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sha256:d581a0b63f22e060af7bf8afccadd64a273fd6fbc52434a9902af0a0e4100b1b +size 24765 diff --git a/parse/train/AuVKs6JmBtY/AuVKs6JmBtY.md b/parse/train/AuVKs6JmBtY/AuVKs6JmBtY.md new file mode 100644 index 0000000000000000000000000000000000000000..a9d78e7d2139c38ad3704d1d789614494aaef11e --- /dev/null +++ b/parse/train/AuVKs6JmBtY/AuVKs6JmBtY.md @@ -0,0 +1,240 @@ +# Towards Robust and Reliable Algorithmic Recourse + +Sohini Upadhyay∗ Harvard University supadhyay@g.harvard.edu + +Shalmali Joshi∗ Harvard University shalmali@seas.harvard.edu + +Himabindu Lakkaraju Harvard University hlakkaraju@hbs.harvard.edu + +# Abstract + +As predictive models are increasingly being deployed in high-stakes decision making (e.g., loan approvals), there has been growing interest in post-hoc techniques which provide recourse to affected individuals. These techniques generate recourses under the assumption that the underlying predictive model does not change. However, in practice, models are often regularly updated for a variety of reasons (e.g., dataset shifts), thereby rendering previously prescribed recourses ineffective. To address this problem, we propose a novel framework, RObust Algorithmic Recourse (ROAR), that leverages adversarial training for finding recourses that are robust to model shifts. To the best of our knowledge, this work proposes the first ever solution to this critical problem. We also carry out theoretical analysis which underscores the importance of constructing recourses that are robust to model shifts: 1) We quantify the probability of invalidation for recourses generated without accounting for model shifts. 2) We prove that the additional cost incurred due to the robust recourses output by our framework is bounded. Experimental evaluation on multiple synthetic and real-world datasets demonstrates the efficacy of the proposed framework. + +# 1 Introduction + +Over the past decade, machine learning (ML) models are increasingly being deployed to make a variety of highly consequential decisions ranging from bail and hiring decisions to loan approvals. Consequently, there is growing emphasis on designing tools and techniques which can provide recourse to individuals who have been adversely impacted by predicted outcomes [30]. For example, when an individual is denied a loan by a predictive model deployed by a bank, they should be provided with reasons for this decision, and also informed about what can be done to reverse it. When providing a recourse to an affected individual, it is absolutely critical to ensure that the corresponding decision making entity (e.g., bank) is able to honor that recourse and approve any re-application that fully implements the recommendations outlined in the prescribed recourse Wachter et al. [31]. + +Several approaches in recent literature tackled the problem of providing recourses by generating local (instance level) counterfactual explanations 2 [31, 26, 12, 21, 18]. For instance, Wachter et al. [31] proposed a gradient based approach which finds the closest modification (counterfactual) that can result in the desired prediction. Ustun et al. [26] proposed an efficient integer programming based approach to obtain actionable recourses in the context of linear classifiers. There has also been some recent research that sheds light on the spuriousness of the recourses generated by counterfactual/contrastive explanation techniques [31, 26] and advocates for causal approaches [3, 14, 15]. + +All the aforementioned approaches generate recourses under the assumption that the underlying predictive models do not change. This assumption, however, may not hold in practice. Real world settings are typically rife with different kinds of distribution shifts (e.g, temporal shifts) [22]. In order to ensure that the deployed models are accurate despite such shifts, these models are periodically retrained and updated. Such model updates, however, pose severe challenges to the validity of recourses because previously prescribed recourses (generated by existing algorithms) may no longer be valid once the model is updated. Recent work by Rawal et al. [24] has, in fact, demonstrated empirically that recourses generated by state-of-the-algorithms are readily invalidated in the face of model shifts resulting from different kinds of dataset shifts (e.g., temporal, geospatial, and data correction shifts). Their work underscores the importance of generating recourses that are robust to changes in models i.e., model shifts, particularly those resulting from dataset shifts. However, none of the existing approaches address this problem. + +In this work, we propose a novel algorithmic framework, RObust Algorithmic Recourse (ROAR) for generating instance level recourses (counterfactual explanations) that are robust to changes in the underlying predictive model. To the best of our knowledge, this work makes the first attempt at generating recourses that are robust to model shifts. To this end, we propose a novel minimax objective that can be used to construct robust actionable recourses while minimizing the recourse costs. Second, we propose a set of model shifts that captures our intuition about the kinds of changes in the models to which recourses should be robust. Next, we outline an algorithm inspired by adversarial training to optimize the proposed objective. We also carry out theoretical analysis to establish the following results: i) we quantify the probability of invalidation for recourses generated without accounting for model shifts, and ii) we derive an upper bound on the relative increase in the costs incurred due to robust recourses (proposed by our framework) to the costs incurred by recourses generated from existing algorithms. Our theoretical results further establish the need for approaches like ours that generate actionable recourses that are robust to model shifts. + +We evaluated our approach ROAR on real world data from financial lending and education domains, focusing on model shifts induced by the following kinds of distribution shifts – data correction shift, temporal shift, and geospatial shift. We also experimented with synthetic data to analyze how the degree of data distribution shifts and consequent model shifts affect the robustness and validity of the recourses output by our framework as well as the baselines. Our results demonstrate that the recourses constructed using our framework, ROAR, are substantially more robust $( 6 7 - 1 0 0 \% )$ to changes in the underlying predictive models compared to those generated using state-of-the-art recourse finding technqiues. We also find that our framework achieves such a high degree of robustness without sacrificing the validity of the recourses w.r.t. the original predictive model or substantially increasing the costs associated with realizing the recourses. + +# 2 Related Work + +Our work lies at the intersection of algorithmic recourse and adversarial robustness. Below, we discuss related work pertaining to each of these topics. + +Algorithmic recourse As discussed in Section 1, several approaches have been proposed to construct algorithmic recourse for predictive models [31, 26, 12, 21, 18, 3, 14, 15, 7]. These approaches can be broadly characterized along the following dimensions [29]: the level of access they require to the underlying predictive model (black box vs. gradients), if and how they enforce sparsity (only a small number of features should be changed) in counterfactuals, if counterfactuals are required to lie on the data manifold or not, if underlying causal relationships should be accounted for when generating counterfactuals or not, whether the output should be multiple diverse counterfactuals or just a single counterfactual. While the aforementioned approaches have focused on generating instance level counterfactuals, there has also been some recent work on generating global summaries of model recourses which can be leveraged to audit ML methods [23]. More recently, Rawal et al. [24] demonstrated that recourses generated by state-of-the-art algorithms are readily invalidated due to model shifts resulting from different kinds of dataset shifts. They argued that model updation is very common place in the real world, and it is important to ensure that recourses provided to affected individuals are robust to such updates. Similar arguments have been echoed in several other recent works [28, 13, 20]. While there has been some recent work that explores the construction of other kinds of explanations (feature attribution and rule based explanations) that are robust to dataset shifts [16], our work makes the first attempt at tackling the problem of constructing recourses that are robust to model shifts. + +Adversarial Robustness The techniques that we leverage in this work are inspired by the adversarial robustness literature. Wachter et al. were the first to remark on similarities between counterfactual generation and adversarial attacks, but did not leverage this connection to develop robust recourse [31]. It is now well established that ML models are vulnerable to adversarial attacks [10, 4, 2]. The adversarial training procedure was recently proposed as a defense against such attacks [19, 1, 32]. This procedure optimizes a minimax objective that captures the worst-case loss over a given set of perturbations to the input data. At a high level, it is based on gradient descent; at each gradient step, it solves an optimization problem to find the worst-case perturbation, and then computes the gradient at this perturbation. In contrast, our training procedure optimizes a minimax objective that captures the worst-case over a given set of model perturbations (thereby simulating model shift) and generates recourses that are valid under the corresponding model shifts. This training procedure is novel and possibly of independent interest. + +# 3 Our Framework: RObust Algorithmic Recourse + +In this section, we detail our framework, RObust Algorithmic Recourse (ROAR). First, we introduce some notation and discuss preliminary details about the algorithmic recourse problem setting. We then introduce our objective function, and discuss how to operationalize and optimize it efficiently. + +# 3.1 Preliminaries + +Let us assume we are given a predictive model $\mathcal { M } : \mathcal { X } \xrightarrow { } \mathcal { Y }$ , where $\mathcal { X } \subseteq \mathbb { R } ^ { d }$ is the feature space, and $\mathcal { V }$ is the space of outcomes. Let $\mathcal { V } = \{ 0 , 1 \}$ where 0 and 1 denote an unfavorable outcome (e.g., loan denied) and a favorable outcome (e.g., loan approved) respectively. Let $x \in \mathcal { X }$ be an instance which received a negative outcome i.e., $\mathcal { M } ( x ) = 0$ . The goal here is to find a recourse for this instance $x$ i.e., to determine a set of changes $\epsilon$ that can be made to $x$ in order to reverse the negative outcome. The problem of finding a recourse for $x$ involves finding a counterfactual $x ^ { \prime } = x + \epsilon$ for which the black box outputs a positive outcome i.e., $\mathcal { M } ( x ^ { \prime } ) = \mathcal { M } \bar { ( } x + \epsilon ) = 1$ . + +There are, however, a few important considerations when finding the counterfactual $x ^ { \prime } = x + \epsilon$ . First, it is desirable to minimize the cost (or effort) required to change $x$ to $x ^ { \prime }$ . To formalize this, let us consider a cost function $c : \mathcal { X } \times \mathcal { X } \to \mathbb { R } _ { + }$ . $c ( \boldsymbol { x } , \boldsymbol { x } ^ { \prime } )$ denotes the cost (or effort) incurred in changing an instance $x$ to $x ^ { \prime }$ . In practice, some of the commonly used cost functions are $\ell _ { 1 }$ or $\ell _ { 2 }$ distance [31], log-percentile shift [26], and costs learned from pairwise feature comparisons input by end users [23]. Furthermore, since recommendations to change features such as gender or race would be unactionable, it is important to restrict the search for counterfactuals in such a way that only actionable changes are allowed. Let $\mathcal { A }$ denote the set of plausible or actionable counterfactuals. + +Putting it all together, the problem of finding a recourse for instance $x$ for which $\mathcal { M } ( x ) = 0$ can be formalized as: + +$$ +x ^ { \prime } = \underset { x ^ { \prime } \in \mathcal { A } } { \arg \operatorname* { m i n } } c ( x , x ^ { \prime } ) \quad \mathrm { s . t } \quad \mathcal { M } ( x ^ { \prime } ) = 1 +$$ + +Eqn. 1 captures the generic formulation leveraged by several of the state-of-the-art recourse finding algorithms. Typically, most approaches optimize the unconstrained and differentiable relaxation of Eqn. 1 which is given below: + +$$ +\boldsymbol { x } ^ { \prime } = \underset { \boldsymbol { x } ^ { \prime } \in \mathcal { A } } { \arg \operatorname* { m i n } } \ell ( \boldsymbol { \mathcal { M } } ( \boldsymbol { x } ^ { \prime } ) , 1 ) + \lambda \boldsymbol { c } ( \boldsymbol { x } , \boldsymbol { x } ^ { \prime } ) +$$ + +where $\ell : \mathcal { V } \times \mathcal { V } \to \mathbb { R } _ { + }$ denotes a differentiable loss function (e.g., binary cross entropy) which ensures that gap between $\mathcal { M } ( \boldsymbol { x } ^ { \prime } )$ and favorable outcome 1 is minimized, and $\lambda > 0$ is a trade-off parameter. + +# 3.2 Formulating Our Objective + +As can be seen from Eqn. 2, state-of-the-art recourse finding algorithms rely heavily on the assumption that the underlying predictive model $\mathcal { M }$ does not change. However, predictive models deployed in the real world often get updated. This implies that individuals who have acted upon previously prescribed recourses are no longer guaranteed a favorable outcome once the model is updated. To address this critical challenge, we propose a novel minimax objective function which generates counterfactuals that minimize the worst-case loss over plausible model shifts. We arrived at this approach after considering the following alternatives: (a) Update the predictive model as desired but ensure that individuals who were previously prescribed recourse will still be guaranteed a favorable outcome. (b) + +Update the predictive model while including constraints to ensure that previously offered recourses are still valid. Note that both of these scenarios would potentially incur huge monetary losses to relevant stakeholders (e.g, banks), hurting the adoption of these approaches. In case (a), banks may be required to guarantee credit to customers that are potentially not creditworthy under the new model and thereby risk losing money. In case (b), access to model training is assumed. Furthermore, training a predictive model under these constraints may be suboptimal and not reflective of the current data distribution, thereby accruing larger errors under the shifted population. There are no incentives for stakeholders such as banks to adopt such practices which could potentially lead to huge monetary losses. While the optimal approach may vary on a case by case basis, we propose our method to avoid the aforementioned pitfalls outlined in cases (a) and (b). + +To formalize our proposed approach, let $\Delta$ denote the set of plausible model shifts and let $\mathcal { M } _ { \delta }$ denote a shifted model where $\delta \in \Delta$ . Our objective function for generating robust recourse $x ^ { \prime \prime }$ for a given instance $x$ can be written as: + +$$ +\boldsymbol { x } ^ { \prime \prime } = \underset { \boldsymbol { x } ^ { \prime \prime } \in \mathcal { A } } { \arg \operatorname* { m i n } } \ \underset { \delta \in \Delta } { \operatorname* { m a x } } \ell ( \boldsymbol { \mathcal { M } } _ { \delta } ( \boldsymbol { x } ^ { \prime \prime } ) , 1 ) + \lambda c ( \boldsymbol { x } , \boldsymbol { x } ^ { \prime \prime } ) +$$ + +where cost function $c$ and loss function $l$ are as defined in Section 3.1. + +Choice of $\Delta$ Predictive models deployed in the real world are often updated regularly to handle data distribution shifts [22]. Since these models are updated regularly, it is likely that they undergo small (and not drastic) shifts each time they are updated. To capture this intuition, we consider the following two choices for the set of plausible model shifts $\Delta$ : + +$$ +\begin{array} { r l } & { \Delta = \{ \delta \in \mathbb { R } ^ { n } \mid \delta _ { m i n } \leq \delta _ { i } \leq \delta _ { m a x } \forall i \in \{ 1 \cdots n \} \} . } \\ & { \Delta = \{ \delta \in \mathbb { R } ^ { n } \mid \| \delta \| _ { p } \leq \delta _ { m a x } \} } \end{array} +$$ + +where $p \geq 1$ . Note that perturbations $\delta \in \Delta$ can be considered as operations either on the parameter space or on the gradient space of $\mathcal { M }$ . While the first choice of $\Delta$ presented above allows us to restrict model shifts within a small range, the second choice allows us to restrict model shifts within a norm-ball. Alternate formulations of the first include incorporating domain knowledge to set $\delta _ { m i n }$ and $\delta _ { m a x }$ per feature. These kinds of shifts can effectively capture small changes to both parameters (e.g., weights of linear models) as well as gradients. Next, we describe how to optimize the objective in Eqn. 3 and construct robust recourses. + +# 3.3 Optimizing Our Objective + +While our objective function, the choice of $\Delta$ , and the perturbations $\delta \in \Delta$ we introduce in Section 3.2 are generic enough to handle shifts to both parameter space as well as the gradient space of any class of predictive models $\mathcal { M }$ , we solve our objective for a linear approximation $f$ of $\mathcal { M }$ . The procedure that we outline here remains generalizable even for non-linear models because local behavior of a given non-linear model can be approximated well by fitting a local linear model [25]. Note that such approximations have already been explored by existing algorithmic recourse methods [26, 23]. Let the linear approximation, which we denote by $f$ be parameterized by $w \in \mathcal { W }$ . We make this parametrization explicit by using a subscript notation: $f _ { w }$ . We consider model shifts represented by perturbations to the model parameters $w \in \mathcal { W }$ . In the case of linear models, these can be operationalized as additive perturbations $\delta \in \Delta$ to $w$ . We will represent the resulting shifted classifier by $f _ { w + \delta }$ . Our objective function (Eqn. 3) can now be written in terms of this linear approximation $f$ as: + +$$ +x ^ { \prime \prime } = \underset { x ^ { \prime \prime } \in \mathcal { A } } { \arg \operatorname* { m i n } } \underset { \delta \in \Delta } { \operatorname* { m a x } } \ell ( f _ { w + \delta } ( x ^ { \prime \prime } ) , 1 ) + \lambda c ( x , x ^ { \prime \prime } ) +$$ + +Notice that the objective function defined in Equation 4 is similar to that of adversarial training [19]. However, in our framework, the perturbations are applied to model parameters as opposed to data samples. These parallels help motivate the optimization procedure for constructing recourses that are robust to model shifts. We outline the optimization procedure that we leverage to optimize our minimax objective (Eqn. 4) in Algorithm 1. + +Algorithm 1 proceeds in an iterative manner where we first find a perturbation $\hat { \delta } \in \Delta$ that maximizes the chance of invalidating the current estimate of the recourse $x ^ { \prime \prime }$ , and then we take appropriate gradient steps on $x ^ { \prime \prime }$ to generate a valid recourse. This procedure is executed iteratively until the objective function value (Eqn. 4) converges. + +# Algorithm 1 Our Optimization Procedure + +
Input:x s.t. fω(x)=O,fw,λ>O,△,learning rate α >0. Initialize x" =x,g =0
repeat = arg maxs∈△ l(fw+8(x"),1)
g =∀[e(fw+8(x"),1)+ λc(x",x)]
x" -=ag
until convergence
Return x"
+ +# 4 Theoretical Analysis + +Here we carry out theoretical analysis to shed light on the benefits of our framework ROAR. More specifically: 1) We quantify the probability that recourses generated without accounting for model shifts are likely to be invalidated. 2) We prove that the additional cost incurred due to the robust recourses output by our framework is bounded. + +We first characterize how recourses that do not account for model shifts (i.e., recourses output by state-of-the-art algorithms) fare when true model shifts can be characterized as additive shifts to model parameters. Specifically, we quantify the likelihood that recourses generated without accounting for model shifts will be invalidated (even if they lie on the original data manifold), under certain conditions. + +Theorem 1. For a given instance $x \sim { \mathcal { N } } ( \mu , \Sigma )$ , let $x ^ { \prime }$ be the recourse that lies on the original data manifold (conditioned on the event that $\mathcal { M } ( x ^ { \prime } ) > 0 . 5 )$ and is obtained without accounting for model shifts. Let $\Sigma = U D U ^ { T }$ . Then, for some true model shift $\delta$ , such that, $\begin{array} { r } { \frac { w ^ { T } \mu } { ( w + \delta ) ^ { T } \mu } \geq \frac { \| \sqrt { D } U w \| } { \| \sqrt { D } U ( w + \delta ) \| } } \end{array}$ , and $\begin{array} { r } { \sqrt { \frac { 2 e } { \pi } } \frac { \sqrt { \beta - 1 } } { \beta } \exp ^ { - \beta \frac { ( w ^ { T } \mu ) ^ { 2 } } { 4 \| \sqrt { D } U w \| ^ { 2 } } } \geq e r f c \bigl ( - \frac { ( w + \delta ) ^ { T } \mu } { \sqrt { 2 } \| w + \delta \| } \bigr ) . } \end{array}$ , for $\beta \geq 1$ , the probability that $x ^ { \prime }$ is invalidated on $f _ { w + \delta }$ is at least: $\begin{array} { r } { \frac { 1 } { 2 } \sqrt { \frac { 2 e } { \pi } } \frac { \sqrt { \beta - 1 } } { \beta } \exp ^ { - \beta \frac { ( w ^ { T } \mu ) ^ { 2 } } { 4 \| \sqrt { D } U w \| ^ { 2 } } } - \frac { 1 } { 2 } e r f c \big ( { - \frac { ( w + \delta ) ^ { T } \mu } { \sqrt { 2 } \| w + \delta \| } } \big ) } \end{array}$ where erfc is the complementary gaussian error function. + +Proof Sketch. Under the assumption that $x ^ { \prime } \sim { \mathcal { N } } ( { \boldsymbol { \mu } } , { \boldsymbol { \Sigma } } )$ , a recourse is invalid under a model shift if it is valid under the original model and invalid under the shifted model. This allows us to define the region where $x ^ { \prime }$ can be invalidated: + +$$ +\Omega = \{ x ^ { \prime } \colon w ^ { T } x ^ { \prime } > 0 \cap ( w + \delta ) ^ { T } x ^ { \prime } \leq 0 \} +$$ + +The probability that $x ^ { \prime }$ is invalidated can be obtained by integrating over $\Omega$ under the PDF of $\mathcal { N } ( \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ . + +We can then transform $x ^ { \prime }$ and correspondingly $\Omega$ , to simplify this integration over a 1-dimensional Gaussian random variable. That is, + +$$ +P ( x { \mathrm { ~ i s ~ i n v a l i d a t e d } } ) = { \frac { 1 } { \sqrt { ( 2 \pi ) } } } \int _ { c _ { 1 } } ^ { c _ { 2 } } \exp { \bigg ( } - { \frac { 1 } { 2 } } s ^ { 2 } { \bigg ) } d s +$$ + +The above quantity can be represented as a difference in the Gaussian error function allowing us to exactly quantify the invalidation probability under our assumptions. Using the lower bounds on the complementary gaussian error function [9] from Chang et al. [5], we obtain our lower bound. To derive the lower bound, we add an extra condition that $\begin{array} { r } { \sqrt { \frac { 2 e } { \pi } } \frac { \sqrt { \beta - 1 } } { \beta } \exp ^ { - \beta \frac { ( w ^ { T } \mu ) ^ { 2 } } { 4 \| \sqrt { D } U w \| ^ { 2 } } } \geq \mathrm { e r f c } ( - \frac { ( w + \delta ) ^ { T } \mu } { \sqrt { 2 } \| w + \delta \| } ) } \end{array}$ , mainly to confirm that the lower bound on the first term still dominates the second term. Both conditions restricts the types of shift for which the bound can be derived. Note that $\beta$ can be optimized away to improve the lower bound. Detailed proof is provided in the Appendix. Discussion about other distributions (e.g., Bernoulli, Uniform, Categorical) is included in the Appendix. □ + +Next we characterize how much more costly recourses can be when they are trained to be robust to model perturbations or model shifts. In the following theorem, we show that the cost of robust recourses is bounded relative to the cost of recourses that do not account for model shifts. + +Theorem 2. Let $x \in \mathcal { X }$ , and $x \sim \nu$ where $\nu$ is a distribution such that $\mathbb { E } _ { \nu } [ x ] = \mu < \infty ,$ , where $\mathcal { X }$ is a metric space $( \mathcal { X } , d ( \cdot , \cdot ) )$ and $d : \mathcal { X } \times \mathcal { X } \to \mathbb { R } _ { + }$ . Let $d \triangleq \ell _ { 2 }$ and assume that $( \mathcal { X } , d )$ has bounded diameter $D = \operatorname* { s u p } _ { x , x ^ { \prime } \in \mathcal { X } } d ( x , x ^ { \prime } )$ . Let recourses obtained without accounting for model shifts and constrained to the manifold be denoted by $x ^ { \prime } \sim \nu$ , and robust recourses be denoted by $x ^ { \prime \prime }$ . Let $\delta > 0$ be the shift that maximizes Eq. 3 for sample $x$ corresponding to $x ^ { \prime \prime }$ . Further assume that the ROAR objective (Equation 3) is convex in $x ^ { \prime \prime }$ for a fixed $\delta$ . For $\ell \triangleq \ell _ { l o g }$ (the cross-entropy loss), some $0 < \eta ^ { \prime } \ll 1$ , w.h.p. $( 1 - \eta ^ { \prime } )$ , we have that: + +$$ +c ( x ^ { \prime \prime } , x ) - c ( x ^ { \prime } , x ) \leq \frac { 1 } { \lambda } \frac { 1 } { \| w + \delta \| } \mathbb { E } _ { \nu } [ \exp { - \phi ( w + \delta ) ^ { T } x ^ { \prime } } ] + \sqrt { \frac { D ^ { 2 } } { 2 } \log { ( \frac { 1 } { \eta ^ { \prime } } ) } } \Bigg ) +$$ + +Proof Sketch. By definition, any recourse $x ^ { \prime }$ generated without accounting for model shifts will have a higher loss for Equation 3 compared to the robust recourse $x ^ { \prime \prime }$ (note that finding the global minimizer is not guaranteed by Algorithm 1). + +Using this insight, and convexity in $x ^ { \prime }$ for fixed $\delta$ , we can bound the cost difference between the robust and non-robust recourse by a 1-Lipschitz function (i.e. the logistic function): + +$$ +c ( x ^ { \prime \prime } , x ) - c ( x ^ { \prime } , x ) \leq \frac { 1 } { \lambda \| w + \delta \| } \log \left\{ 1 + \exp - ( w + \delta ) ^ { T } x ^ { \prime } \right\} +$$ + +Assuming a bounded metric on $\mathcal { X }$ , we can upper bound the RHS using Lemma 2 from van Handel [27] which gives us our bound. Detailed proof including special cases when $\nu$ is Gaussian, is provided in the Appendix. □ + +This result suggests that the additional cost of recourse is bounded by the amount of shift admissible in Equation 3. Note that Theorem 2 applies for general distributions so long as the mean is finite, which is the case for most commonplace distributions like Gaussian, Bernoulli, Multinomial etc. While Theorem 1 demonstrates the probability that a recourse will be invalidated for Gaussian distributions, we refer the reader to the Appendix B.1 for a discussion of other distributions, e.g. Bernoulli, Uniform, Categorical. + +# 5 Experiments + +Here we discuss the detailed experimental evaluation of our framework, ROAR. First, we evaluate how robust the recourses generated by our framework are to model shifts caused by real world data distribution shifts. We also assess the validity of the recourses generated by our framework w.r.t. the original model, and further analyze the average cost of these recourses. Next, using synthetic data, we analyze how varying the degree (magnitude) of data distribution shift impacts the robustness and validity of the recourses output by our framework and other baselines. + +# 5.1 Experimental Setup + +Real world data We evaluate our framework on model shifts induced by real world data distribution shifts. To this end, we leverage three real world datasets which capture different kinds of data distribution shifts, namely, temporal shift, geospatial shift, and data correction shift [24]. Our first dataset is the widely used and publicly available German credit dataset [8] from the UCI repository. This dataset captures demographic (age, gender), personal (marital status), and financial (income, credit duration) details of about 1000 loan applicants. Each applicant is labeled as either a good customer or a bad customer depending on their credit risk. Two versions of this dataset have been released, with the second version incorporating corrections to coding errors in the first dataset [11]. Accordingly, this dataset captures the data correction shift. Our second dataset is the Small Business Administration (SBA) case dataset [17]. This dataset contains information pertaining to 2102 small business loans approved by the state of California during the years of $1 9 8 9 - 2 0 1 2$ , and captures temporal shifts in the data. It comprises of about 24 features capturing various details of the small businesses including zip codes, business category (real estate vs. rental vs. leasing), number of jobs created, and financial status of the business. It also contains information about whether a business has defaulted on a loan or not which we consider as the class label. Our last dataset contains student performance records of 649 students from two Portuguese secondary schools, Gabriel Pereira (GP) and Mousinho da Silveira (MS) [8, 6], and captures geospatial shift. It comprises of information about the academic background (grades, absences, access to internet, failures etc.) of each student along with other demographic attributes (age, gender). Each student is assigned a class label of above average or not depending on their final grade. + +Synthetic data We generate a synthetic dataset with 1K samples and two dimensions to analyze how the degree (magnitude) of data distribution shifts impacts the robustness and validity of the recourses output by our framework and other baselines. Each instance $x$ is generated as follows: First, we randomly sample the class label $y \in \{ 0 , 1 \}$ corresponding to the instance $x$ . Conditioned upon the value of $y$ , we then sample the instance $x$ as: $x \sim \mathrm { \bar { \mathcal { N } } } ( \mu _ { y } , \mathrm { \bar { \Sigma } } _ { y } )$ . We choose $\mu _ { 0 } = [ - 2 , - 2 ] ^ { T }$ and $\mu _ { 1 } = [ + 2 , + 2 ] ^ { T }$ , and $\Sigma _ { 0 } = \Sigma _ { 1 } = 0 . 5 \mathbf { I }$ where $\mu _ { 0 }$ , $\Sigma _ { 0 }$ and $\mu _ { 1 }$ , $\Sigma _ { 1 }$ denote the means and covariance of the Gaussian distributions from which instances in class 0 and class 1 are sampled respectively. A scatter plot of the samples resulting from this generative process and the decision boundary of a logistic regression model fit to this data are shown in Figure 1a. In our experimental evaluation, we consider different kinds of shifts to this synthetic data: + +![](images/98bb3ff9b091767a7781023e340217b6fd8251f08031fdf85213a639fc47f523.jpg) +Figure 1: Synthetic data and examples of model shift. From left to right we have (a) original synthetic dataset, (b) shifted data and decision boundary after mean shift, (c) shifted data and decision boundary after variance shift, and (d) shifted data and decision boundary after mean and variance shift + +(i) Mean shift: To generated shifted data, we leverage the same approach as above but shift the mean of the Gaussian distribution associated with class 0 i.e., $x \sim \mathcal { N } ( \mu _ { y } ^ { \prime } , \Sigma _ { y } )$ where $\mu _ { 0 } ^ { \prime } = \mu _ { 0 } + [ \alpha , 0 ] ^ { T }$ and $\mu _ { 1 } ^ { \prime } = \mu _ { 1 }$ . Note that we only shift the mean of one of the features of class 0 so that the slope of the decision boundary of a linear model we fit to this shifted data changes (relative to the linear model fit on the original data), while the intercept remains the same. Figure 1b shows shifted data with $\alpha = 1 . 5$ . + +(ii) Variance shift: Here, we leverage the same generative process as above, but instead of shifting the mean, we shift the variance of the Gaussian distribution associated with class 0 i.e., i.e., $x \sim$ $\mathcal { N } ( \mu _ { y } , \Sigma _ { y } ^ { \prime } )$ where $\Sigma _ { 0 } ^ { \prime } = ( 1 + \beta ) \Sigma _ { 0 }$ and $\Sigma _ { 1 } ^ { \prime } = \Sigma _ { 1 } ^ { \prime }$ for some increment $\beta \in \mathbb { R }$ . The net result here is that the intercept of the decision boundary of a linear model we fit to this shifted data changes (relative to the linear model fit on the original data), while the slope remains unchanged. Figure 1c shows shifted data with $\beta = 3$ . + +(ii) Mean and variance shift: Here, we change both the mean and variance of the Gaussian distribution associated with class 0 simultaneously (Figure 1d). It can be seen that there are noticeable changes to both the slope and intercept of the decision boundary compared to Figure 1a. + +Predictive models We generate recourses for a variety of linear and non-linear models: deep neural networks (DNNs), SVMs, and logistic regression (LR). Here, we present results for a 3-layer DNN and LR; remaining results are included in the Appendix. Results presented here are representative of those for other model families. + +Baselines We compare our framework, ROAR, to the following state-of-the-art baselines: (i) counterfactual explanations (CFE) framework outlined by Wachter et al. [31], (ii) actionable recourse (AR) in linear classification [26], and (iii) causal recourse framework (MINT) proposed by Karimi et al. [14]. While CFE leverages gradient computations to find counterfactuals, AR employs a mixed integer programming based approach to find counterfactuals that are actionable. The MINT framework operates on top of existing approaches for finding nearby counterfactuals. We use the MINT framework on top of CFE and ROAR and refer to these two approaches as MINT and ROARMINT respectively. As the MINT framework requires access to the underlying causal graph, we experiment with MINT and ROAR-MINT only on the German credit dataset for which such a causal graph is available. + +Cost functions Our framework, ROAR, and all the other baselines we use rely on a cost function $c$ that measures the cost (or effort) required to act upon the prescribed recourse. Furthermore, our approach as well as several other baselines require the cost function to be differentiable. So, we consider two cost functions in our experimentation: $\ell _ { 1 }$ distance between the original instance and the counterfactual, and a cost function learned from pairwise feature comparison inputs (PFC) [13, 26, 23]. PFC uses the Bradley-Terry model to map pairwise feature comparison inputs provided by end users to the cost required to act upon the prescribed recourse for any given instance $x$ . For more details on this cost function, please refer to Rawal and Lakkaraju [23]. In our experiments, we follow the same procedure as Rawal and Lakkaraju [23] and simulate the pairwise feature comparison inputs. + +Setting and implementation details We partition each of our synthetic and real world datasets into two parts: initial data $( D _ { 1 } )$ and shifted data $( D _ { 2 } )$ . In the case of real world datasets, $D _ { 1 }$ and $D _ { 2 }$ can be logically inferred from the data itself – e.g., in case of the German credit dataset, we consider the initial version of the dataset as $D _ { 1 }$ and the corrected version of the dataset as $D _ { 2 }$ . In the case of synthetic datasets, we generate $D _ { 1 }$ and $D _ { 2 }$ as described earlier where $D _ { 2 }$ is generated by shifting $D _ { 1 }$ (See "Synthetic data" in Section 5.1). + +We use 5-fold cross validation throughout our real world and synthetic experiments. On $D _ { 1 }$ , we use 4 folds to train predictive models and the remaining fold to generate and evaluate recourses. We repeat this process 5 times and report averaged values of our evaluation metrics. We leverage $D _ { 2 }$ only to train the shifted models $\mathcal { M } _ { 2 }$ . More details about the data splits, model training, and performance of the predictive models are included in the Appendix. + +We use binary cross entropy loss and the Adam optimizer to operationalize our framework, ROAR. Our framework, ROAR, has the following parameters: the set of acceptable perturbations $\Delta$ (defined in practice by $\delta _ { m a x . }$ ) and the tradeoff parameter $\lambda$ . In our experiments on evaluating robustness to real world shifts, we choose $\delta _ { m a x } = 0 . 1$ given that continuous features are scaled to zero mean and unit variance. Furthermore, in each setting, we choose the $\lambda$ that maximizes the recourse validity of $\mathcal { M } _ { 1 }$ (more details in Section 5.1 "Metrics" and Appendix). In case of our synthetic experiments where we assess the impact of the degree (magnitude) of data distribution shift, features are not normalized, so we do a grid search for both $\delta _ { m a x }$ and $\lambda$ . First, we choose the largest $\delta _ { m a x }$ that maximizes the recourse validity of $\mathcal { M } _ { 1 }$ and then set $\lambda$ in a similar fashion (more details in Appendix). We set the parameters of the baselines using techniques discussed in the original works [31, 14, 26] and employ a similar grid search approach if unspecified. + +Following the precedents set forth in [26] and [23], we adapt AR and ROAR to non-linear models by first generating local linear approximations of these models using LIME [25]. We refer to these variants as AR-LIME and ROAR-LIME respectively. + +Metrics. We consider two metrics in our evaluation: 1) Avg Cost is defined as the average cost incurred to act upon the prescribed recourses where the average is computed over all the instances for which a given algorithm provides recourse. Recall that we consider two notions of cost in our experiments – $\ell _ { 1 }$ distance between the original instance and the counterfactual, costs learned from pairwise feature comparisons (PFC) (See "Cost Functions" in Section 5.1). 2) Validity is defined as the fraction of instances for which acting upon the prescribed recourse results in the desired prediction. Note that validity is computed w.r.t. a given model. + +# 5.2 Robustness to real world shifts + +Here, we evaluate the robustness of the recourses output by our framework, ROAR, as well as the baselines. A recourse finding algorithm can be considered robust if the recourses output by the algorithm remain valid even if the underlying model has changed. To evaluate this, we first leverage our approach and other baselines to find recourses of instances in our test sets w.r.t. the initial model $\mathcal { M } _ { 1 }$ . We then compute the validity of these recourses w.r.t. the shifted model $\mathcal { M } _ { 2 }$ which has been trained on the shifted data. Let us refer to this as $\mathcal { M } _ { 2 }$ validity. The higher the value of $\mathcal { M } _ { 2 }$ validity, the more robust the recourse finding method. Table 1 shows the $\mathcal { M } _ { 2 }$ validity metric computed for different algorithms across different real world datasets. + +It can be seen that recourse methods that use our framework, ROAR and ROAR-MINT, achieve the highest $\mathcal { M } _ { 2 }$ validity across all datasets. In fact, methods that use our framework do almost twice as + +
