mathrm { e x p } } \\triangleq \\mathbb E [ \\mathcal { E } _ { d + 1 } ] + \\eta ^ { 2 } \\mathbb E \\| [ [ A ^ { \\top } ] ^ { + } [ a _ { 1 } ] ] ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            796,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta \\sim \\mathcal { N } ( 0 , \\rho ^ { 2 } I _ { d } )$ and $\\beta _ { 1 } \\sim \\mathcal { N } ( 0 , \\rho ^ { 2 } )$ . The second term in $\\boldsymbol { L } _ { d } ^ { \\mathrm { e x p } }$ and $L _ { d + 1 } ^ { \\mathrm { e x p } }$ is the variance term. Note that ${ \\cal L } _ { d } ^ { \\mathrm { e x p } }$ is the expected value of $L _ { d }$ in (1) and averages over $\\beta$ . Theorem 10 shows that one d can add a Gaussian mixture feature in order to make $L _ { d + 1 } ^ { \\mathrm { e x p } } > L _ { d } ^ { \\mathrm { e x p } }$ , and add a Gaussian feature in order to make Lexpd+1 $L _ { d + 1 } ^ { \\exp } < L _ { d } ^ { \\exp }$ . ",
+        "bbox": [
+            173,
+            847,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 10 (Proof in Appendix C.5). Let $a _ { 1 } , \\beta _ { 1 } \\in \\mathbb { R } ,$ , $x \\in \\mathbb { R } ^ { d \\times 1 }$ , $\\beta \\in \\mathbb { R } ^ { d \\times 1 }$ , $A \\in \\mathbb { R } ^ { n \\times d }$ and $b \\in \\mathbb { R } ^ { n \\times 1 }$ , where $n \\leq d$ . Assume that $x , a _ { 1 } , \\beta _ { 1 } , \\beta , A , b$ are jointly independent, $[ \\beta ^ { \\top } , \\beta _ { 1 } ] ^ { \\top } \\sim$ $\\mathcal { N } ( 0 , \\rho ^ { 2 } I _ { d + 1 } )$ . Moreover, assume that the matrix $[ A , b ]$ has linearly independent rows almost surely. The following statements hold: ",
+        "bbox": [
+            173,
+            89,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "$f a _ { 1 } , b _ { 1 } , \\ldots , b _ { n } \\stackrel { i i d } { \\sim } \\mathcal { N } _ { \\sigma , \\mu } ^ { \\mathrm { m i x } } ,$ , for any $C > 0$ , there exist $\\mu , \\sigma$ such that $L _ { d + 1 } ^ { \\mathrm { e x p } } - L _ { d } ^ { \\mathrm { e x p } } > C$ (b) If $\\cdot _ { a _ { 1 } , b _ { 1 } , . . . , b _ { n } } \\stackrel { i i d } { \\sim } \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ , there exists $\\sigma > 0$ such that for all ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            205,
+            190,
+            658,
+            212
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7eb302e78e08a0fa43c69d7dc10a3d2d535628d58d4627661b31934bc2ac2695.jpg",
+        "text": "$$\n\\rho \\leq \\eta \\sqrt { \\frac { \\mathbb { E } [ \\| ( A ^ { \\top } A ) ^ { + } x \\| ^ { 2 } ] } { \\mathbb { E } \\| A ^ { + \\top } x \\| ^ { 2 } + 1 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            219,
+            619,
+            262
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "we hav e $L _ { d + 1 } ^ { \\exp } < L _ { d } ^ { \\exp }$ ",
+        "bbox": [
+            232,
+            270,
+            380,
+            290
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 10 indicates that for $\\beta$ obeying a normal distribution, one can still construct a generalization curve as desired by adding a Gaussian or Gaussian mixture feature properly. We make this construction explicit for any desired generalization curve in (the proof of) Theorem 11. Similar to the construction in the underparametrized regime (for all $\\beta$ ) and overparametrization regime (for $\\beta = 0$ ), the distribution $\\mathcal { D }$ can be made a product distribution. ",
+        "bbox": [
+            173,
+            297,
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+            369
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 11 (Overparametrized regime, $\\beta$ being Gaussian). Let $n < D - 9$ . Given any sequence $\\Delta _ { n + 8 } , \\Delta _ { n + 9 } , . . . , \\Delta _ { D - 1 }$ where $\\Delta _ { d } \\in \\{ \\uparrow , \\downarrow \\}$ , there exists $\\rho > 0$ and a distribution $\\mathcal { D }$ such that for $\\beta \\sim \\mathcal { N } ( 0 , \\rho ^ { 2 } )$ and every $n + 8 \\leq d \\leq D - 1$ , we have ",
+        "bbox": [
+            173,
+            375,
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+            416
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/432bf31e86eef41a962b09345a5232a7f3d9c603ef2ebfa80af7412a08452350.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { d + 1 } ^ { \\mathrm { e x p } } \\left\\{ { \\stackrel { > } { \\sim } } L _ { d } ^ { \\mathrm { e x p } } , \\quad i f \\Delta _ { d } = \\uparrow \\right. } \\\\ { \\left. < L _ { d } ^ { \\mathrm { e x p } } , \\quad i f \\Delta _ { d } = \\downarrow . \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            428,