Correction ShiftTemporal ShiftGeospatial Shift
Model CostRecourse CFEAvgCost 1.02 ± 0.18MValidity 1.00±0.00MValidity 0.54± 0.27AvgCost 3.57 ± 1.14MValidity 1.00±0.00MValidity 0.31±0.09Avg Cost 8.37±0.73MValidity 0.98±0.03MValidity 0.29±0.09
LRL10.85 ± 0.141.00 ± 0.000.53 ± 0.211.50± 0.281.00 ± 0.000.16 ± 0.065.29 ± 0.281.00 ± 0.000.43 ± 0.14
AR3.14 ± 0.250.99 ± 0.010.98 ±0.0210.88 ± 1.671.00 ± 0.000.67 ± 0.19
ROAR3.13 ± 0.321.00 ± 0.000.94 ± 0.08 0.93 ± 0.07NA
MINT4.73 ± 1.561.00 ± 0.00NANANANANA
ROAR-MINT6.77 ± 0.351.00 ± 0.001.00 ± 0.00NANANANANANA
CFE0.03±0.021.00 ±0.000.56±0.330.24±0.091.00 ± 0.000.26± 0.110.34± 0.041.00±0.000.18 ±0.10
PFCAR0.09 ± 0.021.00 ± 0.000.54 ± 0.270.11 ± 0.021.00 ± 0.000.09 ± 0.050.32 ±0.031.00 ±0.000.24 ± 0.11
ROAR MINT0.36±0.081.00 ± 0.001.00 ± 0.000.44 ± 0.120.99 ± 0.010.98 ± 0.011.20 ± 0.101.00 ± 0.000.91± 0.07
ROAR-MINT1.00 ± 1.151.00 ± 0.00 1.00 ± 0.000.95±0.08 1.00 ± 0.00NANA NANANANANA
L1CFE1.23 ± 0.05 0.55 ±0.101.00± 0.000.47±0.06NA 3.78±0.681.00 ± 0.00NA 0.52±0.09NANANA
AR-LIME0.38 ± 0.150.16 ±0.100.31 ± 0.061.39 ± 0.130.59 ± 0.110.65 ± 0.1710.09± 0.71 9.02 ±1.571.00 ± 0.000.48±0.09 0.83 ±0.10
ROAR-LIME0.76± 0.06
NN1.83 ± 0.190.78 ±0.060.72 ± 0.104.90±0.240.98 ±0.020.97 ±0.0221.05 ± 3.581.00 ± 0.000.97 ±0.03
MINT2.24 ± 1.250.81 ± 0.020.63 ± 0.11NANANANANANA
ROAR-MINT8.59 ± 1.700.90 ±0.030.84 ± 0.04NANANANANANA
CFE0.06±0.021.00± 0.000.51 ± 0.120.19±0.061.00±0.000.50± 0.130.48± 0.061.00±0.000.30±0.14
AR-LIME0.06± 0.030.49 ± 0.110.56± 0.150.11 ± 0.010.54 ± 0.080.62 ± 0.120.78 ± 0.150.84 ± 0.060.82 ± 0.11
PFC ROAR-LIME0.64 ± 0.080.85 ± 0.070.82 ± 0.050.37 ±0.070.99 ± 0.010.99 ±0.01.66 ± 0.211.00 ±0.000.97 ± 0.04
0.60 ± 0.160.82 ±0.070.64 ± 0.15NANANANANA
MINT ROAR-MINT0.60 ±0.070.91± 0.040.81 ± 0.04NANANANANANA NA
+ +Table 1: Avg Cost, $\mathcal { M } _ { 1 }$ (original) validity, and $\mathcal { M } _ { 2 }$ (shifted model) validity of recourses across different real world datasets. Recourses that leverage our framework ROAR are more robust (higher $\mathcal { M } _ { 2 }$ validity) compared to those generated by existing baselines. + +good compared to other baselines on this metric, indicating that ROAR based recourse methods are quite robust. After ROAR, MINT is the next best performing baseline with respect $\mathcal { M } _ { 2 }$ validity. This may be explained by the fact that MINT accounts for the underlying causal graphs when generating recourses. + +We also assess if the robustness achieved by our framework is coming at a cost i.e., by sacrificing validity on the original model or by increasing avg cost. Table 1 shows the results for the same. It can be seen that ROAR based recourses achieve higher than $9 5 \%$ $\mathcal { M } _ { 1 }$ validity in all but two settings. We compute the avg cost of the recourses output by all the algorithms on various datasets and find that ROAR typically has a higher avg cost (both under $\ell _ { 1 }$ and PFC cost functions) compared to CFE and AR baselines. As demonstrated through additional experiments in the Appendix, these relatively higher costs are expected given our Theorem 2 upper bound on ROAR cost. However, overall, MINT and ROAR-MINT seem to exhibit the highest avg costs and are the worst performing algorithms according to this metric. Since non-causal recourse methods assume independent features, and do not have to adhere to the underlying causal structure when finding counterfactuals, they can generate relatively lower cost counterfactuals even if those counterfactuals may not correspond to realistic data instances. This is likely one of the key reasons why we observe higher average costs in the causal recourse methods. + +![](images/79b7233b461f75eae4a6c8dc52c9a20153a7b4b770aa53c7911b559afe6ec4db.jpg) +5.3 Impact of the degree of data distribution shift on recourses +Figure 2: Impact of the degree of data distribution shift on validity of recourse: DNN classifier with $\ell _ { 1 }$ cost function (top row), DNN classifier with PFC cost function (bottom row); Validity of the recourses generated by all methods drops as degree (magnitude) of the shift increases; The drop in the validity is much smaller for our method ROAR-LIME compared to other baselines. + +Here, we assess how different kinds of distribution shifts and the magnitude of these shifts impact the robustness of recourses output by our framework and other baselines. To this end, we leverage our synthetic datasets and introduce mean shifts, variance shifts, and combination shifts (both mean and variance shifts) of different magnitudes by varying $\alpha$ and $\beta$ (See "Synthetic data" in Section 5.1). We then leverage these different kinds of shifted datasets to construct shifted models and then assess the validity of the recourses output by our framework and other baselines w.r.t. the shifted models. + +We generate recourses using our framework and baselines CFE and AR for different predictive models (LR, DNN) and cost functions ( $\ell _ { 1 }$ distance, PFC). Figure 2 captures the results of this experiment for DNN model both with $\ell _ { 1 }$ distance and PFC cost functions. Results with other models are included in the Appendix. It can be seen that the $\mathbf { X }$ -axis of each of these plots captures the magnitude of the dataset shift, and the y-axis captures the validity of the recourses w.r.t. the corresponding shifted model. Standard error bars obtained by averaging the results over 5 runs are also shown. + +It can be seen that as the magnitude of the distribution shift increases, validity of the recourses generated by all the methods starts dropping. This trend prevailed across mean, variance, and combination (mean and variance) shifts. It can also be seen that the rate at which validity of the recourses generated by our method, ROAR-LIME, drops is much smaller compared to that of other baselines CFE and AR-LIME. Furthermore, our method exhibits the highest validity compared to the baselines as the magnitude of the distribution shift increases. CFE seems to be the worst performing baseline and the validity of the recourses generated by CFE drops very sharply even at small magnitudes of distribution shifts. + +# 6 Conclusions & Future Work + +We proposed a novel framework, RObust Algorithmic Recourse (ROAR), to address the critical but under-explored issue of recourse robustness to model updates. To this end, we introduced a novel minimax objective to generate recourses that are robust to model shifts, and leveraged adversarial training to optimize this objective. We also presented novel theoretical results which demonstrate that recourses without accounting for model shifts are likely to be invalidated, underscoring the necessity of ROAR. Furthermore, we also showed that the additional cost incurred by robust recourses generated by ROAR are bounded. Extensive experimentation with real world and synthetic datasets demonstrated that recourses using ROAR are highly robust to model shifts induced by a range of data distribution shifts. Our work also paves the way for further research into techniques for generating robust recourses. For instance, it would be valuable to further analyze the tradeoff between recourse robustness and cost to better understand the impacts to affected individuals. Other interesting future directions include non-linear extensions that leverage novel local linear approximation methods that improve on LIME [33]. + +# Acknowledgements + +We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461 and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, and Google. SJ would like to acknowledge the support of the Center for Research on Computation and Society (CRCS) at the Harvard John A. Paulson School of Engineering and Applied Sciences. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies. + +# References + +[1] Anish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. In International Conference on Machine Learning, pages 274–283. 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A Practical Guide, 1st Ed., Cham: Springer International Publishing, 10:3152676, 2017. +[31] Sandra Wachter, Brent D. Mittelstadt, and Chris Russell. Counterfactual explanations without opening the black box: Automated decisions and the GDPR. CoRR, abs/1711.00399, 2017. URL http://arxiv.org/abs/1711.00399. +[32] Eric Wong and Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In International Conference on Machine Learning, pages 5286–5295. PMLR, 2018. +[33] Xingyu Zhao, Xiaowei Huang, V. Robu, and D. Flynn. Baylime: Bayesian local interpretable model-agnostic explanations. UAI, abs/2012.03058, 2021. \ No newline at end of file diff --git a/parse/train/AuVKs6JmBtY/AuVKs6JmBtY_content_list.json b/parse/train/AuVKs6JmBtY/AuVKs6JmBtY_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..dc724bb0c28265f0d2afd92d4519df83b5500924 --- /dev/null +++ b/parse/train/AuVKs6JmBtY/AuVKs6JmBtY_content_list.json @@ -0,0 +1,1140 @@ +[ + { + "type": "text", + "text": "Towards Robust and Reliable Algorithmic Recourse ", + "text_level": 1, + "bbox": [ + 184, + 122, + 812, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sohini Upadhyay∗ Harvard University supadhyay@g.harvard.edu ", + "bbox": [ + 187, + 200, + 367, + 243 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Shalmali Joshi∗ Harvard University shalmali@seas.harvard.edu ", + "bbox": [ + 390, + 200, + 583, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Himabindu Lakkaraju Harvard University hlakkaraju@hbs.harvard.edu ", + "bbox": [ + 606, + 200, + 808, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 279, + 535, + 295 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As predictive models are increasingly being deployed in high-stakes decision making (e.g., loan approvals), there has been growing interest in post-hoc techniques which provide recourse to affected individuals. These techniques generate recourses under the assumption that the underlying predictive model does not change. However, in practice, models are often regularly updated for a variety of reasons (e.g., dataset shifts), thereby rendering previously prescribed recourses ineffective. To address this problem, we propose a novel framework, RObust Algorithmic Recourse (ROAR), that leverages adversarial training for finding recourses that are robust to model shifts. To the best of our knowledge, this work proposes the first ever solution to this critical problem. We also carry out theoretical analysis which underscores the importance of constructing recourses that are robust to model shifts: 1) We quantify the probability of invalidation for recourses generated without accounting for model shifts. 2) We prove that the additional cost incurred due to the robust recourses output by our framework is bounded. Experimental evaluation on multiple synthetic and real-world datasets demonstrates the efficacy of the proposed framework. ", + "bbox": [ + 233, + 305, + 766, + 523 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 539, + 312, + 555 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Over the past decade, machine learning (ML) models are increasingly being deployed to make a variety of highly consequential decisions ranging from bail and hiring decisions to loan approvals. Consequently, there is growing emphasis on designing tools and techniques which can provide recourse to individuals who have been adversely impacted by predicted outcomes [30]. For example, when an individual is denied a loan by a predictive model deployed by a bank, they should be provided with reasons for this decision, and also informed about what can be done to reverse it. When providing a recourse to an affected individual, it is absolutely critical to ensure that the corresponding decision making entity (e.g., bank) is able to honor that recourse and approve any re-application that fully implements the recommendations outlined in the prescribed recourse Wachter et al. [31]. ", + "bbox": [ + 174, + 563, + 825, + 688 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Several approaches in recent literature tackled the problem of providing recourses by generating local (instance level) counterfactual explanations 2 [31, 26, 12, 21, 18]. For instance, Wachter et al. [31] proposed a gradient based approach which finds the closest modification (counterfactual) that can result in the desired prediction. Ustun et al. [26] proposed an efficient integer programming based approach to obtain actionable recourses in the context of linear classifiers. There has also been some recent research that sheds light on the spuriousness of the recourses generated by counterfactual/contrastive explanation techniques [31, 26] and advocates for causal approaches [3, 14, 15]. ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "All the aforementioned approaches generate recourses under the assumption that the underlying predictive models do not change. This assumption, however, may not hold in practice. Real world settings are typically rife with different kinds of distribution shifts (e.g, temporal shifts) [22]. In order to ensure that the deployed models are accurate despite such shifts, these models are periodically retrained and updated. Such model updates, however, pose severe challenges to the validity of recourses because previously prescribed recourses (generated by existing algorithms) may no longer be valid once the model is updated. Recent work by Rawal et al. [24] has, in fact, demonstrated empirically that recourses generated by state-of-the-algorithms are readily invalidated in the face of model shifts resulting from different kinds of dataset shifts (e.g., temporal, geospatial, and data correction shifts). Their work underscores the importance of generating recourses that are robust to changes in models i.e., model shifts, particularly those resulting from dataset shifts. However, none of the existing approaches address this problem. ", + "bbox": [ + 174, + 797, + 823, + 825 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we propose a novel algorithmic framework, RObust Algorithmic Recourse (ROAR) for generating instance level recourses (counterfactual explanations) that are robust to changes in the underlying predictive model. To the best of our knowledge, this work makes the first attempt at generating recourses that are robust to model shifts. To this end, we propose a novel minimax objective that can be used to construct robust actionable recourses while minimizing the recourse costs. Second, we propose a set of model shifts that captures our intuition about the kinds of changes in the models to which recourses should be robust. Next, we outline an algorithm inspired by adversarial training to optimize the proposed objective. We also carry out theoretical analysis to establish the following results: i) we quantify the probability of invalidation for recourses generated without accounting for model shifts, and ii) we derive an upper bound on the relative increase in the costs incurred due to robust recourses (proposed by our framework) to the costs incurred by recourses generated from existing algorithms. Our theoretical results further establish the need for approaches like ours that generate actionable recourses that are robust to model shifts. ", + "bbox": [ + 174, + 242, + 825, + 421 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluated our approach ROAR on real world data from financial lending and education domains, focusing on model shifts induced by the following kinds of distribution shifts – data correction shift, temporal shift, and geospatial shift. We also experimented with synthetic data to analyze how the degree of data distribution shifts and consequent model shifts affect the robustness and validity of the recourses output by our framework as well as the baselines. Our results demonstrate that the recourses constructed using our framework, ROAR, are substantially more robust $( 6 7 - 1 0 0 \\% )$ to changes in the underlying predictive models compared to those generated using state-of-the-art recourse finding technqiues. We also find that our framework achieves such a high degree of robustness without sacrificing the validity of the recourses w.r.t. the original predictive model or substantially increasing the costs associated with realizing the recourses. ", + "bbox": [ + 174, + 434, + 825, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 592, + 321, + 608 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work lies at the intersection of algorithmic recourse and adversarial robustness. Below, we discuss related work pertaining to each of these topics. ", + "bbox": [ + 176, + 619, + 821, + 647 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithmic recourse As discussed in Section 1, several approaches have been proposed to construct algorithmic recourse for predictive models [31, 26, 12, 21, 18, 3, 14, 15, 7]. These approaches can be broadly characterized along the following dimensions [29]: the level of access they require to the underlying predictive model (black box vs. gradients), if and how they enforce sparsity (only a small number of features should be changed) in counterfactuals, if counterfactuals are required to lie on the data manifold or not, if underlying causal relationships should be accounted for when generating counterfactuals or not, whether the output should be multiple diverse counterfactuals or just a single counterfactual. While the aforementioned approaches have focused on generating instance level counterfactuals, there has also been some recent work on generating global summaries of model recourses which can be leveraged to audit ML methods [23]. More recently, Rawal et al. [24] demonstrated that recourses generated by state-of-the-art algorithms are readily invalidated due to model shifts resulting from different kinds of dataset shifts. They argued that model updation is very common place in the real world, and it is important to ensure that recourses provided to affected individuals are robust to such updates. Similar arguments have been echoed in several other recent works [28, 13, 20]. While there has been some recent work that explores the construction of other kinds of explanations (feature attribution and rule based explanations) that are robust to dataset shifts [16], our work makes the first attempt at tackling the problem of constructing recourses that are robust to model shifts. ", + "bbox": [ + 174, + 661, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Adversarial Robustness The techniques that we leverage in this work are inspired by the adversarial robustness literature. Wachter et al. were the first to remark on similarities between counterfactual generation and adversarial attacks, but did not leverage this connection to develop robust recourse [31]. It is now well established that ML models are vulnerable to adversarial attacks [10, 4, 2]. The adversarial training procedure was recently proposed as a defense against such attacks [19, 1, 32]. This procedure optimizes a minimax objective that captures the worst-case loss over a given set of perturbations to the input data. At a high level, it is based on gradient descent; at each gradient step, it solves an optimization problem to find the worst-case perturbation, and then computes the gradient at this perturbation. In contrast, our training procedure optimizes a minimax objective that captures the worst-case over a given set of model perturbations (thereby simulating model shift) and generates recourses that are valid under the corresponding model shifts. This training procedure is novel and possibly of independent interest. ", + "bbox": [ + 173, + 90, + 826, + 257 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Our Framework: RObust Algorithmic Recourse ", + "text_level": 1, + "bbox": [ + 173, + 270, + 607, + 287 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we detail our framework, RObust Algorithmic Recourse (ROAR). First, we introduce some notation and discuss preliminary details about the algorithmic recourse problem setting. We then introduce our objective function, and discuss how to operationalize and optimize it efficiently. ", + "bbox": [ + 174, + 295, + 825, + 337 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Preliminaries ", + "text_level": 1, + "bbox": [ + 174, + 347, + 307, + 361 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let us assume we are given a predictive model $\\mathcal { M } : \\mathcal { X } \\xrightarrow { } \\mathcal { Y }$ , where $\\mathcal { X } \\subseteq \\mathbb { R } ^ { d }$ is the feature space, and $\\mathcal { V }$ is the space of outcomes. Let $\\mathcal { V } = \\{ 0 , 1 \\}$ where 0 and 1 denote an unfavorable outcome (e.g., loan denied) and a favorable outcome (e.g., loan approved) respectively. Let $x \\in \\mathcal { X }$ be an instance which received a negative outcome i.e., $\\mathcal { M } ( x ) = 0$ . The goal here is to find a recourse for this instance $x$ i.e., to determine a set of changes $\\epsilon$ that can be made to $x$ in order to reverse the negative outcome. The problem of finding a recourse for $x$ involves finding a counterfactual $x ^ { \\prime } = x + \\epsilon$ for which the black box outputs a positive outcome i.e., $\\mathcal { M } ( x ^ { \\prime } ) = \\mathcal { M } \\bar { ( } x + \\epsilon ) = 1$ . ", + "bbox": [ + 173, + 364, + 825, + 463 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There are, however, a few important considerations when finding the counterfactual $x ^ { \\prime } = x + \\epsilon$ . First, it is desirable to minimize the cost (or effort) required to change $x$ to $x ^ { \\prime }$ . To formalize this, let us consider a cost function $c : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }$ . $c ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )$ denotes the cost (or effort) incurred in changing an instance $x$ to $x ^ { \\prime }$ . In practice, some of the commonly used cost functions are $\\ell _ { 1 }$ or $\\ell _ { 2 }$ distance [31], log-percentile shift [26], and costs learned from pairwise feature comparisons input by end users [23]. Furthermore, since recommendations to change features such as gender or race would be unactionable, it is important to restrict the search for counterfactuals in such a way that only actionable changes are allowed. Let $\\mathcal { A }$ denote the set of plausible or actionable counterfactuals. ", + "bbox": [ + 173, + 469, + 826, + 580 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Putting it all together, the problem of finding a recourse for instance $x$ for which $\\mathcal { M } ( x ) = 0$ can be formalized as: ", + "bbox": [ + 174, + 587, + 823, + 613 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1d9289c21d881080b115aaea13075eadddbf6e7a8c5cc2e54cc08ae4750819c6.jpg", + "text": "$$\nx ^ { \\prime } = \\underset { x ^ { \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } c ( x , x ^ { \\prime } ) \\quad \\mathrm { s . t } \\quad \\mathcal { M } ( x ^ { \\prime } ) = 1\n$$", + "text_format": "latex", + "bbox": [ + 364, + 609, + 633, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Eqn. 1 captures the generic formulation leveraged by several of the state-of-the-art recourse finding algorithms. Typically, most approaches optimize the unconstrained and differentiable relaxation of Eqn. 1 which is given below: ", + "bbox": [ + 173, + 645, + 825, + 688 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/acb4c8e7aafc4daaac0921d7954a1dc9584ea329c0078b0299a009725d539ae5.jpg", + "text": "$$\n\\boldsymbol { x } ^ { \\prime } = \\underset { \\boldsymbol { x } ^ { \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\ell ( \\boldsymbol { \\mathcal { M } } ( \\boldsymbol { x } ^ { \\prime } ) , 1 ) + \\lambda \\boldsymbol { c } ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )\n$$", + "text_format": "latex", + "bbox": [ + 369, + 694, + 629, + 723 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\ell : \\mathcal { V } \\times \\mathcal { V } \\to \\mathbb { R } _ { + }$ denotes a differentiable loss function (e.g., binary cross entropy) which ensures that gap between $\\mathcal { M } ( \\boldsymbol { x } ^ { \\prime } )$ and favorable outcome 1 is minimized, and $\\lambda > 0$ is a trade-off parameter. ", + "bbox": [ + 174, + 728, + 825, + 772 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 Formulating Our Objective ", + "text_level": 1, + "bbox": [ + 176, + 781, + 403, + 796 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As can be seen from Eqn. 2, state-of-the-art recourse finding algorithms rely heavily on the assumption that the underlying predictive model $\\mathcal { M }$ does not change. However, predictive models deployed in the real world often get updated. This implies that individuals who have acted upon previously prescribed recourses are no longer guaranteed a favorable outcome once the model is updated. To address this critical challenge, we propose a novel minimax objective function which generates counterfactuals that minimize the worst-case loss over plausible model shifts. We arrived at this approach after considering the following alternatives: (a) Update the predictive model as desired but ensure that individuals who were previously prescribed recourse will still be guaranteed a favorable outcome. (b) ", + "bbox": [ + 173, + 800, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Update the predictive model while including constraints to ensure that previously offered recourses are still valid. Note that both of these scenarios would potentially incur huge monetary losses to relevant stakeholders (e.g, banks), hurting the adoption of these approaches. In case (a), banks may be required to guarantee credit to customers that are potentially not creditworthy under the new model and thereby risk losing money. In case (b), access to model training is assumed. Furthermore, training a predictive model under these constraints may be suboptimal and not reflective of the current data distribution, thereby accruing larger errors under the shifted population. There are no incentives for stakeholders such as banks to adopt such practices which could potentially lead to huge monetary losses. While the optimal approach may vary on a case by case basis, we propose our method to avoid the aforementioned pitfalls outlined in cases (a) and (b). ", + "bbox": [ + 173, + 90, + 825, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To formalize our proposed approach, let $\\Delta$ denote the set of plausible model shifts and let $\\mathcal { M } _ { \\delta }$ denote a shifted model where $\\delta \\in \\Delta$ . Our objective function for generating robust recourse $x ^ { \\prime \\prime }$ for a given instance $x$ can be written as: ", + "bbox": [ + 174, + 236, + 825, + 279 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/00ef4fa002210e4c38b82f3fc94e54c401f680e4fb5f603fc1d76b882c8d4af2.jpg", + "text": "$$\n\\boldsymbol { x } ^ { \\prime \\prime } = \\underset { \\boldsymbol { x } ^ { \\prime \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\ \\underset { \\delta \\in \\Delta } { \\operatorname* { m a x } } \\ell ( \\boldsymbol { \\mathcal { M } } _ { \\delta } ( \\boldsymbol { x } ^ { \\prime \\prime } ) , 1 ) + \\lambda c ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime \\prime } )\n$$", + "text_format": "latex", + "bbox": [ + 344, + 282, + 651, + 310 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where cost function $c$ and loss function $l$ are as defined in Section 3.1. ", + "bbox": [ + 174, + 318, + 630, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Choice of $\\Delta$ Predictive models deployed in the real world are often updated regularly to handle data distribution shifts [22]. Since these models are updated regularly, it is likely that they undergo small (and not drastic) shifts each time they are updated. To capture this intuition, we consider the following two choices for the set of plausible model shifts $\\Delta$ : ", + "bbox": [ + 173, + 340, + 825, + 396 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1cb30dc22d7865eaec0e1c84e8ca40098fbb84971f7235de1182ee5dbdfc1ea0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\delta _ { m i n } \\leq \\delta _ { i } \\leq \\delta _ { m a x } \\forall i \\in \\{ 1 \\cdots n \\} \\} . } \\\\ & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\| \\delta \\| _ { p } \\leq \\delta _ { m a x } \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 400, + 509, + 443 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $p \\geq 1$ . Note that perturbations $\\delta \\in \\Delta$ can be considered as operations either on the parameter space or on the gradient space of $\\mathcal { M }$ . While the first choice of $\\Delta$ presented above allows us to restrict model shifts within a small range, the second choice allows us to restrict model shifts within a norm-ball. Alternate formulations of the first include incorporating domain knowledge to set $\\delta _ { m i n }$ and $\\delta _ { m a x }$ per feature. These kinds of shifts can effectively capture small changes to both parameters (e.g., weights of linear models) as well as gradients. Next, we describe how to optimize the objective in Eqn. 3 and construct robust recourses. ", + "bbox": [ + 173, + 445, + 825, + 542 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 Optimizing Our Objective ", + "text_level": 1, + "bbox": [ + 174, + 553, + 397, + 568 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While our objective function, the choice of $\\Delta$ , and the perturbations $\\delta \\in \\Delta$ we introduce in Section 3.2 are generic enough to handle shifts to both parameter space as well as the gradient space of any class of predictive models $\\mathcal { M }$ , we solve our objective for a linear approximation $f$ of $\\mathcal { M }$ . The procedure that we outline here remains generalizable even for non-linear models because local behavior of a given non-linear model can be approximated well by fitting a local linear model [25]. Note that such approximations have already been explored by existing algorithmic recourse methods [26, 23]. Let the linear approximation, which we denote by $f$ be parameterized by $w \\in \\mathcal { W }$ . We make this parametrization explicit by using a subscript notation: $f _ { w }$ . We consider model shifts represented by perturbations to the model parameters $w \\in \\mathcal { W }$ . In the case of linear models, these can be operationalized as additive perturbations $\\delta \\in \\Delta$ to $w$ . We will represent the resulting shifted classifier by $f _ { w + \\delta }$ . Our objective function (Eqn. 3) can now be written in terms of this linear approximation $f$ as: ", + "bbox": [ + 173, + 571, + 826, + 737 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/770696e463a7193ed4f64b9f7b6f383d8eefc0706c1c46307dbe9a37baeb154d.jpg", + "text": "$$\nx ^ { \\prime \\prime } = \\underset { x ^ { \\prime \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\underset { \\delta \\in \\Delta } { \\operatorname* { m a x } } \\ell ( f _ { w + \\delta } ( x ^ { \\prime \\prime } ) , 1 ) + \\lambda c ( x , x ^ { \\prime \\prime } )\n$$", + "text_format": "latex", + "bbox": [ + 341, + 741, + 656, + 770 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Notice that the objective function defined in Equation 4 is similar to that of adversarial training [19]. However, in our framework, the perturbations are applied to model parameters as opposed to data samples. These parallels help motivate the optimization procedure for constructing recourses that are robust to model shifts. We outline the optimization procedure that we leverage to optimize our minimax objective (Eqn. 4) in Algorithm 1. ", + "bbox": [ + 173, + 776, + 825, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 proceeds in an iterative manner where we first find a perturbation $\\hat { \\delta } \\in \\Delta$ that maximizes the chance of invalidating the current estimate of the recourse $x ^ { \\prime \\prime }$ , and then we take appropriate gradient steps on $x ^ { \\prime \\prime }$ to generate a valid recourse. This procedure is executed iteratively until the objective function value (Eqn. 4) converges. ", + "bbox": [ + 174, + 854, + 823, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 Our Optimization Procedure ", + "text_level": 1, + "bbox": [ + 176, + 90, + 449, + 106 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/24f39766558110ab983e25d51c8dd124b8c2c8f4e4da18a29ccb3bd66b6ba126.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input:x s.t. fω(x)=O,fw,λ>O,△,learning rate α >0. Initialize x" =x,g =0
repeat = arg maxs∈△ l(fw+8(x"),1)
g =∀[e(fw+8(x"),1)+ λc(x",x)]
x" -=ag
until convergence
Return x"
", + "bbox": [ + 179, + 109, + 549, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Theoretical Analysis ", + "text_level": 1, + "bbox": [ + 174, + 253, + 377, + 271 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here we carry out theoretical analysis to shed light on the benefits of our framework ROAR. More specifically: 1) We quantify the probability that recourses generated without accounting for model shifts are likely to be invalidated. 2) We prove that the additional cost incurred due to the robust recourses output by our framework is bounded. ", + "bbox": [ + 173, + 279, + 825, + 335 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We first characterize how recourses that do not account for model shifts (i.e., recourses output by state-of-the-art algorithms) fare when true model shifts can be characterized as additive shifts to model parameters. Specifically, we quantify the likelihood that recourses generated without accounting for model shifts will be invalidated (even if they lie on the original data manifold), under certain conditions. ", + "bbox": [ + 173, + 343, + 825, + 414 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1. For a given instance $x \\sim { \\mathcal { N } } ( \\mu , \\Sigma )$ , let $x ^ { \\prime }$ be the recourse that lies on the original data manifold (conditioned on the event that $\\mathcal { M } ( x ^ { \\prime } ) > 0 . 5 )$ and is obtained without accounting for model shifts. Let $\\Sigma = U D U ^ { T }$ . Then, for some true model shift $\\delta$ , such that, $\\begin{array} { r } { \\frac { w ^ { T } \\mu } { ( w + \\delta ) ^ { T } \\mu } \\geq \\frac { \\| \\sqrt { D } U w \\| } { \\| \\sqrt { D } U ( w + \\delta ) \\| } } \\end{array}$ , and $\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq e r f c \\bigl ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } \\bigr ) . } \\end{array}$ , for $\\beta \\geq 1$ , the probability that $x ^ { \\prime }$ is invalidated on $f _ { w + \\delta }$ is at least: $\\begin{array} { r } { \\frac { 1 } { 2 } \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } - \\frac { 1 } { 2 } e r f c \\big ( { - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } } \\big ) } \\end{array}$ where erfc is the complementary gaussian error function. ", + "bbox": [ + 173, + 420, + 826, + 544 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof Sketch. Under the assumption that $x ^ { \\prime } \\sim { \\mathcal { N } } ( { \\boldsymbol { \\mu } } , { \\boldsymbol { \\Sigma } } )$ , a recourse is invalid under a model shift if it is valid under the original model and invalid under the shifted model. This allows us to define the region where $x ^ { \\prime }$ can be invalidated: ", + "bbox": [ + 173, + 563, + 825, + 606 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d0afb414b31db376a47893a7ce3ed8663ffc4e875d7370634a7700050369b9a7.jpg", + "text": "$$\n\\Omega = \\{ x ^ { \\prime } \\colon w ^ { T } x ^ { \\prime } > 0 \\cap ( w + \\delta ) ^ { T } x ^ { \\prime } \\leq 0 \\}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 614, + 633, + 633 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The probability that $x ^ { \\prime }$ is invalidated can be obtained by integrating over $\\Omega$ under the PDF of $\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )$ . ", + "bbox": [ + 171, + 650, + 825, + 666 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We can then transform $x ^ { \\prime }$ and correspondingly $\\Omega$ , to simplify this integration over a 1-dimensional Gaussian random variable. That is, ", + "bbox": [ + 174, + 672, + 823, + 703 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/9bd556959a393381c310cc85e81f1ce2ea8b97d0d27005e4406ba056b245fa5f.jpg", + "text": "$$\nP ( x { \\mathrm { ~ i s ~ i n v a l i d a t e d } } ) = { \\frac { 1 } { \\sqrt { ( 2 \\pi ) } } } \\int _ { c _ { 1 } } ^ { c _ { 2 } } \\exp { \\bigg ( } - { \\frac { 1 } { 2 } } s ^ { 2 } { \\bigg ) } d s\n$$", + "text_format": "latex", + "bbox": [ + 323, + 709, + 676, + 747 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The above quantity can be represented as a difference in the Gaussian error function allowing us to exactly quantify the invalidation probability under our assumptions. Using the lower bounds on the complementary gaussian error function [9] from Chang et al. [5], we obtain our lower bound. To derive the lower bound, we add an extra condition that $\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq \\mathrm { e r f c } ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } ) } \\end{array}$ , mainly to confirm that the lower bound on the first term still dominates the second term. Both conditions restricts the types of shift for which the bound can be derived. Note that $\\beta$ can be optimized away to improve the lower bound. Detailed proof is provided in the Appendix. Discussion about other distributions (e.g., Bernoulli, Uniform, Categorical) is included in the Appendix. □ ", + "bbox": [ + 173, + 786, + 826, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Next we characterize how much more costly recourses can be when they are trained to be robust to model perturbations or model shifts. In the following theorem, we show that the cost of robust recourses is bounded relative to the cost of recourses that do not account for model shifts. ", + "bbox": [ + 173, + 90, + 825, + 133 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 2. Let $x \\in \\mathcal { X }$ , and $x \\sim \\nu$ where $\\nu$ is a distribution such that $\\mathbb { E } _ { \\nu } [ x ] = \\mu < \\infty ,$ , where $\\mathcal { X }$ is a metric space $( \\mathcal { X } , d ( \\cdot , \\cdot ) )$ and $d : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }$ . Let $d \\triangleq \\ell _ { 2 }$ and assume that $( \\mathcal { X } , d )$ has bounded diameter $D = \\operatorname* { s u p } _ { x , x ^ { \\prime } \\in \\mathcal { X } } d ( x , x ^ { \\prime } )$ . Let recourses obtained without accounting for model shifts and constrained to the manifold be denoted by $x ^ { \\prime } \\sim \\nu$ , and robust recourses be denoted by $x ^ { \\prime \\prime }$ . Let $\\delta > 0$ be the shift that maximizes Eq. 3 for sample $x$ corresponding to $x ^ { \\prime \\prime }$ . Further assume that the ROAR objective (Equation 3) is convex in $x ^ { \\prime \\prime }$ for a fixed $\\delta$ . For $\\ell \\triangleq \\ell _ { l o g }$ (the cross-entropy loss), some $0 < \\eta ^ { \\prime } \\ll 1$ , w.h.p. $( 1 - \\eta ^ { \\prime } )$ , we have that: ", + "bbox": [ + 173, + 138, + 826, + 243 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6a767dbbc087322e15f8dc5c11e384ffd6306ad4ac1a745d87e47f35393f257e.jpg", + "text": "$$\nc ( x ^ { \\prime \\prime } , x ) - c ( x ^ { \\prime } , x ) \\leq \\frac { 1 } { \\lambda } \\frac { 1 } { \\| w + \\delta \\| } \\mathbb { E } _ { \\nu } [ \\exp { - \\phi ( w + \\delta ) ^ { T } x ^ { \\prime } } ] + \\sqrt { \\frac { D ^ { 2 } } { 2 } \\log { ( \\frac { 1 } { \\eta ^ { \\prime } } ) } } \\Bigg )\n$$", + "text_format": "latex", + "bbox": [ + 245, + 248, + 753, + 291 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof Sketch. By definition, any recourse $x ^ { \\prime }$ generated without accounting for model shifts will have a higher loss for Equation 3 compared to the robust recourse $x ^ { \\prime \\prime }$ (note that finding the global minimizer is not guaranteed by Algorithm 1). ", + "bbox": [ + 174, + 303, + 825, + 345 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using this insight, and convexity in $x ^ { \\prime }$ for fixed $\\delta$ , we can bound the cost difference between the robust and non-robust recourse by a 1-Lipschitz function (i.e. the logistic function): ", + "bbox": [ + 171, + 352, + 823, + 381 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f53f509a541745fdd4e7e0d145bc1a46f5a152eaeb4f2901ef59c4c782abd8c9.jpg", + "text": "$$\nc ( x ^ { \\prime \\prime } , x ) - c ( x ^ { \\prime } , x ) \\leq \\frac { 1 } { \\lambda \\| w + \\delta \\| } \\log \\left\\{ 1 + \\exp - ( w + \\delta ) ^ { T } x ^ { \\prime } \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 386, + 702, + 419 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assuming a bounded metric on $\\mathcal { X }$ , we can upper bound the RHS using Lemma 2 from van Handel [27] which gives us our bound. Detailed proof including special cases when $\\nu$ is Gaussian, is provided in the Appendix. □ ", + "bbox": [ + 174, + 430, + 825, + 473 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This result suggests that the additional cost of recourse is bounded by the amount of shift admissible in Equation 3. Note that Theorem 2 applies for general distributions so long as the mean is finite, which is the case for most commonplace distributions like Gaussian, Bernoulli, Multinomial etc. While Theorem 1 demonstrates the probability that a recourse will be invalidated for Gaussian distributions, we refer the reader to the Appendix B.1 for a discussion of other distributions, e.g. Bernoulli, Uniform, Categorical. ", + "bbox": [ + 173, + 486, + 826, + 570 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 582, + 313, + 599 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here we discuss the detailed experimental evaluation of our framework, ROAR. First, we evaluate how robust the recourses generated by our framework are to model shifts caused by real world data distribution shifts. We also assess the validity of the recourses generated by our framework w.r.t. the original model, and further analyze the average cost of these recourses. Next, using synthetic data, we analyze how varying the degree (magnitude) of data distribution shift impacts the robustness and validity of the recourses output by our framework and other baselines. ", + "bbox": [ + 174, + 606, + 825, + 690 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 699, + 352, + 714 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Real world data We evaluate our framework on model shifts induced by real world data distribution shifts. To this end, we leverage three real world datasets which capture different kinds of data distribution shifts, namely, temporal shift, geospatial shift, and data correction shift [24]. Our first dataset is the widely used and publicly available German credit dataset [8] from the UCI repository. This dataset captures demographic (age, gender), personal (marital status), and financial (income, credit duration) details of about 1000 loan applicants. Each applicant is labeled as either a good customer or a bad customer depending on their credit risk. Two versions of this dataset have been released, with the second version incorporating corrections to coding errors in the first dataset [11]. Accordingly, this dataset captures the data correction shift. Our second dataset is the Small Business Administration (SBA) case dataset [17]. This dataset contains information pertaining to 2102 small business loans approved by the state of California during the years of $1 9 8 9 - 2 0 1 2$ , and captures temporal shifts in the data. It comprises of about 24 features capturing various details of the small businesses including zip codes, business category (real estate vs. rental vs. leasing), number of jobs created, and financial status of the business. It also contains information about whether a business has defaulted on a loan or not which we consider as the class label. Our last dataset contains student performance records of 649 students from two Portuguese secondary schools, Gabriel Pereira (GP) and Mousinho da Silveira (MS) [8, 6], and captures geospatial shift. It comprises of information about the academic background (grades, absences, access to internet, failures etc.) of each student along with other demographic attributes (age, gender). Each student is assigned a class label of above average or not depending on their final grade. ", + "bbox": [ + 173, + 717, + 826, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 174 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Synthetic data We generate a synthetic dataset with 1K samples and two dimensions to analyze how the degree (magnitude) of data distribution shifts impacts the robustness and validity of the recourses output by our framework and other baselines. Each instance $x$ is generated as follows: First, we randomly sample the class label $y \\in \\{ 0 , 1 \\}$ corresponding to the instance $x$ . Conditioned upon the value of $y$ , we then sample the instance $x$ as: $x \\sim \\mathrm { \\bar { \\mathcal { N } } } ( \\mu _ { y } , \\mathrm { \\bar { \\Sigma } } _ { y } )$ . We choose $\\mu _ { 0 } = [ - 2 , - 2 ] ^ { T }$ and $\\mu _ { 1 } = [ + 2 , + 2 ] ^ { T }$ , and $\\Sigma _ { 0 } = \\Sigma _ { 1 } = 0 . 5 \\mathbf { I }$ where $\\mu _ { 0 }$ , $\\Sigma _ { 0 }$ and $\\mu _ { 1 }$ , $\\Sigma _ { 1 }$ denote the means and covariance of the Gaussian distributions from which instances in class 0 and class 1 are sampled respectively. A scatter plot of the samples resulting from this generative process and the decision boundary of a logistic regression model fit to this data are shown in Figure 1a. In our experimental evaluation, we consider different kinds of shifts to this synthetic data: ", + "bbox": [ + 173, + 184, + 825, + 324 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/98bb3ff9b091767a7781023e340217b6fd8251f08031fdf85213a639fc47f523.jpg", + "image_caption": [ + "Figure 1: Synthetic data and examples of model shift. From left to right we have (a) original synthetic dataset, (b) shifted data and decision boundary after mean shift, (c) shifted data and decision boundary after variance shift, and (d) shifted data and decision boundary after mean and variance shift " + ], + "image_footnote": [], + "bbox": [ + 191, + 339, + 799, + 446 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(i) Mean shift: To generated shifted data, we leverage the same approach as above but shift the mean of the Gaussian distribution associated with class 0 i.e., $x \\sim \\mathcal { N } ( \\mu _ { y } ^ { \\prime } , \\Sigma _ { y } )$ where $\\mu _ { 0 } ^ { \\prime } = \\mu _ { 0 } + [ \\alpha , 0 ] ^ { T }$ and $\\mu _ { 1 } ^ { \\prime } = \\mu _ { 1 }$ . Note that we only shift the mean of one of the features of class 0 so that the slope of the decision boundary of a linear model we fit to this shifted data changes (relative to the linear model fit on the original data), while the intercept remains the same. Figure 1b shows shifted data with $\\alpha = 1 . 5$ . ", + "bbox": [ + 173, + 501, + 825, + 587 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(ii) Variance shift: Here, we leverage the same generative process as above, but instead of shifting the mean, we shift the variance of the Gaussian distribution associated with class 0 i.e., i.e., $x \\sim$ $\\mathcal { N } ( \\mu _ { y } , \\Sigma _ { y } ^ { \\prime } )$ where $\\Sigma _ { 0 } ^ { \\prime } = ( 1 + \\beta ) \\Sigma _ { 0 }$ and $\\Sigma _ { 1 } ^ { \\prime } = \\Sigma _ { 1 } ^ { \\prime }$ for some increment $\\beta \\in \\mathbb { R }$ . The net result here is that the intercept of the decision boundary of a linear model we fit to this shifted data changes (relative to the linear model fit on the original data), while the slope remains unchanged. Figure 1c shows shifted data with $\\beta = 3$ . ", + "bbox": [ + 173, + 594, + 825, + 678 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(ii) Mean and variance shift: Here, we change both the mean and variance of the Gaussian distribution associated with class 0 simultaneously (Figure 1d). It can be seen that there are noticeable changes to both the slope and intercept of the decision boundary compared to Figure 1a. ", + "bbox": [ + 174, + 684, + 825, + 727 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Predictive models We generate recourses for a variety of linear and non-linear models: deep neural networks (DNNs), SVMs, and logistic regression (LR). Here, we present results for a 3-layer DNN and LR; remaining results are included in the Appendix. Results presented here are representative of those for other model families. ", + "bbox": [ + 174, + 736, + 825, + 791 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines We compare our framework, ROAR, to the following state-of-the-art baselines: (i) counterfactual explanations (CFE) framework outlined by Wachter et al. [31], (ii) actionable recourse (AR) in linear classification [26], and (iii) causal recourse framework (MINT) proposed by Karimi et al. [14]. While CFE leverages gradient computations to find counterfactuals, AR employs a mixed integer programming based approach to find counterfactuals that are actionable. The MINT framework operates on top of existing approaches for finding nearby counterfactuals. We use the MINT framework on top of CFE and ROAR and refer to these two approaches as MINT and ROARMINT respectively. As the MINT framework requires access to the underlying causal graph, we experiment with MINT and ROAR-MINT only on the German credit dataset for which such a causal graph is available. ", + "bbox": [ + 173, + 800, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 90, + 823, + 119 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Cost functions Our framework, ROAR, and all the other baselines we use rely on a cost function $c$ that measures the cost (or effort) required to act upon the prescribed recourse. Furthermore, our approach as well as several other baselines require the cost function to be differentiable. So, we consider two cost functions in our experimentation: $\\ell _ { 1 }$ distance between the original instance and the counterfactual, and a cost function learned from pairwise feature comparison inputs (PFC) [13, 26, 23]. PFC uses the Bradley-Terry model to map pairwise feature comparison inputs provided by end users to the cost required to act upon the prescribed recourse for any given instance $x$ . For more details on this cost function, please refer to Rawal and Lakkaraju [23]. In our experiments, we follow the same procedure as Rawal and Lakkaraju [23] and simulate the pairwise feature comparison inputs. ", + "bbox": [ + 173, + 127, + 825, + 253 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Setting and implementation details We partition each of our synthetic and real world datasets into two parts: initial data $( D _ { 1 } )$ and shifted data $( D _ { 2 } )$ . In the case of real world datasets, $D _ { 1 }$ and $D _ { 2 }$ can be logically inferred from the data itself – e.g., in case of the German credit dataset, we consider the initial version of the dataset as $D _ { 1 }$ and the corrected version of the dataset as $D _ { 2 }$ . In the case of synthetic datasets, we generate $D _ { 1 }$ and $D _ { 2 }$ as described earlier where $D _ { 2 }$ is generated by shifting $D _ { 1 }$ (See \"Synthetic data\" in Section 5.1). ", + "bbox": [ + 174, + 262, + 825, + 344 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use 5-fold cross validation throughout our real world and synthetic experiments. On $D _ { 1 }$ , we use 4 folds to train predictive models and the remaining fold to generate and evaluate recourses. We repeat this process 5 times and report averaged values of our evaluation metrics. We leverage $D _ { 2 }$ only to train the shifted models $\\mathcal { M } _ { 2 }$ . More details about the data splits, model training, and performance of the predictive models are included in the Appendix. ", + "bbox": [ + 173, + 352, + 825, + 421 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use binary cross entropy loss and the Adam optimizer to operationalize our framework, ROAR. Our framework, ROAR, has the following parameters: the set of acceptable perturbations $\\Delta$ (defined in practice by $\\delta _ { m a x . }$ ) and the tradeoff parameter $\\lambda$ . In our experiments on evaluating robustness to real world shifts, we choose $\\delta _ { m a x } = 0 . 1$ given that continuous features are scaled to zero mean and unit variance. Furthermore, in each setting, we choose the $\\lambda$ that maximizes the recourse validity of $\\mathcal { M } _ { 1 }$ (more details in Section 5.1 \"Metrics\" and Appendix). In case of our synthetic experiments where we assess the impact of the degree (magnitude) of data distribution shift, features are not normalized, so we do a grid search for both $\\delta _ { m a x }$ and $\\lambda$ . First, we choose the largest $\\delta _ { m a x }$ that maximizes the recourse validity of $\\mathcal { M } _ { 1 }$ and then set $\\lambda$ in a similar fashion (more details in Appendix). We set the parameters of the baselines using techniques discussed in the original works [31, 14, 26] and employ a similar grid search approach if unspecified. ", + "bbox": [ + 174, + 429, + 825, + 580 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Following the precedents set forth in [26] and [23], we adapt AR and ROAR to non-linear models by first generating local linear approximations of these models using LIME [25]. We refer to these variants as AR-LIME and ROAR-LIME respectively. ", + "bbox": [ + 174, + 588, + 825, + 630 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Metrics. We consider two metrics in our evaluation: 1) Avg Cost is defined as the average cost incurred to act upon the prescribed recourses where the average is computed over all the instances for which a given algorithm provides recourse. Recall that we consider two notions of cost in our experiments – $\\ell _ { 1 }$ distance between the original instance and the counterfactual, costs learned from pairwise feature comparisons (PFC) (See \"Cost Functions\" in Section 5.1). 2) Validity is defined as the fraction of instances for which acting upon the prescribed recourse results in the desired prediction. Note that validity is computed w.r.t. a given model. ", + "bbox": [ + 174, + 638, + 825, + 736 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 Robustness to real world shifts ", + "text_level": 1, + "bbox": [ + 176, + 746, + 426, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Here, we evaluate the robustness of the recourses output by our framework, ROAR, as well as the baselines. A recourse finding algorithm can be considered robust if the recourses output by the algorithm remain valid even if the underlying model has changed. To evaluate this, we first leverage our approach and other baselines to find recourses of instances in our test sets w.r.t. the initial model $\\mathcal { M } _ { 1 }$ . We then compute the validity of these recourses w.r.t. the shifted model $\\mathcal { M } _ { 2 }$ which has been trained on the shifted data. Let us refer to this as $\\mathcal { M } _ { 2 }$ validity. The higher the value of $\\mathcal { M } _ { 2 }$ validity, the more robust the recourse finding method. Table 1 shows the $\\mathcal { M } _ { 2 }$ validity metric computed for different algorithms across different real world datasets. ", + "bbox": [ + 174, + 765, + 825, + 876 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It can be seen that recourse methods that use our framework, ROAR and ROAR-MINT, achieve the highest $\\mathcal { M } _ { 2 }$ validity across all datasets. In fact, methods that use our framework do almost twice as ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8d8b6e9686fadf0f52f07cba7230a70ce8906aeba7c07b660d8b8421265b202f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Correction ShiftTemporal ShiftGeospatial Shift
Model CostRecourse CFEAvgCost 1.02 ± 0.18MValidity 1.00±0.00MValidity 0.54± 0.27AvgCost 3.57 ± 1.14MValidity 1.00±0.00MValidity 0.31±0.09Avg Cost 8.37±0.73MValidity 0.98±0.03MValidity 0.29±0.09
LRL10.85 ± 0.141.00 ± 0.000.53 ± 0.211.50± 0.281.00 ± 0.000.16 ± 0.065.29 ± 0.281.00 ± 0.000.43 ± 0.14
AR3.14 ± 0.250.99 ± 0.010.98 ±0.0210.88 ± 1.671.00 ± 0.000.67 ± 0.19
ROAR3.13 ± 0.321.00 ± 0.000.94 ± 0.08 0.93 ± 0.07NA
MINT4.73 ± 1.561.00 ± 0.00NANANANANA
ROAR-MINT6.77 ± 0.351.00 ± 0.001.00 ± 0.00NANANANANANA
CFE0.03±0.021.00 ±0.000.56±0.330.24±0.091.00 ± 0.000.26± 0.110.34± 0.041.00±0.000.18 ±0.10
PFCAR0.09 ± 0.021.00 ± 0.000.54 ± 0.270.11 ± 0.021.00 ± 0.000.09 ± 0.050.32 ±0.031.00 ±0.000.24 ± 0.11
ROAR MINT0.36±0.081.00 ± 0.001.00 ± 0.000.44 ± 0.120.99 ± 0.010.98 ± 0.011.20 ± 0.101.00 ± 0.000.91± 0.07
ROAR-MINT1.00 ± 1.151.00 ± 0.00 1.00 ± 0.000.95±0.08 1.00 ± 0.00NANA NANANANANA
L1CFE1.23 ± 0.05 0.55 ±0.101.00± 0.000.47±0.06NA 3.78±0.681.00 ± 0.00NA 0.52±0.09NANANA
AR-LIME0.38 ± 0.150.16 ±0.100.31 ± 0.061.39 ± 0.130.59 ± 0.110.65 ± 0.1710.09± 0.71 9.02 ±1.571.00 ± 0.000.48±0.09 0.83 ±0.10
ROAR-LIME0.76± 0.06
NN1.83 ± 0.190.78 ±0.060.72 ± 0.104.90±0.240.98 ±0.020.97 ±0.0221.05 ± 3.581.00 ± 0.000.97 ±0.03
MINT2.24 ± 1.250.81 ± 0.020.63 ± 0.11NANANANANANA
ROAR-MINT8.59 ± 1.700.90 ±0.030.84 ± 0.04NANANANANANA
CFE0.06±0.021.00± 0.000.51 ± 0.120.19±0.061.00±0.000.50± 0.130.48± 0.061.00±0.000.30±0.14
AR-LIME0.06± 0.030.49 ± 0.110.56± 0.150.11 ± 0.010.54 ± 0.080.62 ± 0.120.78 ± 0.150.84 ± 0.060.82 ± 0.11
PFC ROAR-LIME0.64 ± 0.080.85 ± 0.070.82 ± 0.050.37 ±0.070.99 ± 0.010.99 ±0.01.66 ± 0.211.00 ±0.000.97 ± 0.04
0.60 ± 0.160.82 ±0.070.64 ± 0.15NANANANANA
MINT ROAR-MINT0.60 ±0.070.91± 0.040.81 ± 0.04NANANANANANA NA
", + "bbox": [ + 173, + 88, + 823, + 282 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 1: Avg Cost, $\\mathcal { M } _ { 1 }$ (original) validity, and $\\mathcal { M } _ { 2 }$ (shifted model) validity of recourses across different real world datasets. Recourses that leverage our framework ROAR are more robust (higher $\\mathcal { M } _ { 2 }$ validity) compared to those generated by existing baselines. ", + "bbox": [ + 176, + 289, + 823, + 332 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "good compared to other baselines on this metric, indicating that ROAR based recourse methods are quite robust. After ROAR, MINT is the next best performing baseline with respect $\\mathcal { M } _ { 2 }$ validity. This may be explained by the fact that MINT accounts for the underlying causal graphs when generating recourses. ", + "bbox": [ + 174, + 340, + 825, + 396 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also assess if the robustness achieved by our framework is coming at a cost i.e., by sacrificing validity on the original model or by increasing avg cost. Table 1 shows the results for the same. It can be seen that ROAR based recourses achieve higher than $9 5 \\%$ $\\mathcal { M } _ { 1 }$ validity in all but two settings. We compute the avg cost of the recourses output by all the algorithms on various datasets and find that ROAR typically has a higher avg cost (both under $\\ell _ { 1 }$ and PFC cost functions) compared to CFE and AR baselines. As demonstrated through additional experiments in the Appendix, these relatively higher costs are expected given our Theorem 2 upper bound on ROAR cost. However, overall, MINT and ROAR-MINT seem to exhibit the highest avg costs and are the worst performing algorithms according to this metric. Since non-causal recourse methods assume independent features, and do not have to adhere to the underlying causal structure when finding counterfactuals, they can generate relatively lower cost counterfactuals even if those counterfactuals may not correspond to realistic data instances. This is likely one of the key reasons why we observe higher average costs in the causal recourse methods. ", + "bbox": [ + 173, + 404, + 825, + 583 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/79b7233b461f75eae4a6c8dc52c9a20153a7b4b770aa53c7911b559afe6ec4db.jpg", + "image_caption": [ + "5.3 Impact of the degree of data distribution shift on recourses ", + "Figure 2: Impact of the degree of data distribution shift on validity of recourse: DNN classifier with $\\ell _ { 1 }$ cost function (top row), DNN classifier with PFC cost function (bottom row); Validity of the recourses generated by all methods drops as degree (magnitude) of the shift increases; The drop in the validity is much smaller for our method ROAR-LIME compared to other baselines. " + ], + "image_footnote": [], + "bbox": [ + 215, + 632, + 761, + 861 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here, we assess how different kinds of distribution shifts and the magnitude of these shifts impact the robustness of recourses output by our framework and other baselines. To this end, we leverage our synthetic datasets and introduce mean shifts, variance shifts, and combination shifts (both mean and variance shifts) of different magnitudes by varying $\\alpha$ and $\\beta$ (See \"Synthetic data\" in Section 5.1). We then leverage these different kinds of shifted datasets to construct shifted models and then assess the validity of the recourses output by our framework and other baselines w.r.t. the shifted models. ", + "bbox": [ + 174, + 90, + 825, + 174 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We generate recourses using our framework and baselines CFE and AR for different predictive models (LR, DNN) and cost functions ( $\\ell _ { 1 }$ distance, PFC). Figure 2 captures the results of this experiment for DNN model both with $\\ell _ { 1 }$ distance and PFC cost functions. Results with other models are included in the Appendix. It can be seen that the $\\mathbf { X }$ -axis of each of these plots captures the magnitude of the dataset shift, and the y-axis captures the validity of the recourses w.r.t. the corresponding shifted model. Standard error bars obtained by averaging the results over 5 runs are also shown. ", + "bbox": [ + 174, + 181, + 825, + 265 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "It can be seen that as the magnitude of the distribution shift increases, validity of the recourses generated by all the methods starts dropping. This trend prevailed across mean, variance, and combination (mean and variance) shifts. It can also be seen that the rate at which validity of the recourses generated by our method, ROAR-LIME, drops is much smaller compared to that of other baselines CFE and AR-LIME. Furthermore, our method exhibits the highest validity compared to the baselines as the magnitude of the distribution shift increases. CFE seems to be the worst performing baseline and the validity of the recourses generated by CFE drops very sharply even at small magnitudes of distribution shifts. ", + "bbox": [ + 174, + 272, + 825, + 383 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusions & Future Work ", + "text_level": 1, + "bbox": [ + 176, + 396, + 441, + 412 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We proposed a novel framework, RObust Algorithmic Recourse (ROAR), to address the critical but under-explored issue of recourse robustness to model updates. To this end, we introduced a novel minimax objective to generate recourses that are robust to model shifts, and leveraged adversarial training to optimize this objective. We also presented novel theoretical results which demonstrate that recourses without accounting for model shifts are likely to be invalidated, underscoring the necessity of ROAR. Furthermore, we also showed that the additional cost incurred by robust recourses generated by ROAR are bounded. Extensive experimentation with real world and synthetic datasets demonstrated that recourses using ROAR are highly robust to model shifts induced by a range of data distribution shifts. Our work also paves the way for further research into techniques for generating robust recourses. For instance, it would be valuable to further analyze the tradeoff between recourse robustness and cost to better understand the impacts to affected individuals. Other interesting future directions include non-linear extensions that leverage novel local linear approximation methods that improve on LIME [33]. ", + "bbox": [ + 174, + 421, + 825, + 602 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 616, + 338, + 632 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461 and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, and Google. SJ would like to acknowledge the support of the Center for Research on Computation and Society (CRCS) at the Harvard John A. Paulson School of Engineering and Applied Sciences. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies. 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PMLR, 2018. \n[33] Xingyu Zhao, Xiaowei Huang, V. Robu, and D. Flynn. Baylime: Bayesian local interpretable model-agnostic explanations. UAI, abs/2012.03058, 2021. 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There has also been some recent", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "research that sheds light on the spuriousness of the recourses generated by counterfactual/contrastive", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 428, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 428, + 628 + ], + "score": 1.0, + "content": "explanation techniques [31, 26] and advocates for causal approaches [3, 14, 15].", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "All the aforementioned approaches generate recourses under the assumption that the underlying", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "predictive models do not change. 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These techniques generate re-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 274, + 471, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 274, + 471, + 286 + ], + "score": 1.0, + "content": "courses under the assumption that the underlying predictive model does not change.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "score": 1.0, + "content": "However, in practice, models are often regularly updated for a variety of reasons", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "score": 1.0, + "content": "(e.g., dataset shifts), thereby rendering previously prescribed recourses ineffective.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 307, + 470, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 470, + 320 + ], + "score": 1.0, + "content": "To address this problem, we propose a novel framework, RObust Algorithmic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 317, + 469, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 469, + 330 + ], + "score": 1.0, + "content": "Recourse (ROAR), that leverages adversarial training for finding recourses that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 328, + 469, + 341 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 469, + 341 + ], + "score": 1.0, + "content": "are robust to model shifts. To the best of our knowledge, this work proposes the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 339, + 469, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 339, + 469, + 352 + ], + "score": 1.0, + "content": "first ever solution to this critical problem. We also carry out theoretical analysis", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "score": 1.0, + "content": "which underscores the importance of constructing recourses that are robust to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 361, + 470, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 470, + 373 + ], + "score": 1.0, + "content": "model shifts: 1) We quantify the probability of invalidation for recourses generated", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 372, + 470, + 384 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 470, + 384 + ], + "score": 1.0, + "content": "without accounting for model shifts. 2) We prove that the additional cost incurred", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 383, + 469, + 395 + ], + "spans": [ + { + "bbox": [ + 141, + 383, + 469, + 395 + ], + "score": 1.0, + "content": "due to the robust recourses output by our framework is bounded. Experimental", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 394, + 469, + 407 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 469, + 407 + ], + "score": 1.0, + "content": "evaluation on multiple synthetic and real-world datasets demonstrates the efficacy", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 405, + 254, + 417 + ], + "spans": [ + { + "bbox": [ + 142, + 405, + 254, + 417 + ], + "score": 1.0, + "content": "of the proposed framework.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18.5, + "bbox_fs": [ + 141, + 241, + 471, + 417 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 427, + 191, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 192, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 192, + 442 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "Over the past decade, machine learning (ML) models are increasingly being deployed to make a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 458, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 469 + ], + "score": 1.0, + "content": "variety of highly consequential decisions ranging from bail and hiring decisions to loan approvals.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "Consequently, there is growing emphasis on designing tools and techniques which can provide", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "recourse to individuals who have been adversely impacted by predicted outcomes [30]. For example,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "when an individual is denied a loan by a predictive model deployed by a bank, they should be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "provided with reasons for this decision, and also informed about what can be done to reverse it. When", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "providing a recourse to an affected individual, it is absolutely critical to ensure that the corresponding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "decision making entity (e.g., bank) is able to honor that recourse and approve any re-application that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 477, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 477, + 546 + ], + "score": 1.0, + "content": "fully implements the recommendations outlined in the prescribed recourse Wachter et al. [31].", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 446, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "Several approaches in recent literature tackled the problem of providing recourses by generating local", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "(instance level) counterfactual explanations 2 [31, 26, 12, 21, 18]. For instance, Wachter et al. 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There has also been some recent", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "research that sheds light on the spuriousness of the recourses generated by counterfactual/contrastive", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 428, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 428, + 628 + ], + "score": 1.0, + "content": "explanation techniques [31, 26] and advocates for causal approaches [3, 14, 15].", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 550, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "All the aforementioned approaches generate recourses under the assumption that the underlying", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "predictive models do not change. This assumption, however, may not hold in practice. Real world", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "settings are typically rife with different kinds of distribution shifts (e.g, temporal shifts) [22]. In order", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "to ensure that the deployed models are accurate despite such shifts, these models are periodically", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "retrained and updated. Such model updates, however, pose severe challenges to the validity of", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "recourses because previously prescribed recourses (generated by existing algorithms) may no longer", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "be valid once the model is updated. Recent work by Rawal et al. 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Their work underscores the importance of generating recourses that are robust to", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "changes in models i.e., model shifts, particularly those resulting from dataset shifts. However, none", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 300, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 300, + 183 + ], + "score": 1.0, + "content": "of the existing approaches address this problem.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 631, + 505, + 655 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "settings are typically rife with different kinds of distribution shifts (e.g, temporal shifts) [22]. In order", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "to ensure that the deployed models are accurate despite such shifts, these models are periodically", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "retrained and updated. 