+            588,
+            463
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem $I I$ . Define the design matrix $A _ { d } \\triangleq [ x _ { 1 } [ 1 : d ] , \\dots , x _ { n } [ 1 : d ] ] ^ { \\intercal } \\in \\mathbb { R } ^ { n \\times d }$ . Similar to the proof of Theorem 5, we construct the product distribution $\\begin{array} { r } { \\mathcal { D } = \\prod _ { d = 1 } ^ { D } \\mathcal { D } _ { d } } \\end{array}$ . We set $\\mathcal { D } _ { d } = \\mathcal { N } ( 0 , 1 )$ for $d = 1 , \\ldots , n + 8$ . For $n + 8 < d \\leq D$ , $\\mathcal { D } _ { d }$ is either $\\textstyle \\mathcal { N } ( 0 , \\sigma _ { d } ^ { 2 } )$ or $\\sqrt { \\operatorname* { m i x } _ { \\sigma _ { d } , \\mu _ { d } } }$ depending on $\\Delta _ { d }$ being either $\\downarrow$ or $\\uparrow$ . ",
+        "bbox": [
+            173,
+            486,
+            826,
+            549
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "If $\\Delta _ { d - 1 } = \\uparrow$ , by Theorem 10, there exists $\\sigma _ { d }$ and $\\mu _ { d }$ such that $\\mathcal { D } _ { d } = \\mathcal { N } _ { \\sigma _ { d } , \\mu _ { d } } ^ { \\mathrm { m i x } }$ guarantees $L _ { d } ^ { \\exp } > L _ { d - 1 } ^ { \\exp }$ $\\Delta _ { d - 1 } = \\downarrow$ ",
+        "bbox": [
+            173,
+            553,
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+            583
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/883b4b1b7f6595ef74d8cbdc81b871328ca3d0dd5b1b30bd9104660d7e10e23b.jpg",
+        "text": "$$\n\\rho _ { d } \\triangleq \\eta \\sqrt { \\frac { \\mathbb { E } [ \\| ( A _ { d - 1 } ^ { \\top } A _ { d - 1 } ) ^ { + } x _ { \\mathrm { t e s t } } [ 1 : d - 1 ] \\| ^ { 2 } ] } { \\mathbb { E } \\| A _ { d - 1 } ^ { + \\top } x _ { \\mathrm { t e s t } } [ 1 : d - 1 ] \\| ^ { 2 } + 1 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            584,
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+            627
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "By Theorem 10, there exists $\\sigma _ { d } > 0$ such that if $\\rho \\leq \\rho _ { d }$ and $\\mathcal { D } _ { d } = \\mathcal { N } ( 0 , \\sigma _ { d } ^ { 2 } )$ , then $L _ { d } ^ { \\exp } < L _ { d - 1 } ^ { \\exp }$ . We ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9e785045e0b81eb940bada3b6eb7fd32c46e0dc5e63647e7fb4ef6b5885aa0db.jpg",
+        "text": "$$\n\\rho = \\operatorname* { m i n } _ { \\substack { d : \\Delta _ { d - 1 } = \\downarrow } } \\rho _ { d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our work proves that the expected risk of linear regression can manifest multiple descents when the number of features increases and sample size is fixed. This is carried out through an algorithmic construction of a feature-revealing process where the newly revealed feature follows either a Gaussian distribution or a Gaussian mixture distribution. Notably, the construction also enables us to control local maxima in the underparametrized regime and control ascents/descents freely in the overparametrized regime. Overall, this allows us to design the generalization curve away from the interpolation threshold. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We believe that our analysis of linear regression in this paper is a good starting point for explaining non-monotonic generalization curves observed in machine learning studies. Extending these results to more complex problem setups would be a meaningful future direction. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Funding Transparency Statement ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "LC: Funding in direct support of this work: postdoctoral research fellowship by the Simons Institute for the Theory of Computing, University of California, Berkeley, and Google PhD Fellowship by Google. Additional revenues related to this work: internships at Google. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "MB acknowledges support from NSF IIS-1815697, and the support of the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and #814639. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "AK: Funding in direct support of this work: NSF (IIS-1845032) and ONR (N00014-19-1-2406). ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            252,
+            267,
+            268
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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