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Their work underscores the importance of generating recourses that are robust to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "changes in models i.e., model shifts, particularly those resulting from dataset shifts. However, none", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 300, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 300, + 183 + ], + "score": 1.0, + "content": "of the existing approaches address this problem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "In this work, we propose a novel algorithmic framework, RObust Algorithmic Recourse (ROAR)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "for generating instance level recourses (counterfactual explanations) that are robust to changes in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "the underlying predictive model. To the best of our knowledge, this work makes the first attempt", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "at generating recourses that are robust to model shifts. To this end, we propose a novel minimax", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "objective that can be used to construct robust actionable recourses while minimizing the recourse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 261 + ], + "score": 1.0, + "content": "costs. Second, we propose a set of model shifts that captures our intuition about the kinds of changes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 256, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 271 + ], + "score": 1.0, + "content": "in the models to which recourses should be robust. Next, we outline an algorithm inspired by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "adversarial training to optimize the proposed objective. We also carry out theoretical analysis to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "establish the following results: i) we quantify the probability of invalidation for recourses generated", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "without accounting for model shifts, and ii) we derive an upper bound on the relative increase in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "costs incurred due to robust recourses (proposed by our framework) to the costs incurred by recourses", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "generated from existing algorithms. Our theoretical results further establish the need for approaches", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 404, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 404, + 335 + ], + "score": 1.0, + "content": "like ours that generate actionable recourses that are robust to model shifts.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 507, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 507, + 357 + ], + "score": 1.0, + "content": "We evaluated our approach ROAR on real world data from financial lending and education domains,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "focusing on model shifts induced by the following kinds of distribution shifts – data correction shift,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "temporal shift, and geospatial shift. We also experimented with synthetic data to analyze how the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "degree of data distribution shifts and consequent model shifts affect the robustness and validity of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "recourses output by our framework as well as the baselines. Our results demonstrate that the recourses", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 397, + 411 + ], + "score": 1.0, + "content": "constructed using our framework, ROAR, are substantially more robust", + "type": "text" + }, + { + "bbox": [ + 397, + 399, + 447, + 410 + ], + "score": 0.86, + "content": "( 6 7 - 1 0 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "to changes in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "the underlying predictive models compared to those generated using state-of-the-art recourse finding", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "score": 1.0, + "content": "technqiues. We also find that our framework achieves such a high degree of robustness without", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "sacrificing the validity of the recourses w.r.t. the original predictive model or substantially increasing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 441, + 302, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 302, + 455 + ], + "score": 1.0, + "content": "the costs associated with realizing the recourses.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 107, + 469, + 197, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 198, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 198, + 484 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 491, + 503, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "Our work lies at the intersection of algorithmic recourse and adversarial robustness. Below, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 502, + 326, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 502, + 326, + 515 + ], + "score": 1.0, + "content": "discuss related work pertaining to each of these topics.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "Algorithmic recourse As discussed in Section 1, several approaches have been proposed to con-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "struct algorithmic recourse for predictive models [31, 26, 12, 21, 18, 3, 14, 15, 7]. These approaches", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "can be broadly characterized along the following dimensions [29]: the level of access they require", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "to the underlying predictive model (black box vs. gradients), if and how they enforce sparsity (only", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "a small number of features should be changed) in counterfactuals, if counterfactuals are required", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "to lie on the data manifold or not, if underlying causal relationships should be accounted for when", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "generating counterfactuals or not, whether the output should be multiple diverse counterfactuals", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "or just a single counterfactual. While the aforementioned approaches have focused on generating", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "instance level counterfactuals, there has also been some recent work on generating global summaries", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "of model recourses which can be leveraged to audit ML methods [23]. More recently, Rawal et al.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "[24] demonstrated that recourses generated by state-of-the-art algorithms are readily invalidated due", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "to model shifts resulting from different kinds of dataset shifts. They argued that model updation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "is very common place in the real world, and it is important to ensure that recourses provided to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "affected individuals are robust to such updates. Similar arguments have been echoed in several other", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "recent works [28, 13, 20]. While there has been some recent work that explores the construction of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "other kinds of explanations (feature attribution and rule based explanations) that are robust to dataset", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "shifts [16], our work makes the first attempt at tackling the problem of constructing recourses that are", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 711, + 198, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 198, + 722 + ], + "score": 1.0, + "content": "robust to model shifts.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 44.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 105, + 72, + 506, + 183 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "In this work, we propose a novel algorithmic framework, RObust Algorithmic Recourse (ROAR)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "for generating instance level recourses (counterfactual explanations) that are robust to changes in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "the underlying predictive model. 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Our theoretical results further establish the need for approaches", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 404, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 404, + 335 + ], + "score": 1.0, + "content": "like ours that generate actionable recourses that are robust to model shifts.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 192, + 506, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 507, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 507, + 357 + ], + "score": 1.0, + "content": "We evaluated our approach ROAR on real world data from financial lending and education domains,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "focusing on model shifts induced by the following kinds of distribution shifts – data correction shift,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "temporal shift, and geospatial shift. We also experimented with synthetic data to analyze how the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "degree of data distribution shifts and consequent model shifts affect the robustness and validity of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "recourses output by our framework as well as the baselines. Our results demonstrate that the recourses", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 397, + 411 + ], + "score": 1.0, + "content": "constructed using our framework, ROAR, are substantially more robust", + "type": "text" + }, + { + "bbox": [ + 397, + 399, + 447, + 410 + ], + "score": 0.86, + "content": "( 6 7 - 1 0 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "to changes in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "the underlying predictive models compared to those generated using state-of-the-art recourse finding", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "score": 1.0, + "content": "technqiues. We also find that our framework achieves such a high degree of robustness without", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "sacrificing the validity of the recourses w.r.t. the original predictive model or substantially increasing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 441, + 302, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 302, + 455 + ], + "score": 1.0, + "content": "the costs associated with realizing the recourses.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 343, + 507, + 455 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 469, + 197, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 198, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 198, + 484 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 491, + 503, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "Our work lies at the intersection of algorithmic recourse and adversarial robustness. Below, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 502, + 326, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 502, + 326, + 515 + ], + "score": 1.0, + "content": "discuss related work pertaining to each of these topics.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 490, + 505, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "Algorithmic recourse As discussed in Section 1, several approaches have been proposed to con-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "struct algorithmic recourse for predictive models [31, 26, 12, 21, 18, 3, 14, 15, 7]. These approaches", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "can be broadly characterized along the following dimensions [29]: the level of access they require", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "to the underlying predictive model (black box vs. gradients), if and how they enforce sparsity (only", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "a small number of features should be changed) in counterfactuals, if counterfactuals are required", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "to lie on the data manifold or not, if underlying causal relationships should be accounted for when", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "generating counterfactuals or not, whether the output should be multiple diverse counterfactuals", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "or just a single counterfactual. While the aforementioned approaches have focused on generating", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "instance level counterfactuals, there has also been some recent work on generating global summaries", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "of model recourses which can be leveraged to audit ML methods [23]. More recently, Rawal et al.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "[24] demonstrated that recourses generated by state-of-the-art algorithms are readily invalidated due", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "to model shifts resulting from different kinds of dataset shifts. They argued that model updation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "is very common place in the real world, and it is important to ensure that recourses provided to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "affected individuals are robust to such updates. Similar arguments have been echoed in several other", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "recent works [28, 13, 20]. While there has been some recent work that explores the construction of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "other kinds of explanations (feature attribution and rule based explanations) that are robust to dataset", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "shifts [16], our work makes the first attempt at tackling the problem of constructing recourses that are", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 711, + 198, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 198, + 722 + ], + "score": 1.0, + "content": "robust to model shifts.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 525, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 507, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 507, + 85 + ], + "score": 1.0, + "content": "Adversarial Robustness The techniques that we leverage in this work are inspired by the adversar-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "ial robustness literature. Wachter et al. were the first to remark on similarities between counterfactual", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "generation and adversarial attacks, but did not leverage this connection to develop robust recourse", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "[31]. It is now well established that ML models are vulnerable to adversarial attacks [10, 4, 2]. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "adversarial training procedure was recently proposed as a defense against such attacks [19, 1, 32].", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "This procedure optimizes a minimax objective that captures the worst-case loss over a given set of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "perturbations to the input data. At a high level, it is based on gradient descent; at each gradient step,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "it solves an optimization problem to find the worst-case perturbation, and then computes the gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "at this perturbation. In contrast, our training procedure optimizes a minimax objective that captures", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "score": 1.0, + "content": "the worst-case over a given set of model perturbations (thereby simulating model shift) and generates", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "recourses that are valid under the corresponding model shifts. This training procedure is novel and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 193, + 239, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 239, + 205 + ], + "score": 1.0, + "content": "possibly of independent interest.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 106, + 214, + 372, + 228 + ], + "lines": [ + { + "bbox": [ + 104, + 213, + 374, + 231 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 374, + 231 + ], + "score": 1.0, + "content": "3 Our Framework: RObust Algorithmic Recourse", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "In this section, we detail our framework, RObust Algorithmic Recourse (ROAR). First, we introduce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "some notation and discuss preliminary details about the algorithmic recourse problem setting. We then", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 256, + 476, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 476, + 268 + ], + "score": 1.0, + "content": "introduce our objective function, and discuss how to operationalize and optimize it efficiently.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 275, + 188, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 189, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 189, + 289 + ], + "score": 1.0, + "content": "3.1 Preliminaries", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 292, + 303 + ], + "score": 1.0, + "content": "Let us assume we are given a predictive model", + "type": "text" + }, + { + "bbox": [ + 292, + 290, + 342, + 301 + ], + "score": 0.91, + "content": "\\mathcal { M } : \\mathcal { X } \\xrightarrow { } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 289, + 372, + 303 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 289, + 407, + 301 + ], + "score": 0.91, + "content": "\\mathcal { X } \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "is the feature space, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 107, + 301, + 115, + 312 + ], + "score": 0.81, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 300, + 233, + 314 + ], + "score": 1.0, + "content": "is the space of outcomes. Let", + "type": "text" + }, + { + "bbox": [ + 233, + 301, + 279, + 313 + ], + "score": 0.95, + "content": "\\mathcal { V } = \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "where 0 and 1 denote an unfavorable outcome (e.g., loan", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 390, + 324 + ], + "score": 1.0, + "content": "denied) and a favorable outcome (e.g., loan approved) respectively. Let", + "type": "text" + }, + { + "bbox": [ + 391, + 312, + 418, + 322 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "be an instance which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 241, + 335 + ], + "score": 1.0, + "content": "received a negative outcome i.e.,", + "type": "text" + }, + { + "bbox": [ + 241, + 323, + 286, + 335 + ], + "score": 0.93, + "content": "\\mathcal { M } ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 322, + 496, + 335 + ], + "score": 1.0, + "content": ". The goal here is to find a recourse for this instance", + "type": "text" + }, + { + "bbox": [ + 497, + 324, + 504, + 333 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 244, + 346 + ], + "score": 1.0, + "content": "i.e., to determine a set of changes", + "type": "text" + }, + { + "bbox": [ + 244, + 336, + 250, + 344 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 333, + 331, + 346 + ], + "score": 1.0, + "content": "that can be made to", + "type": "text" + }, + { + "bbox": [ + 332, + 336, + 339, + 344 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "in order to reverse the negative outcome.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 260, + 358 + ], + "score": 1.0, + "content": "The problem of finding a recourse for", + "type": "text" + }, + { + "bbox": [ + 261, + 347, + 267, + 354 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 344, + 402, + 358 + ], + "score": 1.0, + "content": "involves finding a counterfactual", + "type": "text" + }, + { + "bbox": [ + 403, + 345, + 448, + 355 + ], + "score": 0.94, + "content": "x ^ { \\prime } = x + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "for which the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 354, + 381, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 274, + 369 + ], + "score": 1.0, + "content": "black box outputs a positive outcome i.e.,", + "type": "text" + }, + { + "bbox": [ + 275, + 355, + 377, + 367 + ], + "score": 0.92, + "content": "\\mathcal { M } ( x ^ { \\prime } ) = \\mathcal { M } \\bar { ( } x + \\epsilon ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 354, + 381, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 434, + 384 + ], + "score": 1.0, + "content": "There are, however, a few important considerations when finding the counterfactual", + "type": "text" + }, + { + "bbox": [ + 435, + 372, + 480, + 383 + ], + "score": 0.91, + "content": "x ^ { \\prime } = x + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". First,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 371, + 395 + ], + "score": 1.0, + "content": "it is desirable to minimize the cost (or effort) required to change", + "type": "text" + }, + { + "bbox": [ + 372, + 385, + 379, + 393 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 383, + 390, + 395 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 391, + 383, + 400, + 393 + ], + "score": 0.85, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 383, + 506, + 395 + ], + "score": 1.0, + "content": ". To formalize this, let us", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 204, + 408 + ], + "score": 1.0, + "content": "consider a cost function", + "type": "text" + }, + { + "bbox": [ + 204, + 394, + 274, + 406 + ], + "score": 0.87, + "content": "c : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 393, + 279, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 279, + 394, + 310, + 406 + ], + "score": 0.91, + "content": "c ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "denotes the cost (or effort) incurred in changing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 153, + 417 + ], + "score": 1.0, + "content": "an instance", + "type": "text" + }, + { + "bbox": [ + 153, + 407, + 160, + 415 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 405, + 171, + 417 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 171, + 405, + 181, + 415 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 405, + 417, + 417 + ], + "score": 1.0, + "content": ". In practice, some of the commonly used cost functions are", + "type": "text" + }, + { + "bbox": [ + 417, + 406, + 427, + 416 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 405, + 439, + 417 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 439, + 405, + 449, + 416 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "distance [31],", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "log-percentile shift [26], and costs learned from pairwise feature comparisons input by end users [23].", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "Furthermore, since recommendations to change features such as gender or race would be unactionable,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "it is important to restrict the search for counterfactuals in such a way that only actionable changes are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 396, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 159, + 461 + ], + "score": 1.0, + "content": "allowed. Let", + "type": "text" + }, + { + "bbox": [ + 159, + 449, + 168, + 459 + ], + "score": 0.78, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 448, + 396, + 461 + ], + "score": 1.0, + "content": "denote the set of plausible or actionable counterfactuals.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 504, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 381, + 478 + ], + "score": 1.0, + "content": "Putting it all together, the problem of finding a recourse for instance", + "type": "text" + }, + { + "bbox": [ + 381, + 468, + 388, + 475 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 464, + 430, + 478 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 431, + 465, + 475, + 478 + ], + "score": 0.93, + "content": "\\mathcal { M } ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 476, + 166, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 166, + 488 + ], + "score": 1.0, + "content": "formalized as:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 483, + 388, + 506 + ], + "lines": [ + { + "bbox": [ + 223, + 483, + 388, + 506 + ], + "spans": [ + { + "bbox": [ + 223, + 483, + 388, + 506 + ], + "score": 0.93, + "content": "x ^ { \\prime } = \\underset { x ^ { \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } c ( x , x ^ { \\prime } ) \\quad \\mathrm { s . t } \\quad \\mathcal { M } ( x ^ { \\prime } ) = 1", + "type": "interline_equation", + "image_path": "1d9289c21d881080b115aaea13075eadddbf6e7a8c5cc2e54cc08ae4750819c6.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 223, + 483, + 388, + 506 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Eqn. 1 captures the generic formulation leveraged by several of the state-of-the-art recourse finding", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "algorithms. Typically, most approaches optimize the unconstrained and differentiable relaxation of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 534, + 226, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 226, + 546 + ], + "score": 1.0, + "content": "Eqn. 1 which is given below:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 550, + 385, + 573 + ], + "lines": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "spans": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "score": 0.93, + "content": "\\boldsymbol { x } ^ { \\prime } = \\underset { \\boldsymbol { x } ^ { \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\ell ( \\boldsymbol { \\mathcal { M } } ( \\boldsymbol { x } ^ { \\prime } ) , 1 ) + \\lambda \\boldsymbol { c } ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )", + "type": "interline_equation", + "image_path": "acb4c8e7aafc4daaac0921d7954a1dc9584ea329c0078b0299a009725d539ae5.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 134, + 591 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 578, + 205, + 590 + ], + "score": 0.91, + "content": "\\ell : \\mathcal { V } \\times \\mathcal { V } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "denotes a differentiable loss function (e.g., binary cross entropy) which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 212, + 601 + ], + "score": 1.0, + "content": "ensures that gap between", + "type": "text" + }, + { + "bbox": [ + 212, + 589, + 241, + 601 + ], + "score": 0.92, + "content": "\\mathcal { M } ( \\boldsymbol { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 590, + 422, + 601 + ], + "score": 1.0, + "content": "and favorable outcome 1 is minimized, and", + "type": "text" + }, + { + "bbox": [ + 423, + 590, + 449, + 600 + ], + "score": 0.9, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 590, + 506, + 601 + ], + "score": 1.0, + "content": "is a trade-off", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 601, + 151, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 151, + 612 + ], + "score": 1.0, + "content": "parameter.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 247, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 248, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 248, + 633 + ], + "score": 1.0, + "content": "3.2 Formulating Our Objective", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "As can be seen from Eqn. 2, state-of-the-art recourse finding algorithms rely heavily on the assumption", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 250, + 658 + ], + "score": 1.0, + "content": "that the underlying predictive model", + "type": "text" + }, + { + "bbox": [ + 250, + 646, + 263, + 655 + ], + "score": 0.76, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "does not change. However, predictive models deployed in the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "real world often get updated. This implies that individuals who have acted upon previously prescribed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "recourses are no longer guaranteed a favorable outcome once the model is updated. To address this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "critical challenge, we propose a novel minimax objective function which generates counterfactuals", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "that minimize the worst-case loss over plausible model shifts. We arrived at this approach after", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "considering the following alternatives: (a) Update the predictive model as desired but ensure that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "individuals who were previously prescribed recourse will still be guaranteed a favorable outcome. (b)", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 507, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 507, + 85 + ], + "score": 1.0, + "content": "Adversarial Robustness The techniques that we leverage in this work are inspired by the adversar-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "ial robustness literature. Wachter et al. were the first to remark on similarities between counterfactual", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "generation and adversarial attacks, but did not leverage this connection to develop robust recourse", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "[31]. It is now well established that ML models are vulnerable to adversarial attacks [10, 4, 2]. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "adversarial training procedure was recently proposed as a defense against such attacks [19, 1, 32].", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "This procedure optimizes a minimax objective that captures the worst-case loss over a given set of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "perturbations to the input data. At a high level, it is based on gradient descent; at each gradient step,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "it solves an optimization problem to find the worst-case perturbation, and then computes the gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "at this perturbation. In contrast, our training procedure optimizes a minimax objective that captures", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "score": 1.0, + "content": "the worst-case over a given set of model perturbations (thereby simulating model shift) and generates", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "recourses that are valid under the corresponding model shifts. This training procedure is novel and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 193, + 239, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 239, + 205 + ], + "score": 1.0, + "content": "possibly of independent interest.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 73, + 507, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 214, + 372, + 228 + ], + "lines": [ + { + "bbox": [ + 104, + 213, + 374, + 231 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 374, + 231 + ], + "score": 1.0, + "content": "3 Our Framework: RObust Algorithmic Recourse", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "In this section, we detail our framework, RObust Algorithmic Recourse (ROAR). First, we introduce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "some notation and discuss preliminary details about the algorithmic recourse problem setting. We then", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 256, + 476, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 476, + 268 + ], + "score": 1.0, + "content": "introduce our objective function, and discuss how to operationalize and optimize it efficiently.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 234, + 505, + 268 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 275, + 188, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 189, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 189, + 289 + ], + "score": 1.0, + "content": "3.1 Preliminaries", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 292, + 303 + ], + "score": 1.0, + "content": "Let us assume we are given a predictive model", + "type": "text" + }, + { + "bbox": [ + 292, + 290, + 342, + 301 + ], + "score": 0.91, + "content": "\\mathcal { M } : \\mathcal { X } \\xrightarrow { } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 289, + 372, + 303 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 289, + 407, + 301 + ], + "score": 0.91, + "content": "\\mathcal { X } \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "is the feature space, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 107, + 301, + 115, + 312 + ], + "score": 0.81, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 300, + 233, + 314 + ], + "score": 1.0, + "content": "is the space of outcomes. Let", + "type": "text" + }, + { + "bbox": [ + 233, + 301, + 279, + 313 + ], + "score": 0.95, + "content": "\\mathcal { V } = \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "where 0 and 1 denote an unfavorable outcome (e.g., loan", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 390, + 324 + ], + "score": 1.0, + "content": "denied) and a favorable outcome (e.g., loan approved) respectively. Let", + "type": "text" + }, + { + "bbox": [ + 391, + 312, + 418, + 322 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "be an instance which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 241, + 335 + ], + "score": 1.0, + "content": "received a negative outcome i.e.,", + "type": "text" + }, + { + "bbox": [ + 241, + 323, + 286, + 335 + ], + "score": 0.93, + "content": "\\mathcal { M } ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 322, + 496, + 335 + ], + "score": 1.0, + "content": ". The goal here is to find a recourse for this instance", + "type": "text" + }, + { + "bbox": [ + 497, + 324, + 504, + 333 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 244, + 346 + ], + "score": 1.0, + "content": "i.e., to determine a set of changes", + "type": "text" + }, + { + "bbox": [ + 244, + 336, + 250, + 344 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 333, + 331, + 346 + ], + "score": 1.0, + "content": "that can be made to", + "type": "text" + }, + { + "bbox": [ + 332, + 336, + 339, + 344 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "in order to reverse the negative outcome.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 260, + 358 + ], + "score": 1.0, + "content": "The problem of finding a recourse for", + "type": "text" + }, + { + "bbox": [ + 261, + 347, + 267, + 354 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 344, + 402, + 358 + ], + "score": 1.0, + "content": "involves finding a counterfactual", + "type": "text" + }, + { + "bbox": [ + 403, + 345, + 448, + 355 + ], + "score": 0.94, + "content": "x ^ { \\prime } = x + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "for which the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 354, + 381, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 274, + 369 + ], + "score": 1.0, + "content": "black box outputs a positive outcome i.e.,", + "type": "text" + }, + { + "bbox": [ + 275, + 355, + 377, + 367 + ], + "score": 0.92, + "content": "\\mathcal { M } ( x ^ { \\prime } ) = \\mathcal { M } \\bar { ( } x + \\epsilon ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 354, + 381, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 289, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 434, + 384 + ], + "score": 1.0, + "content": "There are, however, a few important considerations when finding the counterfactual", + "type": "text" + }, + { + "bbox": [ + 435, + 372, + 480, + 383 + ], + "score": 0.91, + "content": "x ^ { \\prime } = x + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". First,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 371, + 395 + ], + "score": 1.0, + "content": "it is desirable to minimize the cost (or effort) required to change", + "type": "text" + }, + { + "bbox": [ + 372, + 385, + 379, + 393 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 383, + 390, + 395 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 391, + 383, + 400, + 393 + ], + "score": 0.85, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 383, + 506, + 395 + ], + "score": 1.0, + "content": ". To formalize this, let us", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 204, + 408 + ], + "score": 1.0, + "content": "consider a cost function", + "type": "text" + }, + { + "bbox": [ + 204, + 394, + 274, + 406 + ], + "score": 0.87, + "content": "c : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 393, + 279, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 279, + 394, + 310, + 406 + ], + "score": 0.91, + "content": "c ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "denotes the cost (or effort) incurred in changing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 153, + 417 + ], + "score": 1.0, + "content": "an instance", + "type": "text" + }, + { + "bbox": [ + 153, + 407, + 160, + 415 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 405, + 171, + 417 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 171, + 405, + 181, + 415 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 405, + 417, + 417 + ], + "score": 1.0, + "content": ". 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Typically, most approaches optimize the unconstrained and differentiable relaxation of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 534, + 226, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 226, + 546 + ], + "score": 1.0, + "content": "Eqn. 1 which is given below:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 511, + 506, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 550, + 385, + 573 + ], + "lines": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "spans": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "score": 0.93, + "content": "\\boldsymbol { x } ^ { \\prime } = \\underset { \\boldsymbol { x } ^ { \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\ell ( \\boldsymbol { \\mathcal { M } } ( \\boldsymbol { x } ^ { \\prime } ) , 1 ) + \\lambda \\boldsymbol { c } ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )", + "type": "interline_equation", + "image_path": "acb4c8e7aafc4daaac0921d7954a1dc9584ea329c0078b0299a009725d539ae5.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 226, + 550, + 385, + 573 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 134, + 591 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 578, + 205, + 590 + ], + "score": 0.91, + "content": "\\ell : \\mathcal { V } \\times \\mathcal { V } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "denotes a differentiable loss function (e.g., binary cross entropy) which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 212, + 601 + ], + "score": 1.0, + "content": "ensures that gap between", + "type": "text" + }, + { + "bbox": [ + 212, + 589, + 241, + 601 + ], + "score": 0.92, + "content": "\\mathcal { M } ( \\boldsymbol { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 590, + 422, + 601 + ], + "score": 1.0, + "content": "and favorable outcome 1 is minimized, and", + "type": "text" + }, + { + "bbox": [ + 423, + 590, + 449, + 600 + ], + "score": 0.9, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 590, + 506, + 601 + ], + "score": 1.0, + "content": "is a trade-off", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 601, + 151, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 151, + 612 + ], + "score": 1.0, + "content": "parameter.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 577, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 247, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 248, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 248, + 633 + ], + "score": 1.0, + "content": "3.2 Formulating Our Objective", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "As can be seen from Eqn. 2, state-of-the-art recourse finding algorithms rely heavily on the assumption", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 250, + 658 + ], + "score": 1.0, + "content": "that the underlying predictive model", + "type": "text" + }, + { + "bbox": [ + 250, + 646, + 263, + 655 + ], + "score": 0.76, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "does not change. However, predictive models deployed in the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "real world often get updated. This implies that individuals who have acted upon previously prescribed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "recourses are no longer guaranteed a favorable outcome once the model is updated. To address this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "critical challenge, we propose a novel minimax objective function which generates counterfactuals", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "that minimize the worst-case loss over plausible model shifts. We arrived at this approach after", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "considering the following alternatives: (a) Update the predictive model as desired but ensure that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "individuals who were previously prescribed recourse will still be guaranteed a favorable outcome. (b)", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 634, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "Update the predictive model while including constraints to ensure that previously offered recourses", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "are still valid. Note that both of these scenarios would potentially incur huge monetary losses to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "relevant stakeholders (e.g, banks), hurting the adoption of these approaches. In case (a), banks may", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "be required to guarantee credit to customers that are potentially not creditworthy under the new model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "score": 1.0, + "content": "and thereby risk losing money. In case (b), access to model training is assumed. Furthermore, training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "a predictive model under these constraints may be suboptimal and not reflective of the current data", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "distribution, thereby accruing larger errors under the shifted population. There are no incentives for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "stakeholders such as banks to adopt such practices which could potentially lead to huge monetary", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "losses. While the optimal approach may vary on a case by case basis, we propose our method to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "score": 1.0, + "content": "avoid the aforementioned pitfalls outlined in cases (a) and (b).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 264, + 199 + ], + "score": 1.0, + "content": "To formalize our proposed approach, let", + "type": "text" + }, + { + "bbox": [ + 264, + 188, + 274, + 198 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 187, + 458, + 199 + ], + "score": 1.0, + "content": "denote the set of plausible model shifts and let", + "type": "text" + }, + { + "bbox": [ + 458, + 188, + 475, + 199 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 198, + 211 + ], + "score": 1.0, + "content": "a shifted model where", + "type": "text" + }, + { + "bbox": [ + 199, + 199, + 225, + 209 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 198, + 446, + 211 + ], + "score": 1.0, + "content": ". Our objective function for generating robust recourse", + "type": "text" + }, + { + "bbox": [ + 446, + 199, + 458, + 208 + ], + "score": 0.87, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "for a given", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 223, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 141, + 221 + ], + "score": 1.0, + "content": "instance", + "type": "text" + }, + { + "bbox": [ + 142, + 212, + 149, + 219 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 209, + 223, + 221 + ], + "score": 1.0, + "content": "can be written as:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 224, + 399, + 246 + ], + "lines": [ + { + "bbox": [ + 211, + 224, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 211, + 224, + 399, + 246 + ], + "score": 0.93, + "content": "\\boldsymbol { x } ^ { \\prime \\prime } = \\underset { \\boldsymbol { x } ^ { \\prime \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\ \\underset { \\delta \\in \\Delta } { \\operatorname* { m a x } } \\ell ( \\boldsymbol { \\mathcal { M } } _ { \\delta } ( \\boldsymbol { x } ^ { \\prime \\prime } ) , 1 ) + \\lambda c ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime \\prime } )", + "type": "interline_equation", + "image_path": "00ef4fa002210e4c38b82f3fc94e54c401f680e4fb5f603fc1d76b882c8d4af2.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 211, + 224, + 399, + 246 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 386, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 387, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 187, + 264 + ], + "score": 1.0, + "content": "where cost function", + "type": "text" + }, + { + "bbox": [ + 187, + 254, + 193, + 262 + ], + "score": 0.78, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 251, + 264, + 264 + ], + "score": 1.0, + "content": "and loss function", + "type": "text" + }, + { + "bbox": [ + 265, + 253, + 269, + 262 + ], + "score": 0.76, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 251, + 387, + 264 + ], + "score": 1.0, + "content": "are as defined in Section 3.1.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 149, + 282 + ], + "score": 1.0, + "content": "Choice of", + "type": "text" + }, + { + "bbox": [ + 150, + 271, + 159, + 280 + ], + "score": 0.75, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "Predictive models deployed in the real world are often updated regularly to handle", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "data distribution shifts [22]. Since these models are updated regularly, it is likely that they undergo", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "small (and not drastic) shifts each time they are updated. To capture this intuition, we consider the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 303, + 354, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 340, + 315 + ], + "score": 1.0, + "content": "following two choices for the set of plausible model shifts", + "type": "text" + }, + { + "bbox": [ + 340, + 304, + 350, + 313 + ], + "score": 0.83, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 303, + 354, + 315 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 317, + 312, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 312, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 312, + 351 + ], + "score": 0.63, + "content": "\\begin{array} { r l } & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\delta _ { m i n } \\leq \\delta _ { i } \\leq \\delta _ { m a x } \\forall i \\in \\{ 1 \\cdots n \\} \\} . } \\\\ & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\| \\delta \\| _ { p } \\leq \\delta _ { m a x } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "1cb30dc22d7865eaec0e1c84e8ca40098fbb84971f7235de1182ee5dbdfc1ea0.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 105, + 317, + 312, + 334.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 105, + 334.0, + 312, + 351.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 132, + 366 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 354, + 157, + 365 + ], + "score": 0.91, + "content": "p \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 353, + 255, + 366 + ], + "score": 1.0, + "content": ". Note that perturbations", + "type": "text" + }, + { + "bbox": [ + 255, + 354, + 282, + 364 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "can be considered as operations either on the parameter", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 247, + 376 + ], + "score": 1.0, + "content": "space or on the gradient space of", + "type": "text" + }, + { + "bbox": [ + 248, + 365, + 261, + 374 + ], + "score": 0.82, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 364, + 372, + 376 + ], + "score": 1.0, + "content": ". While the first choice of", + "type": "text" + }, + { + "bbox": [ + 372, + 365, + 381, + 375 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "presented above allows us to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "restrict model shifts within a small range, the second choice allows us to restrict model shifts within a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 385, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 483, + 401 + ], + "score": 1.0, + "content": "norm-ball. Alternate formulations of the first include incorporating domain knowledge to set", + "type": "text" + }, + { + "bbox": [ + 483, + 387, + 504, + 398 + ], + "score": 0.89, + "content": "\\delta _ { m i n }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 123, + 410 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 397, + 145, + 408 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "per feature. These kinds of shifts can effectively capture small changes to both parameters", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "(e.g., weights of linear models) as well as gradients. Next, we describe how to optimize the objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "score": 1.0, + "content": "in Eqn. 3 and construct robust recourses.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 243, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 243, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 243, + 453 + ], + "score": 1.0, + "content": "3.3 Optimizing Our Objective", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 506, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 273, + 466 + ], + "score": 1.0, + "content": "While our objective function, the choice of", + "type": "text" + }, + { + "bbox": [ + 273, + 454, + 282, + 464 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 453, + 370, + 466 + ], + "score": 1.0, + "content": ", and the perturbations", + "type": "text" + }, + { + "bbox": [ + 370, + 453, + 396, + 464 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "we introduce in Section 3.2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "are generic enough to handle shifts to both parameter space as well as the gradient space of any class", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 191, + 488 + ], + "score": 1.0, + "content": "of predictive models", + "type": "text" + }, + { + "bbox": [ + 191, + 475, + 204, + 485 + ], + "score": 0.79, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 475, + 407, + 488 + ], + "score": 1.0, + "content": ", we solve our objective for a linear approximation", + "type": "text" + }, + { + "bbox": [ + 407, + 475, + 415, + 487 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 475, + 427, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 427, + 475, + 440, + 485 + ], + "score": 0.78, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 475, + 505, + 488 + ], + "score": 1.0, + "content": ". The procedure", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "that we outline here remains generalizable even for non-linear models because local behavior of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "a given non-linear model can be approximated well by fitting a local linear model [25]. Note that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 507, + 507, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 507, + 520 + ], + "score": 1.0, + "content": "such approximations have already been explored by existing algorithmic recourse methods [26, 23].", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 314, + 531 + ], + "score": 1.0, + "content": "Let the linear approximation, which we denote by", + "type": "text" + }, + { + "bbox": [ + 314, + 519, + 322, + 530 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 518, + 408, + 531 + ], + "score": 1.0, + "content": "be parameterized by", + "type": "text" + }, + { + "bbox": [ + 409, + 519, + 442, + 529 + ], + "score": 0.9, + "content": "w \\in \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 518, + 506, + 531 + ], + "score": 1.0, + "content": ". We make this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 331, + 543 + ], + "score": 1.0, + "content": "parametrization explicit by using a subscript notation:", + "type": "text" + }, + { + "bbox": [ + 331, + 530, + 343, + 541 + ], + "score": 0.88, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". We consider model shifts represented", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 282, + 552 + ], + "score": 1.0, + "content": "by perturbations to the model parameters", + "type": "text" + }, + { + "bbox": [ + 283, + 541, + 318, + 551 + ], + "score": 0.91, + "content": "w \\in \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 541, + 505, + 552 + ], + "score": 1.0, + "content": ". In the case of linear models, these can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 267, + 564 + ], + "score": 1.0, + "content": "operationalized as additive perturbations", + "type": "text" + }, + { + "bbox": [ + 267, + 551, + 294, + 561 + ], + "score": 0.91, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 551, + 304, + 564 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 305, + 553, + 313, + 561 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 551, + 506, + 564 + ], + "score": 1.0, + "content": ". We will represent the resulting shifted classifier", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 562, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 119, + 576 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 563, + 141, + 574 + ], + "score": 0.91, + "content": "f _ { w + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 562, + 496, + 576 + ], + "score": 1.0, + "content": ". Our objective function (Eqn. 3) can now be written in terms of this linear approximation", + "type": "text" + }, + { + "bbox": [ + 497, + 563, + 504, + 574 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 575, + 121, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 121, + 585 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 587, + 402, + 610 + ], + "lines": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "spans": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "score": 0.93, + "content": "x ^ { \\prime \\prime } = \\underset { x ^ { \\prime \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\underset { \\delta \\in \\Delta } { \\operatorname* { m a x } } \\ell ( f _ { w + \\delta } ( x ^ { \\prime \\prime } ) , 1 ) + \\lambda c ( x , x ^ { \\prime \\prime } )", + "type": "interline_equation", + "image_path": "770696e463a7193ed4f64b9f7b6f383d8eefc0706c1c46307dbe9a37baeb154d.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "Notice that the objective function defined in Equation 4 is similar to that of adversarial training [19].", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "However, in our framework, the perturbations are applied to model parameters as opposed to data", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "samples. These parallels help motivate the optimization procedure for constructing recourses that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "are robust to model shifts. We outline the optimization procedure that we leverage to optimize our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 282, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 282, + 672 + ], + "score": 1.0, + "content": "minimax objective (Eqn. 4) in Algorithm 1.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 415, + 690 + ], + "score": 1.0, + "content": "Algorithm 1 proceeds in an iterative manner where we first find a perturbation", + "type": "text" + }, + { + "bbox": [ + 415, + 676, + 442, + 688 + ], + "score": 0.92, + "content": "\\hat { \\delta } \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "that maximizes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 366, + 700 + ], + "score": 1.0, + "content": "the chance of invalidating the current estimate of the recourse", + "type": "text" + }, + { + "bbox": [ + 366, + 689, + 378, + 699 + ], + "score": 0.86, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 689, + 506, + 700 + ], + "score": 1.0, + "content": ", and then we take appropriate", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 179, + 712 + ], + "score": 1.0, + "content": "gradient steps on", + "type": "text" + }, + { + "bbox": [ + 179, + 700, + 191, + 710 + ], + "score": 0.9, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "to generate a valid recourse. This procedure is executed iteratively until the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 285, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 285, + 723 + ], + "score": 1.0, + "content": "objective function value (Eqn. 4) converges.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "Update the predictive model while including constraints to ensure that previously offered recourses", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "are still valid. Note that both of these scenarios would potentially incur huge monetary losses to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "relevant stakeholders (e.g, banks), hurting the adoption of these approaches. In case (a), banks may", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "be required to guarantee credit to customers that are potentially not creditworthy under the new model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "score": 1.0, + "content": "and thereby risk losing money. In case (b), access to model training is assumed. Furthermore, training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "a predictive model under these constraints may be suboptimal and not reflective of the current data", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "distribution, thereby accruing larger errors under the shifted population. There are no incentives for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "stakeholders such as banks to adopt such practices which could potentially lead to huge monetary", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "losses. While the optimal approach may vary on a case by case basis, we propose our method to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "score": 1.0, + "content": "avoid the aforementioned pitfalls outlined in cases (a) and (b).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 72, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 264, + 199 + ], + "score": 1.0, + "content": "To formalize our proposed approach, let", + "type": "text" + }, + { + "bbox": [ + 264, + 188, + 274, + 198 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 187, + 458, + 199 + ], + "score": 1.0, + "content": "denote the set of plausible model shifts and let", + "type": "text" + }, + { + "bbox": [ + 458, + 188, + 475, + 199 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 198, + 211 + ], + "score": 1.0, + "content": "a shifted model where", + "type": "text" + }, + { + "bbox": [ + 199, + 199, + 225, + 209 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 198, + 446, + 211 + ], + "score": 1.0, + "content": ". 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Since these models are updated regularly, it is likely that they undergo", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "small (and not drastic) shifts each time they are updated. To capture this intuition, we consider the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 303, + 354, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 340, + 315 + ], + "score": 1.0, + "content": "following two choices for the set of plausible model shifts", + "type": "text" + }, + { + "bbox": [ + 340, + 304, + 350, + 313 + ], + "score": 0.83, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 303, + 354, + 315 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 270, + 505, + 315 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 317, + 312, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 312, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 312, + 351 + ], + "score": 0.63, + "content": "\\begin{array} { r l } & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\delta _ { m i n } \\leq \\delta _ { i } \\leq \\delta _ { m a x } \\forall i \\in \\{ 1 \\cdots n \\} \\} . } \\\\ & { \\Delta = \\{ \\delta \\in \\mathbb { R } ^ { n } \\mid \\| \\delta \\| _ { p } \\leq \\delta _ { m a x } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "1cb30dc22d7865eaec0e1c84e8ca40098fbb84971f7235de1182ee5dbdfc1ea0.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 105, + 317, + 312, + 334.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 105, + 334.0, + 312, + 351.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 132, + 366 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 354, + 157, + 365 + ], + "score": 0.91, + "content": "p \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 353, + 255, + 366 + ], + "score": 1.0, + "content": ". Note that perturbations", + "type": "text" + }, + { + "bbox": [ + 255, + 354, + 282, + 364 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "can be considered as operations either on the parameter", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 247, + 376 + ], + "score": 1.0, + "content": "space or on the gradient space of", + "type": "text" + }, + { + "bbox": [ + 248, + 365, + 261, + 374 + ], + "score": 0.82, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 364, + 372, + 376 + ], + "score": 1.0, + "content": ". While the first choice of", + "type": "text" + }, + { + "bbox": [ + 372, + 365, + 381, + 375 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "presented above allows us to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "restrict model shifts within a small range, the second choice allows us to restrict model shifts within a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 385, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 483, + 401 + ], + "score": 1.0, + "content": "norm-ball. Alternate formulations of the first include incorporating domain knowledge to set", + "type": "text" + }, + { + "bbox": [ + 483, + 387, + 504, + 398 + ], + "score": 0.89, + "content": "\\delta _ { m i n }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 123, + 410 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 397, + 145, + 408 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "per feature. These kinds of shifts can effectively capture small changes to both parameters", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "(e.g., weights of linear models) as well as gradients. Next, we describe how to optimize the objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "score": 1.0, + "content": "in Eqn. 3 and construct robust recourses.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 353, + 506, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 243, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 243, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 243, + 453 + ], + "score": 1.0, + "content": "3.3 Optimizing Our Objective", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 506, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 273, + 466 + ], + "score": 1.0, + "content": "While our objective function, the choice of", + "type": "text" + }, + { + "bbox": [ + 273, + 454, + 282, + 464 + ], + "score": 0.8, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 453, + 370, + 466 + ], + "score": 1.0, + "content": ", and the perturbations", + "type": "text" + }, + { + "bbox": [ + 370, + 453, + 396, + 464 + ], + "score": 0.9, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "we introduce in Section 3.2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "are generic enough to handle shifts to both parameter space as well as the gradient space of any class", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 191, + 488 + ], + "score": 1.0, + "content": "of predictive models", + "type": "text" + }, + { + "bbox": [ + 191, + 475, + 204, + 485 + ], + "score": 0.79, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 475, + 407, + 488 + ], + "score": 1.0, + "content": ", we solve our objective for a linear approximation", + "type": "text" + }, + { + "bbox": [ + 407, + 475, + 415, + 487 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 475, + 427, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 427, + 475, + 440, + 485 + ], + "score": 0.78, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 475, + 505, + 488 + ], + "score": 1.0, + "content": ". The procedure", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "that we outline here remains generalizable even for non-linear models because local behavior of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "a given non-linear model can be approximated well by fitting a local linear model [25]. Note that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 507, + 507, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 507, + 520 + ], + "score": 1.0, + "content": "such approximations have already been explored by existing algorithmic recourse methods [26, 23].", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 314, + 531 + ], + "score": 1.0, + "content": "Let the linear approximation, which we denote by", + "type": "text" + }, + { + "bbox": [ + 314, + 519, + 322, + 530 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 518, + 408, + 531 + ], + "score": 1.0, + "content": "be parameterized by", + "type": "text" + }, + { + "bbox": [ + 409, + 519, + 442, + 529 + ], + "score": 0.9, + "content": "w \\in \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 518, + 506, + 531 + ], + "score": 1.0, + "content": ". We make this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 331, + 543 + ], + "score": 1.0, + "content": "parametrization explicit by using a subscript notation:", + "type": "text" + }, + { + "bbox": [ + 331, + 530, + 343, + 541 + ], + "score": 0.88, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". We consider model shifts represented", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 282, + 552 + ], + "score": 1.0, + "content": "by perturbations to the model parameters", + "type": "text" + }, + { + "bbox": [ + 283, + 541, + 318, + 551 + ], + "score": 0.91, + "content": "w \\in \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 541, + 505, + 552 + ], + "score": 1.0, + "content": ". In the case of linear models, these can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 267, + 564 + ], + "score": 1.0, + "content": "operationalized as additive perturbations", + "type": "text" + }, + { + "bbox": [ + 267, + 551, + 294, + 561 + ], + "score": 0.91, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 551, + 304, + 564 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 305, + 553, + 313, + 561 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 551, + 506, + 564 + ], + "score": 1.0, + "content": ". We will represent the resulting shifted classifier", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 562, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 119, + 576 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 563, + 141, + 574 + ], + "score": 0.91, + "content": "f _ { w + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 562, + 496, + 576 + ], + "score": 1.0, + "content": ". Our objective function (Eqn. 3) can now be written in terms of this linear approximation", + "type": "text" + }, + { + "bbox": [ + 497, + 563, + 504, + 574 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 575, + 121, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 121, + 585 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 453, + 507, + 585 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 587, + 402, + 610 + ], + "lines": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "spans": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "score": 0.93, + "content": "x ^ { \\prime \\prime } = \\underset { x ^ { \\prime \\prime } \\in \\mathcal { A } } { \\arg \\operatorname* { m i n } } \\underset { \\delta \\in \\Delta } { \\operatorname* { m a x } } \\ell ( f _ { w + \\delta } ( x ^ { \\prime \\prime } ) , 1 ) + \\lambda c ( x , x ^ { \\prime \\prime } )", + "type": "interline_equation", + "image_path": "770696e463a7193ed4f64b9f7b6f383d8eefc0706c1c46307dbe9a37baeb154d.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 209, + 587, + 402, + 610 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "Notice that the objective function defined in Equation 4 is similar to that of adversarial training [19].", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "However, in our framework, the perturbations are applied to model parameters as opposed to data", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "samples. These parallels help motivate the optimization procedure for constructing recourses that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "are robust to model shifts. We outline the optimization procedure that we leverage to optimize our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 282, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 282, + 672 + ], + "score": 1.0, + "content": "minimax objective (Eqn. 4) in Algorithm 1.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 614, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 415, + 690 + ], + "score": 1.0, + "content": "Algorithm 1 proceeds in an iterative manner where we first find a perturbation", + "type": "text" + }, + { + "bbox": [ + 415, + 676, + 442, + 688 + ], + "score": 0.92, + "content": "\\hat { \\delta } \\in \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "that maximizes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 366, + 700 + ], + "score": 1.0, + "content": "the chance of invalidating the current estimate of the recourse", + "type": "text" + }, + { + "bbox": [ + 366, + 689, + 378, + 699 + ], + "score": 0.86, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 689, + 506, + 700 + ], + "score": 1.0, + "content": ", and then we take appropriate", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 179, + 712 + ], + "score": 1.0, + "content": "gradient steps on", + "type": "text" + }, + { + "bbox": [ + 179, + 700, + 191, + 710 + ], + "score": 0.9, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "to generate a valid recourse. This procedure is executed iteratively until the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 285, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 285, + 723 + ], + "score": 1.0, + "content": "objective function value (Eqn. 4) converges.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 676, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 72, + 275, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 276, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 276, + 85 + ], + "score": 1.0, + "content": "Algorithm 1 Our Optimization Procedure", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 110, + 87, + 336, + 180 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 110, + 87, + 336, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 87, + 336, + 180 + ], + "spans": [ + { + "bbox": [ + 110, + 87, + 336, + 180 + ], + "score": 0.436, + "html": "
Input:x s.t. fω(x)=O,fw,λ>O,△,learning rate α >0. Initialize x" =x,g =0
repeat = arg maxs∈△ l(fw+8(x"),1)
g =∀[e(fw+8(x"),1)+ λc(x",x)]
x" -=ag
until convergence
Return x"
", + "type": "table", + "image_path": "24f39766558110ab983e25d51c8dd124b8c2c8f4e4da18a29ccb3bd66b6ba126.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 110, + 87, + 336, + 100.28571428571429 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 100.28571428571429, + 336, + 113.57142857142858 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 110, + 113.57142857142858, + 336, + 126.85714285714288 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 110, + 126.85714285714288, + 336, + 140.14285714285717 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 110, + 140.14285714285717, + 336, + 153.42857142857144 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 110, + 153.42857142857144, + 336, + 166.71428571428572 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 110, + 166.71428571428572, + 336, + 180.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 201, + 231, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 231, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 231, + 217 + ], + "score": 1.0, + "content": "4 Theoretical Analysis", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "Here we carry out theoretical analysis to shed light on the benefits of our framework ROAR. More", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "specifically: 1) We quantify the probability that recourses generated without accounting for model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "shifts are likely to be invalidated. 2) We prove that the additional cost incurred due to the robust", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 297, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 297, + 266 + ], + "score": 1.0, + "content": "recourses output by our framework is bounded.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "We first characterize how recourses that do not account for model shifts (i.e., recourses output by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "state-of-the-art algorithms) fare when true model shifts can be characterized as additive shifts to model", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "parameters. Specifically, we quantify the likelihood that recourses generated without accounting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "for model shifts will be invalidated (even if they lie on the original data manifold), under certain", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 316, + 153, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 153, + 328 + ], + "score": 1.0, + "content": "conditions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 506, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 243, + 347 + ], + "score": 1.0, + "content": "Theorem 1. For a given instance", + "type": "text" + }, + { + "bbox": [ + 243, + 334, + 298, + 346 + ], + "score": 0.92, + "content": "x \\sim { \\mathcal { N } } ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 334, + 315, + 347 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 315, + 334, + 324, + 344 + ], + "score": 0.84, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "be the recourse that lies on the original data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 264, + 358 + ], + "score": 1.0, + "content": "manifold (conditioned on the event that", + "type": "text" + }, + { + "bbox": [ + 264, + 346, + 321, + 357 + ], + "score": 0.9, + "content": "\\mathcal { M } ( x ^ { \\prime } ) > 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "and is obtained without accounting for model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 148, + 377 + ], + "score": 1.0, + "content": "shifts. Let", + "type": "text" + }, + { + "bbox": [ + 149, + 359, + 200, + 370 + ], + "score": 0.9, + "content": "\\Sigma = U D U ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 353, + 331, + 377 + ], + "score": 1.0, + "content": ". Then, for some true model shift", + "type": "text" + }, + { + "bbox": [ + 331, + 361, + 337, + 370 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 353, + 383, + 377 + ], + "score": 1.0, + "content": ", such that,", + "type": "text" + }, + { + "bbox": [ + 383, + 357, + 483, + 376 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { w ^ { T } \\mu } { ( w + \\delta ) ^ { T } \\mu } \\geq \\frac { \\| \\sqrt { D } U w \\| } { \\| \\sqrt { D } U ( w + \\delta ) \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 353, + 506, + 377 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 108, + 375, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 108, + 375, + 300, + 397 + ], + "score": 0.53, + "content": "\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq e r f c \\bigl ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } \\bigr ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 375, + 318, + 399 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 318, + 382, + 343, + 394 + ], + "score": 0.82, + "content": "\\beta \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 375, + 426, + 399 + ], + "score": 1.0, + "content": ", the probability that", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 435, + 392 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 375, + 506, + 399 + ], + "score": 1.0, + "content": "is invalidated on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 398, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 405, + 129, + 416 + ], + "score": 0.89, + "content": "f _ { w + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 403, + 173, + 418 + ], + "score": 1.0, + "content": "is at least:", + "type": "text" + }, + { + "bbox": [ + 173, + 398, + 375, + 420 + ], + "score": 0.81, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } - \\frac { 1 } { 2 } e r f c \\big ( { - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 398, + 505, + 418 + ], + "score": 1.0, + "content": "where erfc is the complementary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 205, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 205, + 431 + ], + "score": 1.0, + "content": "gaussian error function.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 272, + 460 + ], + "score": 1.0, + "content": "Proof Sketch. Under the assumption that", + "type": "text" + }, + { + "bbox": [ + 273, + 447, + 330, + 459 + ], + "score": 0.93, + "content": "x ^ { \\prime } \\sim { \\mathcal { N } } ( { \\boldsymbol { \\mu } } , { \\boldsymbol { \\Sigma } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ", a recourse is invalid under a model shift if", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "score": 1.0, + "content": "it is valid under the original model and invalid under the shifted model. This allows us to define the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 249, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 161, + 480 + ], + "score": 1.0, + "content": "region where", + "type": "text" + }, + { + "bbox": [ + 162, + 469, + 171, + 479 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 470, + 249, + 480 + ], + "score": 1.0, + "content": "can be invalidated:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 487, + 388, + 502 + ], + "lines": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "spans": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "score": 0.9, + "content": "\\Omega = \\{ x ^ { \\prime } \\colon w ^ { T } x ^ { \\prime } > 0 \\cap ( w + \\delta ) ^ { T } x ^ { \\prime } \\leq 0 \\}", + "type": "interline_equation", + "image_path": "d0afb414b31db376a47893a7ce3ed8663ffc4e875d7370634a7700050369b9a7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 515, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 185, + 529 + ], + "score": 1.0, + "content": "The probability that", + "type": "text" + }, + { + "bbox": [ + 186, + 516, + 195, + 526 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 515, + 388, + 529 + ], + "score": 1.0, + "content": "is invalidated can be obtained by integrating over", + "type": "text" + }, + { + "bbox": [ + 389, + 517, + 397, + 526 + ], + "score": 0.84, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 515, + 467, + 529 + ], + "score": 1.0, + "content": "under the PDF of", + "type": "text" + }, + { + "bbox": [ + 467, + 516, + 503, + 528 + ], + "score": 0.93, + "content": "\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 515, + 506, + 529 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 199, + 547 + ], + "score": 1.0, + "content": "We can then transform", + "type": "text" + }, + { + "bbox": [ + 200, + 534, + 209, + 544 + ], + "score": 0.87, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 533, + 296, + 547 + ], + "score": 1.0, + "content": "and correspondingly", + "type": "text" + }, + { + "bbox": [ + 296, + 535, + 304, + 544 + ], + "score": 0.83, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 533, + 505, + 547 + ], + "score": 1.0, + "content": ", to simplify this integration over a 1-dimensional", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 546, + 248, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 248, + 558 + ], + "score": 1.0, + "content": "Gaussian random variable. That is,", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 562, + 414, + 592 + ], + "lines": [ + { + "bbox": [ + 198, + 562, + 414, + 592 + ], + "spans": [ + { + "bbox": [ + 198, + 562, + 414, + 592 + ], + "score": 0.92, + "content": "P ( x { \\mathrm { ~ i s ~ i n v a l i d a t e d } } ) = { \\frac { 1 } { \\sqrt { ( 2 \\pi ) } } } \\int _ { c _ { 1 } } ^ { c _ { 2 } } \\exp { \\bigg ( } - { \\frac { 1 } { 2 } } s ^ { 2 } { \\bigg ) } d s", + "type": "interline_equation", + "image_path": "9bd556959a393381c310cc85e81f1ce2ea8b97d0d27005e4406ba056b245fa5f.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 562, + 414, + 577.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 198, + 577.0, + 414, + 592.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "The above quantity can be represented as a difference in the Gaussian error function allowing us to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "exactly quantify the invalidation probability under our assumptions. Using the lower bounds on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "complementary gaussian error function [9] from Chang et al. [5], we obtain our lower bound. To derive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 656, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 307, + 677 + ], + "score": 1.0, + "content": "the lower bound, we add an extra condition that", + "type": "text" + }, + { + "bbox": [ + 307, + 658, + 503, + 680 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq \\mathrm { e r f c } ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 656, + 506, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "mainly to confirm that the lower bound on the first term still dominates the second term. Both", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 465, + 701 + ], + "score": 1.0, + "content": "conditions restricts the types of shift for which the bound can be derived. Note that", + "type": "text" + }, + { + "bbox": [ + 465, + 689, + 473, + 700 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "optimized away to improve the lower bound. Detailed proof is provided in the Appendix. Discussion", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 477, + 723 + ], + "score": 1.0, + "content": "about other distributions (e.g., Bernoulli, Uniform, Categorical) is included in the Appendix.", + "type": "text" + }, + { + "bbox": [ + 494, + 712, + 505, + 721 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 72, + 275, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 276, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 276, + 85 + ], + "score": 1.0, + "content": "Algorithm 1 Our Optimization Procedure", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 110, + 87, + 336, + 180 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 110, + 87, + 336, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 87, + 336, + 180 + ], + "spans": [ + { + "bbox": [ + 110, + 87, + 336, + 180 + ], + "score": 0.436, + "html": "
Input:x s.t. fω(x)=O,fw,λ>O,△,learning rate α >0. Initialize x" =x,g =0
repeat = arg maxs∈△ l(fw+8(x"),1)
g =∀[e(fw+8(x"),1)+ λc(x",x)]
x" -=ag
until convergence
Return x"
", + "type": "table", + "image_path": "24f39766558110ab983e25d51c8dd124b8c2c8f4e4da18a29ccb3bd66b6ba126.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 110, + 87, + 336, + 100.28571428571429 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 100.28571428571429, + 336, + 113.57142857142858 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 110, + 113.57142857142858, + 336, + 126.85714285714288 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 110, + 126.85714285714288, + 336, + 140.14285714285717 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 110, + 140.14285714285717, + 336, + 153.42857142857144 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 110, + 153.42857142857144, + 336, + 166.71428571428572 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 110, + 166.71428571428572, + 336, + 180.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 201, + 231, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 231, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 231, + 217 + ], + "score": 1.0, + "content": "4 Theoretical Analysis", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "Here we carry out theoretical analysis to shed light on the benefits of our framework ROAR. More", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "specifically: 1) We quantify the probability that recourses generated without accounting for model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "shifts are likely to be invalidated. 2) We prove that the additional cost incurred due to the robust", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 297, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 297, + 266 + ], + "score": 1.0, + "content": "recourses output by our framework is bounded.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 222, + 506, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "We first characterize how recourses that do not account for model shifts (i.e., recourses output by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "state-of-the-art algorithms) fare when true model shifts can be characterized as additive shifts to model", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "parameters. Specifically, we quantify the likelihood that recourses generated without accounting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "for model shifts will be invalidated (even if they lie on the original data manifold), under certain", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 316, + 153, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 153, + 328 + ], + "score": 1.0, + "content": "conditions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 271, + 506, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 506, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 243, + 347 + ], + "score": 1.0, + "content": "Theorem 1. For a given instance", + "type": "text" + }, + { + "bbox": [ + 243, + 334, + 298, + 346 + ], + "score": 0.92, + "content": "x \\sim { \\mathcal { N } } ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 334, + 315, + 347 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 315, + 334, + 324, + 344 + ], + "score": 0.84, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "be the recourse that lies on the original data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 264, + 358 + ], + "score": 1.0, + "content": "manifold (conditioned on the event that", + "type": "text" + }, + { + "bbox": [ + 264, + 346, + 321, + 357 + ], + "score": 0.9, + "content": "\\mathcal { M } ( x ^ { \\prime } ) > 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "and is obtained without accounting for model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 148, + 377 + ], + "score": 1.0, + "content": "shifts. Let", + "type": "text" + }, + { + "bbox": [ + 149, + 359, + 200, + 370 + ], + "score": 0.9, + "content": "\\Sigma = U D U ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 353, + 331, + 377 + ], + "score": 1.0, + "content": ". Then, for some true model shift", + "type": "text" + }, + { + "bbox": [ + 331, + 361, + 337, + 370 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 353, + 383, + 377 + ], + "score": 1.0, + "content": ", such that,", + "type": "text" + }, + { + "bbox": [ + 383, + 357, + 483, + 376 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { w ^ { T } \\mu } { ( w + \\delta ) ^ { T } \\mu } \\geq \\frac { \\| \\sqrt { D } U w \\| } { \\| \\sqrt { D } U ( w + \\delta ) \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 353, + 506, + 377 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 108, + 375, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 108, + 375, + 300, + 397 + ], + "score": 0.53, + "content": "\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq e r f c \\bigl ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } \\bigr ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 375, + 318, + 399 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 318, + 382, + 343, + 394 + ], + "score": 0.82, + "content": "\\beta \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 375, + 426, + 399 + ], + "score": 1.0, + "content": ", the probability that", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 435, + 392 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 375, + 506, + 399 + ], + "score": 1.0, + "content": "is invalidated on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 398, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 405, + 129, + 416 + ], + "score": 0.89, + "content": "f _ { w + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 403, + 173, + 418 + ], + "score": 1.0, + "content": "is at least:", + "type": "text" + }, + { + "bbox": [ + 173, + 398, + 375, + 420 + ], + "score": 0.81, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } - \\frac { 1 } { 2 } e r f c \\big ( { - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 398, + 505, + 418 + ], + "score": 1.0, + "content": "where erfc is the complementary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 205, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 205, + 431 + ], + "score": 1.0, + "content": "gaussian error function.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 334, + 506, + 431 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 272, + 460 + ], + "score": 1.0, + "content": "Proof Sketch. Under the assumption that", + "type": "text" + }, + { + "bbox": [ + 273, + 447, + 330, + 459 + ], + "score": 0.93, + "content": "x ^ { \\prime } \\sim { \\mathcal { N } } ( { \\boldsymbol { \\mu } } , { \\boldsymbol { \\Sigma } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ", a recourse is invalid under a model shift if", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "score": 1.0, + "content": "it is valid under the original model and invalid under the shifted model. This allows us to define the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 249, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 161, + 480 + ], + "score": 1.0, + "content": "region where", + "type": "text" + }, + { + "bbox": [ + 162, + 469, + 171, + 479 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 470, + 249, + 480 + ], + "score": 1.0, + "content": "can be invalidated:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 446, + 506, + 480 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 487, + 388, + 502 + ], + "lines": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "spans": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "score": 0.9, + "content": "\\Omega = \\{ x ^ { \\prime } \\colon w ^ { T } x ^ { \\prime } > 0 \\cap ( w + \\delta ) ^ { T } x ^ { \\prime } \\leq 0 \\}", + "type": "interline_equation", + "image_path": "d0afb414b31db376a47893a7ce3ed8663ffc4e875d7370634a7700050369b9a7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 487, + 388, + 502 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 515, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 185, + 529 + ], + "score": 1.0, + "content": "The probability that", + "type": "text" + }, + { + "bbox": [ + 186, + 516, + 195, + 526 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 515, + 388, + 529 + ], + "score": 1.0, + "content": "is invalidated can be obtained by integrating over", + "type": "text" + }, + { + "bbox": [ + 389, + 517, + 397, + 526 + ], + "score": 0.84, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 515, + 467, + 529 + ], + "score": 1.0, + "content": "under the PDF of", + "type": "text" + }, + { + "bbox": [ + 467, + 516, + 503, + 528 + ], + "score": 0.93, + "content": "\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 515, + 506, + 529 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 515, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 199, + 547 + ], + "score": 1.0, + "content": "We can then transform", + "type": "text" + }, + { + "bbox": [ + 200, + 534, + 209, + 544 + ], + "score": 0.87, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 533, + 296, + 547 + ], + "score": 1.0, + "content": "and correspondingly", + "type": "text" + }, + { + "bbox": [ + 296, + 535, + 304, + 544 + ], + "score": 0.83, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 533, + 505, + 547 + ], + "score": 1.0, + "content": ", to simplify this integration over a 1-dimensional", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 546, + 248, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 248, + 558 + ], + "score": 1.0, + "content": "Gaussian random variable. That is,", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 533, + 505, + 558 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 562, + 414, + 592 + ], + "lines": [ + { + "bbox": [ + 198, + 562, + 414, + 592 + ], + "spans": [ + { + "bbox": [ + 198, + 562, + 414, + 592 + ], + "score": 0.92, + "content": "P ( x { \\mathrm { ~ i s ~ i n v a l i d a t e d } } ) = { \\frac { 1 } { \\sqrt { ( 2 \\pi ) } } } \\int _ { c _ { 1 } } ^ { c _ { 2 } } \\exp { \\bigg ( } - { \\frac { 1 } { 2 } } s ^ { 2 } { \\bigg ) } d s", + "type": "interline_equation", + "image_path": "9bd556959a393381c310cc85e81f1ce2ea8b97d0d27005e4406ba056b245fa5f.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 562, + 414, + 577.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 198, + 577.0, + 414, + 592.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "The above quantity can be represented as a difference in the Gaussian error function allowing us to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "exactly quantify the invalidation probability under our assumptions. Using the lower bounds on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "complementary gaussian error function [9] from Chang et al. [5], we obtain our lower bound. To derive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 656, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 307, + 677 + ], + "score": 1.0, + "content": "the lower bound, we add an extra condition that", + "type": "text" + }, + { + "bbox": [ + 307, + 658, + 503, + 680 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sqrt { \\frac { 2 e } { \\pi } } \\frac { \\sqrt { \\beta - 1 } } { \\beta } \\exp ^ { - \\beta \\frac { ( w ^ { T } \\mu ) ^ { 2 } } { 4 \\| \\sqrt { D } U w \\| ^ { 2 } } } \\geq \\mathrm { e r f c } ( - \\frac { ( w + \\delta ) ^ { T } \\mu } { \\sqrt { 2 } \\| w + \\delta \\| } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 656, + 506, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "mainly to confirm that the lower bound on the first term still dominates the second term. Both", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 465, + 701 + ], + "score": 1.0, + "content": "conditions restricts the types of shift for which the bound can be derived. Note that", + "type": "text" + }, + { + "bbox": [ + 465, + 689, + 473, + 700 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "optimized away to improve the lower bound. Detailed proof is provided in the Appendix. Discussion", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 477, + 723 + ], + "score": 1.0, + "content": "about other distributions (e.g., Bernoulli, Uniform, Categorical) is included in the Appendix.", + "type": "text" + }, + { + "bbox": [ + 494, + 712, + 505, + 721 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 623, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "Next we characterize how much more costly recourses can be when they are trained to be robust", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "to model perturbations or model shifts. In the following theorem, we show that the cost of robust", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 466, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 466, + 105 + ], + "score": 1.0, + "content": "recourses is bounded relative to the cost of recourses that do not account for model shifts.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 506, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 176, + 124 + ], + "score": 1.0, + "content": "Theorem 2. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 111, + 205, + 121 + ], + "score": 0.88, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 110, + 229, + 124 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 229, + 113, + 257, + 121 + ], + "score": 0.86, + "content": "x \\sim \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 110, + 286, + 124 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 286, + 113, + 293, + 121 + ], + "score": 0.64, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 110, + 403, + 124 + ], + "score": 1.0, + "content": "is a distribution such that", + "type": "text" + }, + { + "bbox": [ + 403, + 111, + 474, + 123 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\nu } [ x ] = \\mu < \\infty ,", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 110, + 505, + 124 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 122, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 107, + 124, + 117, + 134 + ], + "score": 0.72, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 122, + 192, + 137 + ], + "score": 1.0, + "content": "is a metric space", + "type": "text" + }, + { + "bbox": [ + 192, + 123, + 236, + 136 + ], + "score": 0.9, + "content": "( \\mathcal { X } , d ( \\cdot , \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 122, + 256, + 137 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 123, + 334, + 135 + ], + "score": 0.87, + "content": "d : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 122, + 357, + 137 + ], + "score": 1.0, + "content": ". 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Further assume that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 290, + 182 + ], + "score": 1.0, + "content": "the ROAR objective (Equation 3) is convex in", + "type": "text" + }, + { + "bbox": [ + 291, + 169, + 302, + 180 + ], + "score": 0.85, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 169, + 347, + 182 + ], + "score": 1.0, + "content": "for a fixed", + "type": "text" + }, + { + "bbox": [ + 347, + 170, + 353, + 180 + ], + "score": 0.52, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 169, + 374, + 182 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 374, + 169, + 406, + 182 + ], + "score": 0.81, + "content": "\\ell \\triangleq \\ell _ { l o g }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "(the cross-entropy loss),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 302, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 129, + 193 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 181, + 177, + 192 + ], + "score": 0.88, + "content": "0 < \\eta ^ { \\prime } \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 181, + 208, + 193 + ], + "score": 1.0, + "content": ", w.h.p.", + "type": "text" + }, + { + "bbox": [ + 208, + 181, + 241, + 193 + ], + "score": 0.89, + "content": "( 1 - \\eta ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 181, + 302, + 193 + ], + "score": 1.0, + "content": ", we have that:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 197, + 461, + 231 + ], + "lines": [ + { + "bbox": [ + 150, + 197, + 461, + 231 + ], + "spans": [ + { + "bbox": [ + 150, + 197, + 461, + 231 + ], + "score": 0.94, + "content": "c ( x ^ { \\prime \\prime } , x ) - c ( x ^ { \\prime } , x ) \\leq \\frac { 1 } { \\lambda } \\frac { 1 } { \\| w + \\delta \\| } \\mathbb { E } _ { \\nu } [ \\exp { - \\phi ( w + \\delta ) ^ { T } x ^ { \\prime } } ] + \\sqrt { \\frac { D ^ { 2 } } { 2 } \\log { ( \\frac { 1 } { \\eta ^ { \\prime } } ) } } \\Bigg )", + "type": "interline_equation", + "image_path": "6a767dbbc087322e15f8dc5c11e384ffd6306ad4ac1a745d87e47f35393f257e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 150, + 197, + 461, + 208.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 150, + 208.33333333333334, + 461, + 219.66666666666669 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 150, + 219.66666666666669, + 461, + 231.00000000000003 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 280, + 253 + ], + "score": 1.0, + "content": "Proof Sketch. By definition, any recourse", + "type": "text" + }, + { + "bbox": [ + 280, + 241, + 290, + 251 + ], + "score": 0.85, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "generated without accounting for model shifts will", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 376, + 264 + ], + "score": 1.0, + "content": "have a higher loss for Equation 3 compared to the robust recourse", + "type": "text" + }, + { + "bbox": [ + 376, + 252, + 388, + 262 + ], + "score": 0.86, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "(note that finding the global", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 290, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 290, + 275 + ], + "score": 1.0, + "content": "minimizer is not guaranteed by Algorithm 1).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 279, + 504, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 255, + 291 + ], + "score": 1.0, + "content": "Using this insight, and convexity in", + "type": "text" + }, + { + "bbox": [ + 255, + 279, + 265, + 289 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 279, + 304, + 291 + ], + "score": 1.0, + "content": "for fixed", + "type": "text" + }, + { + "bbox": [ + 304, + 280, + 311, + 289 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 279, + 505, + 291 + ], + "score": 1.0, + "content": ", we can bound the cost difference between the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 290, + 441, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 441, + 302 + ], + "score": 1.0, + "content": "robust and non-robust recourse by a 1-Lipschitz function (i.e. the logistic function):", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 306, + 430, + 332 + ], + "lines": [ + { + "bbox": [ + 181, + 306, + 430, + 332 + ], + "spans": [ + { + "bbox": [ + 181, + 306, + 430, + 332 + ], + "score": 0.92, + "content": "c ( x ^ { \\prime \\prime } , x ) - c ( x ^ { \\prime } , x ) \\leq \\frac { 1 } { \\lambda \\| w + \\delta \\| } \\log \\left\\{ 1 + \\exp - ( w + \\delta ) ^ { T } x ^ { \\prime } \\right\\}", + "type": "interline_equation", + "image_path": "f53f509a541745fdd4e7e0d145bc1a46f5a152eaeb4f2901ef59c4c782abd8c9.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 181, + 306, + 430, + 332 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 235, + 354 + ], + "score": 1.0, + "content": "Assuming a bounded metric on", + "type": "text" + }, + { + "bbox": [ + 235, + 342, + 244, + 352 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 342, + 505, + 354 + ], + "score": 1.0, + "content": ", we can upper bound the RHS using Lemma 2 from van Handel", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 402, + 366 + ], + "score": 1.0, + "content": "[27] which gives us our bound. Detailed proof including special cases when", + "type": "text" + }, + { + "bbox": [ + 403, + 355, + 410, + 363 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "is Gaussian, is provided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 176, + 376 + ], + "score": 1.0, + "content": "in the Appendix.", + "type": "text" + }, + { + "bbox": [ + 494, + 364, + 505, + 375 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 506, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "This result suggests that the additional cost of recourse is bounded by the amount of shift admissible", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "in Equation 3. Note that Theorem 2 applies for general distributions so long as the mean is finite,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 507, + 420 + ], + "score": 1.0, + "content": "which is the case for most commonplace distributions like Gaussian, Bernoulli, Multinomial etc.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "While Theorem 1 demonstrates the probability that a recourse will be invalidated for Gaussian", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 507, + 444 + ], + "score": 1.0, + "content": "distributions, we refer the reader to the Appendix B.1 for a discussion of other distributions, e.g.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 441, + 239, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 239, + 453 + ], + "score": 1.0, + "content": "Bernoulli, Uniform, Categorical.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 192, + 475 + ], + "lines": [ + { + "bbox": [ + 103, + 459, + 193, + 479 + ], + "spans": [ + { + "bbox": [ + 103, + 459, + 193, + 479 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "Here we discuss the detailed experimental evaluation of our framework, ROAR. First, we evaluate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "how robust the recourses generated by our framework are to model shifts caused by real world data", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "distribution shifts. We also assess the validity of the recourses generated by our framework w.r.t. the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "original model, and further analyze the average cost of these recourses. Next, using synthetic data,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "we analyze how varying the degree (magnitude) of data distribution shift impacts the robustness and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 388, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 388, + 548 + ], + "score": 1.0, + "content": "validity of the recourses output by our framework and other baselines.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 554, + 216, + 566 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 217, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 217, + 570 + ], + "score": 1.0, + "content": "5.1 Experimental Setup", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "Real world data We evaluate our framework on model shifts induced by real world data distribution", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "shifts. To this end, we leverage three real world datasets which capture different kinds of data", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "distribution shifts, namely, temporal shift, geospatial shift, and data correction shift [24]. Our first", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 507, + 615 + ], + "score": 1.0, + "content": "dataset is the widely used and publicly available German credit dataset [8] from the UCI repository.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "This dataset captures demographic (age, gender), personal (marital status), and financial (income,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "credit duration) details of about 1000 loan applicants. Each applicant is labeled as either a good", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "customer or a bad customer depending on their credit risk. Two versions of this dataset have been", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "released, with the second version incorporating corrections to coding errors in the first dataset [11].", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Accordingly, this dataset captures the data correction shift. Our second dataset is the Small Business", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Administration (SBA) case dataset [17]. This dataset contains information pertaining to 2102 small", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 393, + 691 + ], + "score": 1.0, + "content": "business loans approved by the state of California during the years of", + "type": "text" + }, + { + "bbox": [ + 394, + 678, + 447, + 689 + ], + "score": 0.48, + "content": "1 9 8 9 - 2 0 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 678, + 506, + 691 + ], + "score": 1.0, + "content": ", and captures", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "temporal shifts in the data. It comprises of about 24 features capturing various details of the small", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "businesses including zip codes, business category (real estate vs. rental vs. leasing), number of jobs", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "created, and financial status of the business. It also contains information about whether a business", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "Next we characterize how much more costly recourses can be when they are trained to be robust", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "to model perturbations or model shifts. In the following theorem, we show that the cost of robust", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 466, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 466, + 105 + ], + "score": 1.0, + "content": "recourses is bounded relative to the cost of recourses that do not account for model shifts.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 73, + 506, + 105 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 506, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 176, + 124 + ], + "score": 1.0, + "content": "Theorem 2. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 111, + 205, + 121 + ], + "score": 0.88, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 110, + 229, + 124 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 229, + 113, + 257, + 121 + ], + "score": 0.86, + "content": "x \\sim \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 110, + 286, + 124 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 286, + 113, + 293, + 121 + ], + "score": 0.64, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 110, + 403, + 124 + ], + "score": 1.0, + "content": "is a distribution such that", + "type": "text" + }, + { + "bbox": [ + 403, + 111, + 474, + 123 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\nu } [ x ] = \\mu < \\infty ,", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 110, + 505, + 124 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 122, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 107, + 124, + 117, + 134 + ], + "score": 0.72, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 122, + 192, + 137 + ], + "score": 1.0, + "content": "is a metric space", + "type": "text" + }, + { + "bbox": [ + 192, + 123, + 236, + 136 + ], + "score": 0.9, + "content": "( \\mathcal { X } , d ( \\cdot , \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 122, + 256, + 137 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 123, + 334, + 135 + ], + "score": 0.87, + "content": "d : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 122, + 357, + 137 + ], + "score": 1.0, + "content": ". 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Let recourses obtained without accounting for model", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 145, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 316, + 159 + ], + "score": 1.0, + "content": "shifts and constrained to the manifold be denoted by", + "type": "text" + }, + { + "bbox": [ + 316, + 146, + 344, + 157 + ], + "score": 0.9, + "content": "x ^ { \\prime } \\sim \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 145, + 491, + 159 + ], + "score": 1.0, + "content": ", and robust recourses be denoted by", + "type": "text" + }, + { + "bbox": [ + 491, + 148, + 502, + 156 + ], + "score": 0.85, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 145, + 506, + 159 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 121, + 170 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 158, + 146, + 168 + ], + "score": 0.88, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 157, + 325, + 170 + ], + "score": 1.0, + "content": "be the shift that maximizes Eq. 3 for sample", + "type": "text" + }, + { + "bbox": [ + 325, + 160, + 332, + 167 + ], + "score": 0.32, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 157, + 405, + 170 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 405, + 158, + 416, + 167 + ], + "score": 0.84, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 157, + 506, + 170 + ], + "score": 1.0, + "content": ". 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Detailed proof including special cases when", + "type": "text" + }, + { + "bbox": [ + 403, + 355, + 410, + 363 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "is Gaussian, is provided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 176, + 376 + ], + "score": 1.0, + "content": "in the Appendix.", + "type": "text" + }, + { + "bbox": [ + 494, + 364, + 505, + 375 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 342, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 506, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "This result suggests that the additional cost of recourse is bounded by the amount of shift admissible", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "in Equation 3. Note that Theorem 2 applies for general distributions so long as the mean is finite,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 507, + 420 + ], + "score": 1.0, + "content": "which is the case for most commonplace distributions like Gaussian, Bernoulli, Multinomial etc.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "While Theorem 1 demonstrates the probability that a recourse will be invalidated for Gaussian", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 507, + 444 + ], + "score": 1.0, + "content": "distributions, we refer the reader to the Appendix B.1 for a discussion of other distributions, e.g.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 441, + 239, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 239, + 453 + ], + "score": 1.0, + "content": "Bernoulli, Uniform, Categorical.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 385, + 507, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 192, + 475 + ], + "lines": [ + { + "bbox": [ + 103, + 459, + 193, + 479 + ], + "spans": [ + { + "bbox": [ + 103, + 459, + 193, + 479 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "Here we discuss the detailed experimental evaluation of our framework, ROAR. First, we evaluate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "how robust the recourses generated by our framework are to model shifts caused by real world data", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "distribution shifts. We also assess the validity of the recourses generated by our framework w.r.t. the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "original model, and further analyze the average cost of these recourses. Next, using synthetic data,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "we analyze how varying the degree (magnitude) of data distribution shift impacts the robustness and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 388, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 388, + 548 + ], + "score": 1.0, + "content": "validity of the recourses output by our framework and other baselines.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 480, + 506, + 548 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 554, + 216, + 566 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 217, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 217, + 570 + ], + "score": 1.0, + "content": "5.1 Experimental Setup", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "Real world data We evaluate our framework on model shifts induced by real world data distribution", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "shifts. To this end, we leverage three real world datasets which capture different kinds of data", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "distribution shifts, namely, temporal shift, geospatial shift, and data correction shift [24]. Our first", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 507, + 615 + ], + "score": 1.0, + "content": "dataset is the widely used and publicly available German credit dataset [8] from the UCI repository.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "This dataset captures demographic (age, gender), personal (marital status), and financial (income,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "credit duration) details of about 1000 loan applicants. Each applicant is labeled as either a good", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "customer or a bad customer depending on their credit risk. Two versions of this dataset have been", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "released, with the second version incorporating corrections to coding errors in the first dataset [11].", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Accordingly, this dataset captures the data correction shift. Our second dataset is the Small Business", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Administration (SBA) case dataset [17]. This dataset contains information pertaining to 2102 small", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 393, + 691 + ], + "score": 1.0, + "content": "business loans approved by the state of California during the years of", + "type": "text" + }, + { + "bbox": [ + 394, + 678, + 447, + 689 + ], + "score": 0.48, + "content": "1 9 8 9 - 2 0 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 678, + 506, + 691 + ], + "score": 1.0, + "content": ", and captures", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "temporal shifts in the data. It comprises of about 24 features capturing various details of the small", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "businesses including zip codes, business category (real estate vs. rental vs. leasing), number of jobs", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "created, and financial status of the business. It also contains information about whether a business", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "has defaulted on a loan or not which we consider as the class label. Our last dataset contains student", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "performance records of 649 students from two Portuguese secondary schools, Gabriel Pereira (GP)", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "and Mousinho da Silveira (MS) [8, 6], and captures geospatial shift. It comprises of information", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "about the academic background (grades, absences, access to internet, failures etc.) of each student", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "along with other demographic attributes (age, gender). Each student is assigned a class label of above", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 290, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 290, + 140 + ], + "score": 1.0, + "content": "average or not depending on their final grade.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 568, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "has defaulted on a loan or not which we consider as the class label. Our last dataset contains student", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "performance records of 649 students from two Portuguese secondary schools, Gabriel Pereira (GP)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "and Mousinho da Silveira (MS) [8, 6], and captures geospatial shift. It comprises of information", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "about the academic background (grades, absences, access to internet, failures etc.) of each student", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "along with other demographic attributes (age, gender). Each student is assigned a class label of above", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 290, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 290, + 140 + ], + "score": 1.0, + "content": "average or not depending on their final grade.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "Synthetic data We generate a synthetic dataset with 1K samples and two dimensions to analyze", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "how the degree (magnitude) of data distribution shifts impacts the robustness and validity of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 381, + 180 + ], + "score": 1.0, + "content": "recourses output by our framework and other baselines. Each instance", + "type": "text" + }, + { + "bbox": [ + 381, + 169, + 388, + 177 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "is generated as follows: First,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 252, + 192 + ], + "score": 1.0, + "content": "we randomly sample the class label", + "type": "text" + }, + { + "bbox": [ + 253, + 178, + 295, + 191 + ], + "score": 0.93, + "content": "y \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 179, + 418, + 192 + ], + "score": 1.0, + "content": "corresponding to the instance", + "type": "text" + }, + { + "bbox": [ + 418, + 181, + 425, + 189 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 179, + 505, + 192 + ], + "score": 1.0, + "content": ". Conditioned upon", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 156, + 203 + ], + "score": 1.0, + "content": "the value of", + "type": "text" + }, + { + "bbox": [ + 156, + 191, + 163, + 201 + ], + "score": 0.76, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 189, + 282, + 203 + ], + "score": 1.0, + "content": ", we then sample the instance", + "type": "text" + }, + { + "bbox": [ + 282, + 192, + 289, + 199 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 189, + 304, + 203 + ], + "score": 1.0, + "content": "as:", + "type": "text" + }, + { + "bbox": [ + 305, + 189, + 369, + 202 + ], + "score": 0.93, + "content": "x \\sim \\mathrm { \\bar { \\mathcal { N } } } ( \\mu _ { y } , \\mathrm { \\bar { \\Sigma } } _ { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 189, + 420, + 203 + ], + "score": 1.0, + "content": ". We choose", + "type": "text" + }, + { + "bbox": [ + 420, + 189, + 486, + 201 + ], + "score": 0.94, + "content": "\\mu _ { 0 } = [ - 2 , - 2 ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 173, + 213 + ], + "score": 0.92, + "content": "\\mu _ { 1 } = [ + 2 , + 2 ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 201, + 194, + 215 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 195, + 201, + 263, + 213 + ], + "score": 0.92, + "content": "\\Sigma _ { 0 } = \\Sigma _ { 1 } = 0 . 5 \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 201, + 291, + 215 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 292, + 203, + 303, + 213 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 201, + 307, + 215 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 307, + 202, + 320, + 213 + ], + "score": 0.8, + "content": "\\Sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 201, + 338, + 215 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 203, + 350, + 213 + ], + "score": 0.83, + "content": "\\mu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 201, + 354, + 215 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 354, + 203, + 367, + 213 + ], + "score": 0.82, + "content": "\\Sigma _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "denote the means and covariance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "of the Gaussian distributions from which instances in class 0 and class 1 are sampled respectively.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "A scatter plot of the samples resulting from this generative process and the decision boundary of a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "logistic regression model fit to this data are shown in Figure 1a. In our experimental evaluation, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 326, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 326, + 258 + ], + "score": 1.0, + "content": "consider different kinds of shifts to this synthetic data:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "image", + "bbox": [ + 117, + 269, + 489, + 354 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 269, + 489, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 269, + 489, + 354 + ], + "spans": [ + { + "bbox": [ + 117, + 269, + 489, + 354 + ], + "score": 0.965, + "type": "image", + "image_path": "98bb3ff9b091767a7781023e340217b6fd8251f08031fdf85213a639fc47f523.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 117, + 269, + 489, + 297.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 117, + 297.3333333333333, + 489, + 325.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 117, + 325.66666666666663, + 489, + 353.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 359, + 503, + 393 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "Figure 1: Synthetic data and examples of model shift. From left to right we have (a) original synthetic", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "dataset, (b) shifted data and decision boundary after mean shift, (c) shifted data and decision boundary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 381, + 476, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 476, + 394 + ], + "score": 1.0, + "content": "after variance shift, and (d) shifted data and decision boundary after mean and variance shift", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "(i) Mean shift: To generated shifted data, we leverage the same approach as above but shift the mean", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 407, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 334, + 423 + ], + "score": 1.0, + "content": "of the Gaussian distribution associated with class 0 i.e.,", + "type": "text" + }, + { + "bbox": [ + 334, + 409, + 399, + 422 + ], + "score": 0.93, + "content": "x \\sim \\mathcal { N } ( \\mu _ { y } ^ { \\prime } , \\Sigma _ { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 407, + 428, + 423 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 428, + 408, + 504, + 421 + ], + "score": 0.93, + "content": "\\mu _ { 0 } ^ { \\prime } = \\mu _ { 0 } + [ \\alpha , 0 ] ^ { T }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 124, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 421, + 161, + 433 + ], + "score": 0.92, + "content": "\\mu _ { 1 } ^ { \\prime } = \\mu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ". Note that we only shift the mean of one of the features of class 0 so that the slope", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "score": 1.0, + "content": "of the decision boundary of a linear model we fit to this shifted data changes (relative to the linear", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "model fit on the original data), while the intercept remains the same. Figure 1b shows shifted data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 454, + 164, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 126, + 465 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 454, + 160, + 464 + ], + "score": 0.88, + "content": "\\alpha = 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 454, + 164, + 465 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "(ii) Variance shift: Here, we leverage the same generative process as above, but instead of shifting", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 485, + 494 + ], + "score": 1.0, + "content": "the mean, we shift the variance of the Gaussian distribution associated with class 0 i.e., i.e.,", + "type": "text" + }, + { + "bbox": [ + 486, + 483, + 505, + 493 + ], + "score": 0.8, + "content": "x \\sim", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 152, + 506 + ], + "score": 0.92, + "content": "\\mathcal { N } ( \\mu _ { y } , \\Sigma _ { y } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 492, + 181, + 507 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 182, + 493, + 251, + 505 + ], + "score": 0.93, + "content": "\\Sigma _ { 0 } ^ { \\prime } = ( 1 + \\beta ) \\Sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 492, + 270, + 507 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 493, + 309, + 505 + ], + "score": 0.93, + "content": "\\Sigma _ { 1 } ^ { \\prime } = \\Sigma _ { 1 } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 492, + 393, + 507 + ], + "score": 1.0, + "content": "for some increment", + "type": "text" + }, + { + "bbox": [ + 393, + 493, + 421, + 504 + ], + "score": 0.92, + "content": "\\beta \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 492, + 506, + 507 + ], + "score": 1.0, + "content": ". The net result here", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "score": 1.0, + "content": "is that the intercept of the decision boundary of a linear model we fit to this shifted data changes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "(relative to the linear model fit on the original data), while the slope remains unchanged. Figure 1c", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 526, + 232, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 202, + 537 + ], + "score": 1.0, + "content": "shows shifted data with", + "type": "text" + }, + { + "bbox": [ + 203, + 526, + 228, + 537 + ], + "score": 0.91, + "content": "\\beta = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 526, + 232, + 537 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "(ii) Mean and variance shift: Here, we change both the mean and variance of the Gaussian distri-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 504, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 504, + 566 + ], + "score": 1.0, + "content": "bution associated with class 0 simultaneously (Figure 1d). It can be seen that there are noticeable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 565, + 460, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 460, + 577 + ], + "score": 1.0, + "content": "changes to both the slope and intercept of the decision boundary compared to Figure 1a.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Predictive models We generate recourses for a variety of linear and non-linear models: deep neural", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "networks (DNNs), SVMs, and logistic regression (LR). Here, we present results for a 3-layer DNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "and LR; remaining results are included in the Appendix. Results presented here are representative of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 231, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 231, + 628 + ], + "score": 1.0, + "content": "those for other model families.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "Baselines We compare our framework, ROAR, to the following state-of-the-art baselines: (i)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "counterfactual explanations (CFE) framework outlined by Wachter et al. 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From left to right we have (a) original synthetic", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "dataset, (b) shifted data and decision boundary after mean shift, (c) shifted data and decision boundary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 381, + 476, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 476, + 394 + ], + "score": 1.0, + "content": "after variance shift, and (d) shifted data and decision boundary after mean and variance shift", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "(i) Mean shift: To generated shifted data, we leverage the same approach as above but shift the mean", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 407, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 334, + 423 + ], + "score": 1.0, + "content": "of the Gaussian distribution associated with class 0 i.e.,", + "type": "text" + }, + { + "bbox": [ + 334, + 409, + 399, + 422 + ], + "score": 0.93, + "content": "x \\sim \\mathcal { N } ( \\mu _ { y } ^ { \\prime } , \\Sigma _ { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 407, + 428, + 423 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 428, + 408, + 504, + 421 + ], + "score": 0.93, + "content": "\\mu _ { 0 } ^ { \\prime } = \\mu _ { 0 } + [ \\alpha , 0 ] ^ { T }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 124, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 421, + 161, + 433 + ], + "score": 0.92, + "content": "\\mu _ { 1 } ^ { \\prime } = \\mu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ". 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The net result here", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "score": 1.0, + "content": "is that the intercept of the decision boundary of a linear model we fit to this shifted data changes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "(relative to the linear model fit on the original data), while the slope remains unchanged. Figure 1c", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 526, + 232, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 202, + 537 + ], + "score": 1.0, + "content": "shows shifted data with", + "type": "text" + }, + { + "bbox": [ + 203, + 526, + 228, + 537 + ], + "score": 0.91, + "content": "\\beta = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 526, + 232, + 537 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 469, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "(ii) Mean and variance shift: Here, we change both the mean and variance of the Gaussian distri-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 504, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 504, + 566 + ], + "score": 1.0, + "content": "bution associated with class 0 simultaneously (Figure 1d). It can be seen that there are noticeable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 565, + 460, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 460, + 577 + ], + "score": 1.0, + "content": "changes to both the slope and intercept of the decision boundary compared to Figure 1a.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 543, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Predictive models We generate recourses for a variety of linear and non-linear models: deep neural", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "networks (DNNs), SVMs, and logistic regression (LR). Here, we present results for a 3-layer DNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "and LR; remaining results are included in the Appendix. Results presented here are representative of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 231, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 231, + 628 + ], + "score": 1.0, + "content": "those for other model families.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 583, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "Baselines We compare our framework, ROAR, to the following state-of-the-art baselines: (i)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "counterfactual explanations (CFE) framework outlined by Wachter et al. [31], (ii) actionable recourse", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "(AR) in linear classification [26], and (iii) causal recourse framework (MINT) proposed by Karimi", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "et al. [14]. While CFE leverages gradient computations to find counterfactuals, AR employs a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "mixed integer programming based approach to find counterfactuals that are actionable. The MINT", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "framework operates on top of existing approaches for finding nearby counterfactuals. We use the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "MINT framework on top of CFE and ROAR and refer to these two approaches as MINT and ROAR-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "MINT respectively. As the MINT framework requires access to the underlying causal graph, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "experiment with MINT and ROAR-MINT only on the German credit dataset for which such a causal", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 181, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 181, + 95 + ], + "score": 1.0, + "content": "graph is available.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 633, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "experiment with MINT and ROAR-MINT only on the German credit dataset for which such a causal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 181, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 181, + 95 + ], + "score": 1.0, + "content": "graph is available.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "Cost functions Our framework, ROAR, and all the other baselines we use rely on a cost function", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 107, + 114, + 113, + 123 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "that measures the cost (or effort) required to act upon the prescribed recourse. Furthermore, our", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "score": 1.0, + "content": "approach as well as several other baselines require the cost function to be differentiable. So, we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 311, + 147 + ], + "score": 1.0, + "content": "consider two cost functions in our experimentation:", + "type": "text" + }, + { + "bbox": [ + 311, + 135, + 321, + 146 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "distance between the original instance and the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "counterfactual, and a cost function learned from pairwise feature comparison inputs (PFC) [13, 26, 23].", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "PFC uses the Bradley-Terry model to map pairwise feature comparison inputs provided by end users", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 414, + 180 + ], + "score": 1.0, + "content": "to the cost required to act upon the prescribed recourse for any given instance", + "type": "text" + }, + { + "bbox": [ + 415, + 170, + 421, + 177 + ], + "score": 0.6, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 168, + 505, + 180 + ], + "score": 1.0, + "content": ". For more details on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "this cost function, please refer to Rawal and Lakkaraju [23]. In our experiments, we follow the same", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 477, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 477, + 203 + ], + "score": 1.0, + "content": "procedure as Rawal and Lakkaraju [23] and simulate the pairwise feature comparison inputs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Setting and implementation details We partition each of our synthetic and real world datasets", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 212, + 231 + ], + "score": 1.0, + "content": "into two parts: initial data", + "type": "text" + }, + { + "bbox": [ + 212, + 219, + 232, + 230 + ], + "score": 0.83, + "content": "( D _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 218, + 298, + 231 + ], + "score": 1.0, + "content": "and shifted data", + "type": "text" + }, + { + "bbox": [ + 299, + 219, + 318, + 230 + ], + "score": 0.84, + "content": "( D _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 218, + 458, + 231 + ], + "score": 1.0, + "content": ". In the case of real world datasets,", + "type": "text" + }, + { + "bbox": [ + 458, + 219, + 472, + 230 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 218, + 490, + 231 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 219, + 504, + 230 + ], + "score": 0.88, + "content": "D _ { 2 }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "can be logically inferred from the data itself – e.g., in case of the German credit dataset, we consider", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 245, + 252 + ], + "score": 1.0, + "content": "the initial version of the dataset as", + "type": "text" + }, + { + "bbox": [ + 245, + 241, + 259, + 252 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 241, + 430, + 252 + ], + "score": 1.0, + "content": "and the corrected version of the dataset as", + "type": "text" + }, + { + "bbox": [ + 430, + 241, + 444, + 252 + ], + "score": 0.9, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 241, + 505, + 252 + ], + "score": 1.0, + "content": ". In the case of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 230, + 264 + ], + "score": 1.0, + "content": "synthetic datasets, we generate", + "type": "text" + }, + { + "bbox": [ + 230, + 252, + 244, + 262 + ], + "score": 0.9, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 251, + 261, + 264 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 252, + 275, + 262 + ], + "score": 0.89, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 251, + 381, + 264 + ], + "score": 1.0, + "content": "as described earlier where", + "type": "text" + }, + { + "bbox": [ + 381, + 252, + 395, + 262 + ], + "score": 0.89, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 251, + 491, + 264 + ], + "score": 1.0, + "content": "is generated by shifting", + "type": "text" + }, + { + "bbox": [ + 491, + 252, + 505, + 263 + ], + "score": 0.88, + "content": "D _ { 1 }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 262, + 257, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 257, + 274 + ], + "score": 1.0, + "content": "(See \"Synthetic data\" in Section 5.1).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 451, + 291 + ], + "score": 1.0, + "content": "We use 5-fold cross validation throughout our real world and synthetic experiments. On", + "type": "text" + }, + { + "bbox": [ + 451, + 280, + 464, + 290 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 279, + 505, + 291 + ], + "score": 1.0, + "content": ", we use 4", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "folds to train predictive models and the remaining fold to generate and evaluate recourses. We repeat", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 459, + 314 + ], + "score": 1.0, + "content": "this process 5 times and report averaged values of our evaluation metrics. We leverage", + "type": "text" + }, + { + "bbox": [ + 459, + 301, + 473, + 312 + ], + "score": 0.88, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "only to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 203, + 325 + ], + "score": 1.0, + "content": "train the shifted models", + "type": "text" + }, + { + "bbox": [ + 203, + 312, + 220, + 323 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ". More details about the data splits, model training, and performance of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 315, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 315, + 335 + ], + "score": 1.0, + "content": "the predictive models are included in the Appendix.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 507, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 507, + 353 + ], + "score": 1.0, + "content": "We use binary cross entropy loss and the Adam optimizer to operationalize our framework, ROAR.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 459, + 364 + ], + "score": 1.0, + "content": "Our framework, ROAR, has the following parameters: the set of acceptable perturbations", + "type": "text" + }, + { + "bbox": [ + 460, + 351, + 469, + 361 + ], + "score": 0.78, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "(defined", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 162, + 375 + ], + "score": 1.0, + "content": "in practice by", + "type": "text" + }, + { + "bbox": [ + 162, + 362, + 184, + 373 + ], + "score": 0.9, + "content": "\\delta _ { m a x . }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 361, + 293, + 375 + ], + "score": 1.0, + "content": ") and the tradeoff parameter", + "type": "text" + }, + { + "bbox": [ + 293, + 362, + 300, + 372 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 361, + 506, + 375 + ], + "score": 1.0, + "content": ". In our experiments on evaluating robustness to real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 203, + 385 + ], + "score": 1.0, + "content": "world shifts, we choose", + "type": "text" + }, + { + "bbox": [ + 203, + 373, + 251, + 384 + ], + "score": 0.92, + "content": "\\delta _ { m a x } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "given that continuous features are scaled to zero mean and unit", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 384, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 321, + 396 + ], + "score": 1.0, + "content": "variance. Furthermore, in each setting, we choose the", + "type": "text" + }, + { + "bbox": [ + 321, + 384, + 329, + 393 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 384, + 487, + 396 + ], + "score": 1.0, + "content": "that maximizes the recourse validity of", + "type": "text" + }, + { + "bbox": [ + 487, + 384, + 504, + 395 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "(more details in Section 5.1 \"Metrics\" and Appendix). In case of our synthetic experiments where we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "assess the impact of the degree (magnitude) of data distribution shift, features are not normalized,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 235, + 429 + ], + "score": 1.0, + "content": "so we do a grid search for both", + "type": "text" + }, + { + "bbox": [ + 236, + 417, + 257, + 428 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 416, + 276, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 417, + 283, + 426 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 416, + 402, + 429 + ], + "score": 1.0, + "content": ". First, we choose the largest", + "type": "text" + }, + { + "bbox": [ + 402, + 416, + 424, + 428 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "that maximizes the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 187, + 440 + ], + "score": 1.0, + "content": "recourse validity of", + "type": "text" + }, + { + "bbox": [ + 188, + 428, + 205, + 438 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 427, + 257, + 440 + ], + "score": 1.0, + "content": "and then set", + "type": "text" + }, + { + "bbox": [ + 257, + 428, + 264, + 437 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "in a similar fashion (more details in Appendix). We set the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "parameters of the baselines using techniques discussed in the original works [31, 14, 26] and employ", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 288, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 288, + 462 + ], + "score": 1.0, + "content": "a similar grid search approach if unspecified.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "Following the precedents set forth in [26] and [23], we adapt AR and ROAR to non-linear models", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "by first generating local linear approximations of these models using LIME [25]. We refer to these", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 487, + 321, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 321, + 501 + ], + "score": 1.0, + "content": "variants as AR-LIME and ROAR-LIME respectively.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Metrics. We consider two metrics in our evaluation: 1) Avg Cost is defined as the average cost", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "incurred to act upon the prescribed recourses where the average is computed over all the instances", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "for which a given algorithm provides recourse. Recall that we consider two notions of cost in our", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 165, + 552 + ], + "score": 1.0, + "content": "experiments –", + "type": "text" + }, + { + "bbox": [ + 165, + 539, + 176, + 550 + ], + "score": 0.56, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "distance between the original instance and the counterfactual, costs learned from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "pairwise feature comparisons (PFC) (See \"Cost Functions\" in Section 5.1). 2) Validity is defined", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "as the fraction of instances for which acting upon the prescribed recourse results in the desired", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 572, + 359, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 359, + 584 + ], + "score": 1.0, + "content": "prediction. Note that validity is computed w.r.t. a given model.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 261, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 262, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 262, + 604 + ], + "score": 1.0, + "content": "5.2 Robustness to real world shifts", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "Here, we evaluate the robustness of the recourses output by our framework, ROAR, as well as the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "baselines. A recourse finding algorithm can be considered robust if the recourses output by the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "algorithm remain valid even if the underlying model has changed. To evaluate this, we first leverage", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "our approach and other baselines to find recourses of instances in our test sets w.r.t. the initial model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 650, + 124, + 661 + ], + "score": 0.88, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 650, + 421, + 662 + ], + "score": 1.0, + "content": ". We then compute the validity of these recourses w.r.t. the shifted model", + "type": "text" + }, + { + "bbox": [ + 421, + 650, + 439, + 661 + ], + "score": 0.91, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "which has been", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 302, + 674 + ], + "score": 1.0, + "content": "trained on the shifted data. Let us refer to this as", + "type": "text" + }, + { + "bbox": [ + 303, + 661, + 320, + 672 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 659, + 452, + 674 + ], + "score": 1.0, + "content": "validity. The higher the value of", + "type": "text" + }, + { + "bbox": [ + 453, + 661, + 470, + 672 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "validity,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 367, + 684 + ], + "score": 1.0, + "content": "the more robust the recourse finding method. Table 1 shows the", + "type": "text" + }, + { + "bbox": [ + 368, + 672, + 385, + 683 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "validity metric computed for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 682, + 332, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 332, + 695 + ], + "score": 1.0, + "content": "different algorithms across different real world datasets.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "It can be seen that recourse methods that use our framework, ROAR and ROAR-MINT, achieve the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 138, + 723 + ], + "score": 1.0, + "content": "highest", + "type": "text" + }, + { + "bbox": [ + 138, + 711, + 155, + 722 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "validity across all datasets. In fact, methods that use our framework do almost twice as", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 72, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 506, + 95 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "Cost functions Our framework, ROAR, and all the other baselines we use rely on a cost function", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 107, + 114, + 113, + 123 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "that measures the cost (or effort) required to act upon the prescribed recourse. Furthermore, our", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "score": 1.0, + "content": "approach as well as several other baselines require the cost function to be differentiable. So, we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 311, + 147 + ], + "score": 1.0, + "content": "consider two cost functions in our experimentation:", + "type": "text" + }, + { + "bbox": [ + 311, + 135, + 321, + 146 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "distance between the original instance and the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "counterfactual, and a cost function learned from pairwise feature comparison inputs (PFC) [13, 26, 23].", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "PFC uses the Bradley-Terry model to map pairwise feature comparison inputs provided by end users", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 414, + 180 + ], + "score": 1.0, + "content": "to the cost required to act upon the prescribed recourse for any given instance", + "type": "text" + }, + { + "bbox": [ + 415, + 170, + 421, + 177 + ], + "score": 0.6, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 168, + 505, + 180 + ], + "score": 1.0, + "content": ". For more details on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "this cost function, please refer to Rawal and Lakkaraju [23]. In our experiments, we follow the same", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 477, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 477, + 203 + ], + "score": 1.0, + "content": "procedure as Rawal and Lakkaraju [23] and simulate the pairwise feature comparison inputs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 101, + 506, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Setting and implementation details We partition each of our synthetic and real world datasets", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 212, + 231 + ], + "score": 1.0, + "content": "into two parts: initial data", + "type": "text" + }, + { + "bbox": [ + 212, + 219, + 232, + 230 + ], + "score": 0.83, + "content": "( D _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 218, + 298, + 231 + ], + "score": 1.0, + "content": "and shifted data", + "type": "text" + }, + { + "bbox": [ + 299, + 219, + 318, + 230 + ], + "score": 0.84, + "content": "( D _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 218, + 458, + 231 + ], + "score": 1.0, + "content": ". In the case of real world datasets,", + "type": "text" + }, + { + "bbox": [ + 458, + 219, + 472, + 230 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 218, + 490, + 231 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 219, + 504, + 230 + ], + "score": 0.88, + "content": "D _ { 2 }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "can be logically inferred from the data itself – e.g., in case of the German credit dataset, we consider", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 245, + 252 + ], + "score": 1.0, + "content": "the initial version of the dataset as", + "type": "text" + }, + { + "bbox": [ + 245, + 241, + 259, + 252 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 241, + 430, + 252 + ], + "score": 1.0, + "content": "and the corrected version of the dataset as", + "type": "text" + }, + { + "bbox": [ + 430, + 241, + 444, + 252 + ], + "score": 0.9, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 241, + 505, + 252 + ], + "score": 1.0, + "content": ". In the case of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 230, + 264 + ], + "score": 1.0, + "content": "synthetic datasets, we generate", + "type": "text" + }, + { + "bbox": [ + 230, + 252, + 244, + 262 + ], + "score": 0.9, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 251, + 261, + 264 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 252, + 275, + 262 + ], + "score": 0.89, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 251, + 381, + 264 + ], + "score": 1.0, + "content": "as described earlier where", + "type": "text" + }, + { + "bbox": [ + 381, + 252, + 395, + 262 + ], + "score": 0.89, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 251, + 491, + 264 + ], + "score": 1.0, + "content": "is generated by shifting", + "type": "text" + }, + { + "bbox": [ + 491, + 252, + 505, + 263 + ], + "score": 0.88, + "content": "D _ { 1 }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 262, + 257, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 257, + 274 + ], + "score": 1.0, + "content": "(See \"Synthetic data\" in Section 5.1).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 207, + 505, + 274 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 451, + 291 + ], + "score": 1.0, + "content": "We use 5-fold cross validation throughout our real world and synthetic experiments. On", + "type": "text" + }, + { + "bbox": [ + 451, + 280, + 464, + 290 + ], + "score": 0.89, + "content": "D _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 279, + 505, + 291 + ], + "score": 1.0, + "content": ", we use 4", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "folds to train predictive models and the remaining fold to generate and evaluate recourses. We repeat", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 459, + 314 + ], + "score": 1.0, + "content": "this process 5 times and report averaged values of our evaluation metrics. We leverage", + "type": "text" + }, + { + "bbox": [ + 459, + 301, + 473, + 312 + ], + "score": 0.88, + "content": "D _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "only to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 203, + 325 + ], + "score": 1.0, + "content": "train the shifted models", + "type": "text" + }, + { + "bbox": [ + 203, + 312, + 220, + 323 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ". More details about the data splits, model training, and performance of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 315, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 315, + 335 + ], + "score": 1.0, + "content": "the predictive models are included in the Appendix.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 279, + 506, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 507, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 507, + 353 + ], + "score": 1.0, + "content": "We use binary cross entropy loss and the Adam optimizer to operationalize our framework, ROAR.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 459, + 364 + ], + "score": 1.0, + "content": "Our framework, ROAR, has the following parameters: the set of acceptable perturbations", + "type": "text" + }, + { + "bbox": [ + 460, + 351, + 469, + 361 + ], + "score": 0.78, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "(defined", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 162, + 375 + ], + "score": 1.0, + "content": "in practice by", + "type": "text" + }, + { + "bbox": [ + 162, + 362, + 184, + 373 + ], + "score": 0.9, + "content": "\\delta _ { m a x . }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 361, + 293, + 375 + ], + "score": 1.0, + "content": ") and the tradeoff parameter", + "type": "text" + }, + { + "bbox": [ + 293, + 362, + 300, + 372 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 361, + 506, + 375 + ], + "score": 1.0, + "content": ". In our experiments on evaluating robustness to real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 203, + 385 + ], + "score": 1.0, + "content": "world shifts, we choose", + "type": "text" + }, + { + "bbox": [ + 203, + 373, + 251, + 384 + ], + "score": 0.92, + "content": "\\delta _ { m a x } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "given that continuous features are scaled to zero mean and unit", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 384, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 321, + 396 + ], + "score": 1.0, + "content": "variance. Furthermore, in each setting, we choose the", + "type": "text" + }, + { + "bbox": [ + 321, + 384, + 329, + 393 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 384, + 487, + 396 + ], + "score": 1.0, + "content": "that maximizes the recourse validity of", + "type": "text" + }, + { + "bbox": [ + 487, + 384, + 504, + 395 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "(more details in Section 5.1 \"Metrics\" and Appendix). In case of our synthetic experiments where we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "assess the impact of the degree (magnitude) of data distribution shift, features are not normalized,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 235, + 429 + ], + "score": 1.0, + "content": "so we do a grid search for both", + "type": "text" + }, + { + "bbox": [ + 236, + 417, + 257, + 428 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 416, + 276, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 417, + 283, + 426 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 416, + 402, + 429 + ], + "score": 1.0, + "content": ". First, we choose the largest", + "type": "text" + }, + { + "bbox": [ + 402, + 416, + 424, + 428 + ], + "score": 0.91, + "content": "\\delta _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "that maximizes the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 187, + 440 + ], + "score": 1.0, + "content": "recourse validity of", + "type": "text" + }, + { + "bbox": [ + 188, + 428, + 205, + 438 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 427, + 257, + 440 + ], + "score": 1.0, + "content": "and then set", + "type": "text" + }, + { + "bbox": [ + 257, + 428, + 264, + 437 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "in a similar fashion (more details in Appendix). We set the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "parameters of the baselines using techniques discussed in the original works [31, 14, 26] and employ", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 288, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 288, + 462 + ], + "score": 1.0, + "content": "a similar grid search approach if unspecified.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 339, + 507, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "Following the precedents set forth in [26] and [23], we adapt AR and ROAR to non-linear models", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "by first generating local linear approximations of these models using LIME [25]. We refer to these", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 487, + 321, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 321, + 501 + ], + "score": 1.0, + "content": "variants as AR-LIME and ROAR-LIME respectively.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 466, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Metrics. We consider two metrics in our evaluation: 1) Avg Cost is defined as the average cost", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "incurred to act upon the prescribed recourses where the average is computed over all the instances", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "for which a given algorithm provides recourse. Recall that we consider two notions of cost in our", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 165, + 552 + ], + "score": 1.0, + "content": "experiments –", + "type": "text" + }, + { + "bbox": [ + 165, + 539, + 176, + 550 + ], + "score": 0.56, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "distance between the original instance and the counterfactual, costs learned from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "pairwise feature comparisons (PFC) (See \"Cost Functions\" in Section 5.1). 2) Validity is defined", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "as the fraction of instances for which acting upon the prescribed recourse results in the desired", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 572, + 359, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 359, + 584 + ], + "score": 1.0, + "content": "prediction. Note that validity is computed w.r.t. a given model.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 505, + 506, + 584 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 261, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 262, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 262, + 604 + ], + "score": 1.0, + "content": "5.2 Robustness to real world shifts", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "Here, we evaluate the robustness of the recourses output by our framework, ROAR, as well as the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "baselines. A recourse finding algorithm can be considered robust if the recourses output by the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "algorithm remain valid even if the underlying model has changed. To evaluate this, we first leverage", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "our approach and other baselines to find recourses of instances in our test sets w.r.t. the initial model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 650, + 124, + 661 + ], + "score": 0.88, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 650, + 421, + 662 + ], + "score": 1.0, + "content": ". We then compute the validity of these recourses w.r.t. the shifted model", + "type": "text" + }, + { + "bbox": [ + 421, + 650, + 439, + 661 + ], + "score": 0.91, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "which has been", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 302, + 674 + ], + "score": 1.0, + "content": "trained on the shifted data. Let us refer to this as", + "type": "text" + }, + { + "bbox": [ + 303, + 661, + 320, + 672 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 659, + 452, + 674 + ], + "score": 1.0, + "content": "validity. The higher the value of", + "type": "text" + }, + { + "bbox": [ + 453, + 661, + 470, + 672 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "validity,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 367, + 684 + ], + "score": 1.0, + "content": "the more robust the recourse finding method. Table 1 shows the", + "type": "text" + }, + { + "bbox": [ + 368, + 672, + 385, + 683 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "validity metric computed for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 682, + 332, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 332, + 695 + ], + "score": 1.0, + "content": "different algorithms across different real world datasets.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 606, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "It can be seen that recourse methods that use our framework, ROAR and ROAR-MINT, achieve the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 138, + 723 + ], + "score": 1.0, + "content": "highest", + "type": "text" + }, + { + "bbox": [ + 138, + 711, + 155, + 722 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "validity across all datasets. In fact, methods that use our framework do almost twice as", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 700, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 70, + 504, + 224 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 70, + 504, + 224 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 70, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 504, + 224 + ], + "score": 0.984, + "html": "
Correction ShiftTemporal ShiftGeospatial Shift
Model CostRecourse CFEAvgCost 1.02 ± 0.18MValidity 1.00±0.00MValidity 0.54± 0.27AvgCost 3.57 ± 1.14MValidity 1.00±0.00MValidity 0.31±0.09Avg Cost 8.37±0.73MValidity 0.98±0.03MValidity 0.29±0.09
LRL10.85 ± 0.141.00 ± 0.000.53 ± 0.211.50± 0.281.00 ± 0.000.16 ± 0.065.29 ± 0.281.00 ± 0.000.43 ± 0.14
AR3.14 ± 0.250.99 ± 0.010.98 ±0.0210.88 ± 1.671.00 ± 0.000.67 ± 0.19
ROAR3.13 ± 0.321.00 ± 0.000.94 ± 0.08 0.93 ± 0.07NA
MINT4.73 ± 1.561.00 ± 0.00NANANANANA
ROAR-MINT6.77 ± 0.351.00 ± 0.001.00 ± 0.00NANANANANANA
CFE0.03±0.021.00 ±0.000.56±0.330.24±0.091.00 ± 0.000.26± 0.110.34± 0.041.00±0.000.18 ±0.10
PFCAR0.09 ± 0.021.00 ± 0.000.54 ± 0.270.11 ± 0.021.00 ± 0.000.09 ± 0.050.32 ±0.031.00 ±0.000.24 ± 0.11
ROAR MINT0.36±0.081.00 ± 0.001.00 ± 0.000.44 ± 0.120.99 ± 0.010.98 ± 0.011.20 ± 0.101.00 ± 0.000.91± 0.07
ROAR-MINT1.00 ± 1.151.00 ± 0.00 1.00 ± 0.000.95±0.08 1.00 ± 0.00NANA NANANANANA
L1CFE1.23 ± 0.05 0.55 ±0.101.00± 0.000.47±0.06NA 3.78±0.681.00 ± 0.00NA 0.52±0.09NANANA
AR-LIME0.38 ± 0.150.16 ±0.100.31 ± 0.061.39 ± 0.130.59 ± 0.110.65 ± 0.1710.09± 0.71 9.02 ±1.571.00 ± 0.000.48±0.09 0.83 ±0.10
ROAR-LIME0.76± 0.06
NN1.83 ± 0.190.78 ±0.060.72 ± 0.104.90±0.240.98 ±0.020.97 ±0.0221.05 ± 3.581.00 ± 0.000.97 ±0.03
MINT2.24 ± 1.250.81 ± 0.020.63 ± 0.11NANANANANANA
ROAR-MINT8.59 ± 1.700.90 ±0.030.84 ± 0.04NANANANANANA
CFE0.06±0.021.00± 0.000.51 ± 0.120.19±0.061.00±0.000.50± 0.130.48± 0.061.00±0.000.30±0.14
AR-LIME0.06± 0.030.49 ± 0.110.56± 0.150.11 ± 0.010.54 ± 0.080.62 ± 0.120.78 ± 0.150.84 ± 0.060.82 ± 0.11
PFC ROAR-LIME0.64 ± 0.080.85 ± 0.070.82 ± 0.050.37 ±0.070.99 ± 0.010.99 ±0.01.66 ± 0.211.00 ±0.000.97 ± 0.04
0.60 ± 0.160.82 ±0.070.64 ± 0.15NANANANANA
MINT ROAR-MINT0.60 ±0.070.91± 0.040.81 ± 0.04NANANANANANA NA
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Recourses that leverage our framework ROAR are more robust (higher", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 252, + 366, + 264 + ], + "spans": [ + { + "bbox": [ + 107, + 252, + 124, + 262 + ], + "score": 0.88, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 252, + 366, + 264 + ], + "score": 1.0, + "content": "validity) compared to those generated by existing baselines.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "good compared to other baselines on this metric, indicating that ROAR based recourse methods are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 432, + 294 + ], + "score": 1.0, + "content": "quite robust. After ROAR, MINT is the next best performing baseline with respect", + "type": "text" + }, + { + "bbox": [ + 432, + 282, + 450, + 293 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "validity. This", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "score": 1.0, + "content": "may be explained by the fact that MINT accounts for the underlying causal graphs when generating", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 305, + 149, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 149, + 315 + ], + "score": 1.0, + "content": "recourses.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "score": 1.0, + "content": "We also assess if the robustness achieved by our framework is coming at a cost i.e., by sacrificing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "validity on the original model or by increasing avg cost. Table 1 shows the results for the same. It", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 345, + 355 + ], + "score": 1.0, + "content": "can be seen that ROAR based recourses achieve higher than", + "type": "text" + }, + { + "bbox": [ + 345, + 342, + 365, + 353 + ], + "score": 0.72, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 342, + 384, + 353 + ], + "score": 0.42, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "validity in all but two settings.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 354, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 365 + ], + "score": 1.0, + "content": "We compute the avg cost of the recourses output by all the algorithms on various datasets and find", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 323, + 377 + ], + "score": 1.0, + "content": "that ROAR typically has a higher avg cost (both under", + "type": "text" + }, + { + "bbox": [ + 324, + 365, + 334, + 375 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "and PFC cost functions) compared to CFE", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "and AR baselines. As demonstrated through additional experiments in the Appendix, these relatively", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "higher costs are expected given our Theorem 2 upper bound on ROAR cost. However, overall, MINT", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "and ROAR-MINT seem to exhibit the highest avg costs and are the worst performing algorithms", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "according to this metric. Since non-causal recourse methods assume independent features, and do not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "have to adhere to the underlying causal structure when finding counterfactuals, they can generate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "relatively lower cost counterfactuals even if those counterfactuals may not correspond to realistic data", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "instances. This is likely one of the key reasons why we observe higher average costs in the causal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 181, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 181, + 462 + ], + "score": 1.0, + "content": "recourse methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "image", + "bbox": [ + 132, + 501, + 466, + 682 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 106, + 470, + 380, + 483 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 470, + 381, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 381, + 484 + ], + "score": 1.0, + "content": "5.3 Impact of the degree of data distribution shift on recourses", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "image_body", + "bbox": [ + 132, + 501, + 466, + 682 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 501, + 466, + 682 + ], + "spans": [ + { + "bbox": [ + 132, + 501, + 466, + 682 + ], + "score": 0.976, + "type": "image", + "image_path": "79b7233b461f75eae4a6c8dc52c9a20153a7b4b770aa53c7911b559afe6ec4db.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 132, + 501, + 466, + 561.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 132, + 561.3333333333334, + 466, + 621.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 132, + 621.6666666666667, + 466, + 682.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 689, + 505, + 734 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "Figure 2: Impact of the degree of data distribution shift on validity of recourse: DNN classifier with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 117, + 711 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "cost function (top row), DNN classifier with PFC cost function (bottom row); Validity of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "recourses generated by all methods drops as degree (magnitude) of the shift increases; The drop in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 721, + 456, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 456, + 735 + ], + "score": 1.0, + "content": "the validity is much smaller for our method ROAR-LIME compared to other baselines.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + } + ], + "index": 25 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 70, + 504, + 224 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 70, + 504, + 224 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 70, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 504, + 224 + ], + "score": 0.984, + "html": "
Correction ShiftTemporal ShiftGeospatial Shift
Model CostRecourse CFEAvgCost 1.02 ± 0.18MValidity 1.00±0.00MValidity 0.54± 0.27AvgCost 3.57 ± 1.14MValidity 1.00±0.00MValidity 0.31±0.09Avg Cost 8.37±0.73MValidity 0.98±0.03MValidity 0.29±0.09
LRL10.85 ± 0.141.00 ± 0.000.53 ± 0.211.50± 0.281.00 ± 0.000.16 ± 0.065.29 ± 0.281.00 ± 0.000.43 ± 0.14
AR3.14 ± 0.250.99 ± 0.010.98 ±0.0210.88 ± 1.671.00 ± 0.000.67 ± 0.19
ROAR3.13 ± 0.321.00 ± 0.000.94 ± 0.08 0.93 ± 0.07NA
MINT4.73 ± 1.561.00 ± 0.00NANANANANA
ROAR-MINT6.77 ± 0.351.00 ± 0.001.00 ± 0.00NANANANANANA
CFE0.03±0.021.00 ±0.000.56±0.330.24±0.091.00 ± 0.000.26± 0.110.34± 0.041.00±0.000.18 ±0.10
PFCAR0.09 ± 0.021.00 ± 0.000.54 ± 0.270.11 ± 0.021.00 ± 0.000.09 ± 0.050.32 ±0.031.00 ±0.000.24 ± 0.11
ROAR MINT0.36±0.081.00 ± 0.001.00 ± 0.000.44 ± 0.120.99 ± 0.010.98 ± 0.011.20 ± 0.101.00 ± 0.000.91± 0.07
ROAR-MINT1.00 ± 1.151.00 ± 0.00 1.00 ± 0.000.95±0.08 1.00 ± 0.00NANA NANANANANA
L1CFE1.23 ± 0.05 0.55 ±0.101.00± 0.000.47±0.06NA 3.78±0.681.00 ± 0.00NA 0.52±0.09NANANA
AR-LIME0.38 ± 0.150.16 ±0.100.31 ± 0.061.39 ± 0.130.59 ± 0.110.65 ± 0.1710.09± 0.71 9.02 ±1.571.00 ± 0.000.48±0.09 0.83 ±0.10
ROAR-LIME0.76± 0.06
NN1.83 ± 0.190.78 ±0.060.72 ± 0.104.90±0.240.98 ±0.020.97 ±0.0221.05 ± 3.581.00 ± 0.000.97 ±0.03
MINT2.24 ± 1.250.81 ± 0.020.63 ± 0.11NANANANANANA
ROAR-MINT8.59 ± 1.700.90 ±0.030.84 ± 0.04NANANANANANA
CFE0.06±0.021.00± 0.000.51 ± 0.120.19±0.061.00±0.000.50± 0.130.48± 0.061.00±0.000.30±0.14
AR-LIME0.06± 0.030.49 ± 0.110.56± 0.150.11 ± 0.010.54 ± 0.080.62 ± 0.120.78 ± 0.150.84 ± 0.060.82 ± 0.11
PFC ROAR-LIME0.64 ± 0.080.85 ± 0.070.82 ± 0.050.37 ±0.070.99 ± 0.010.99 ±0.01.66 ± 0.211.00 ±0.000.97 ± 0.04
0.60 ± 0.160.82 ±0.070.64 ± 0.15NANANANANA
MINT ROAR-MINT0.60 ±0.070.91± 0.040.81 ± 0.04NANANANANANA NA
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Recourses that leverage our framework ROAR are more robust (higher", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 252, + 366, + 264 + ], + "spans": [ + { + "bbox": [ + 107, + 252, + 124, + 262 + ], + "score": 0.88, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 252, + 366, + 264 + ], + "score": 1.0, + "content": "validity) compared to those generated by existing baselines.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 228, + 505, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "good compared to other baselines on this metric, indicating that ROAR based recourse methods are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 432, + 294 + ], + "score": 1.0, + "content": "quite robust. After ROAR, MINT is the next best performing baseline with respect", + "type": "text" + }, + { + "bbox": [ + 432, + 282, + 450, + 293 + ], + "score": 0.89, + "content": "\\mathcal { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "validity. This", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "score": 1.0, + "content": "may be explained by the fact that MINT accounts for the underlying causal graphs when generating", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 305, + 149, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 149, + 315 + ], + "score": 1.0, + "content": "recourses.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 270, + 505, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "score": 1.0, + "content": "We also assess if the robustness achieved by our framework is coming at a cost i.e., by sacrificing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "validity on the original model or by increasing avg cost. Table 1 shows the results for the same. It", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 345, + 355 + ], + "score": 1.0, + "content": "can be seen that ROAR based recourses achieve higher than", + "type": "text" + }, + { + "bbox": [ + 345, + 342, + 365, + 353 + ], + "score": 0.72, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 342, + 384, + 353 + ], + "score": 0.42, + "content": "\\mathcal { M } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "validity in all but two settings.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 354, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 365 + ], + "score": 1.0, + "content": "We compute the avg cost of the recourses output by all the algorithms on various datasets and find", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 323, + 377 + ], + "score": 1.0, + "content": "that ROAR typically has a higher avg cost (both under", + "type": "text" + }, + { + "bbox": [ + 324, + 365, + 334, + 375 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "and PFC cost functions) compared to CFE", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "and AR baselines. As demonstrated through additional experiments in the Appendix, these relatively", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "higher costs are expected given our Theorem 2 upper bound on ROAR cost. However, overall, MINT", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "and ROAR-MINT seem to exhibit the highest avg costs and are the worst performing algorithms", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "according to this metric. Since non-causal recourse methods assume independent features, and do not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "have to adhere to the underlying causal structure when finding counterfactuals, they can generate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "relatively lower cost counterfactuals even if those counterfactuals may not correspond to realistic data", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "instances. This is likely one of the key reasons why we observe higher average costs in the causal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 181, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 181, + 462 + ], + "score": 1.0, + "content": "recourse methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 319, + 506, + 462 + ] + }, + { + "type": "image", + "bbox": [ + 132, + 501, + 466, + 682 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 106, + 470, + 380, + 483 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 470, + 381, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 381, + 484 + ], + "score": 1.0, + "content": "5.3 Impact of the degree of data distribution shift on recourses", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "image_body", + "bbox": [ + 132, + 501, + 466, + 682 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 501, + 466, + 682 + ], + "spans": [ + { + "bbox": [ + 132, + 501, + 466, + 682 + ], + "score": 0.976, + "type": "image", + "image_path": "79b7233b461f75eae4a6c8dc52c9a20153a7b4b770aa53c7911b559afe6ec4db.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 132, + 501, + 466, + 561.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 132, + 561.3333333333334, + 466, + 621.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 132, + 621.6666666666667, + 466, + 682.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 689, + 505, + 734 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "Figure 2: Impact of the degree of data distribution shift on validity of recourse: DNN classifier with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 117, + 711 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "cost function (top row), DNN classifier with PFC cost function (bottom row); Validity of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "recourses generated by all methods drops as degree (magnitude) of the shift increases; The drop in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 721, + 456, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 456, + 735 + ], + "score": 1.0, + "content": "the validity is much smaller for our method ROAR-LIME compared to other baselines.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + } + ], + "index": 25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "Here, we assess how different kinds of distribution shifts and the magnitude of these shifts impact the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "robustness of recourses output by our framework and other baselines. To this end, we leverage our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "synthetic datasets and introduce mean shifts, variance shifts, and combination shifts (both mean and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 307, + 118 + ], + "score": 1.0, + "content": "variance shifts) of different magnitudes by varying", + "type": "text" + }, + { + "bbox": [ + 307, + 107, + 315, + 116 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 106, + 332, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 106, + 340, + 117 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "(See \"Synthetic data\" in Section 5.1). We", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "then leverage these different kinds of shifted datasets to construct shifted models and then assess the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 476, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 476, + 139 + ], + "score": 1.0, + "content": "validity of the recourses output by our framework and other baselines w.r.t. the shifted models.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "We generate recourses using our framework and baselines CFE and AR for different predictive models", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 231, + 168 + ], + "score": 1.0, + "content": "(LR, DNN) and cost functions (", + "type": "text" + }, + { + "bbox": [ + 231, + 156, + 241, + 167 + ], + "score": 0.83, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 155, + 506, + 168 + ], + "score": 1.0, + "content": "distance, PFC). Figure 2 captures the results of this experiment for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 199, + 178 + ], + "score": 1.0, + "content": "DNN model both with", + "type": "text" + }, + { + "bbox": [ + 199, + 167, + 209, + 177 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "distance and PFC cost functions. Results with other models are included", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 267, + 189 + ], + "score": 1.0, + "content": "in the Appendix. It can be seen that the", + "type": "text" + }, + { + "bbox": [ + 267, + 179, + 274, + 187 + ], + "score": 0.35, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 178, + 505, + 189 + ], + "score": 1.0, + "content": "-axis of each of these plots captures the magnitude of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "dataset shift, and the y-axis captures the validity of the recourses w.r.t. the corresponding shifted", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "score": 1.0, + "content": "model. Standard error bars obtained by averaging the results over 5 runs are also shown.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "It can be seen that as the magnitude of the distribution shift increases, validity of the recourses", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "score": 1.0, + "content": "generated by all the methods starts dropping. This trend prevailed across mean, variance, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 238, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 504, + 250 + ], + "score": 1.0, + "content": "combination (mean and variance) shifts. It can also be seen that the rate at which validity of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "recourses generated by our method, ROAR-LIME, drops is much smaller compared to that of other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "baselines CFE and AR-LIME. Furthermore, our method exhibits the highest validity compared", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "to the baselines as the magnitude of the distribution shift increases. CFE seems to be the worst", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "performing baseline and the validity of the recourses generated by CFE drops very sharply even at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 293, + 265, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 265, + 304 + ], + "score": 1.0, + "content": "small magnitudes of distribution shifts.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 270, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 272, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 272, + 329 + ], + "score": 1.0, + "content": "6 Conclusions & Future Work", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "We proposed a novel framework, RObust Algorithmic Recourse (ROAR), to address the critical but", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "under-explored issue of recourse robustness to model updates. To this end, we introduced a novel", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "minimax objective to generate recourses that are robust to model shifts, and leveraged adversarial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "training to optimize this objective. We also presented novel theoretical results which demonstrate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "that recourses without accounting for model shifts are likely to be invalidated, underscoring the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "necessity of ROAR. Furthermore, we also showed that the additional cost incurred by robust recourses", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "generated by ROAR are bounded. Extensive experimentation with real world and synthetic datasets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "demonstrated that recourses using ROAR are highly robust to model shifts induced by a range of data", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "distribution shifts. Our work also paves the way for further research into techniques for generating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "robust recourses. For instance, it would be valuable to further analyze the tradeoff between recourse", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 443, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 504, + 456 + ], + "score": 1.0, + "content": "robustness and cost to better understand the impacts to affected individuals. Other interesting future", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "directions include non-linear extensions that leverage novel local linear approximation methods that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 466, + 203, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 203, + 479 + ], + "score": 1.0, + "content": "improve on LIME [33].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 488, + 207, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 208, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 208, + 504 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "in part by the NSF awards #IIS-2008461 and #IIS-2040989, and research awards from the Harvard", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "Data Science Institute, Amazon, Bayer, and Google. SJ would like to acknowledge the support of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Center for Research on Computation and Society (CRCS) at the Harvard John A. Paulson School of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Engineering and Applied Sciences. 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The hidden assumptions behind", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 128, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 128, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "counterfactual explanations and principal reasons. Proceedings of the 2020 Conference on", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 127, + 701, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 127, + 701, + 506, + 712 + ], + "score": 1.0, + "content": "Fairness, Accountability, and Transparency, Jan 2020. doi: 10.1145/3351095.3372830. 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Figure 2 captures the results of this experiment for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 199, + 178 + ], + "score": 1.0, + "content": "DNN model both with", + "type": "text" + }, + { + "bbox": [ + 199, + 167, + 209, + 177 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "distance and PFC cost functions. Results with other models are included", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 267, + 189 + ], + "score": 1.0, + "content": "in the Appendix. It can be seen that the", + "type": "text" + }, + { + "bbox": [ + 267, + 179, + 274, + 187 + ], + "score": 0.35, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 178, + 505, + 189 + ], + "score": 1.0, + "content": "-axis of each of these plots captures the magnitude of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "dataset shift, and the y-axis captures the validity of the recourses w.r.t. the corresponding shifted", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "score": 1.0, + "content": "model. Standard error bars obtained by averaging the results over 5 runs are also shown.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 144, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "It can be seen that as the magnitude of the distribution shift increases, validity of the recourses", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "score": 1.0, + "content": "generated by all the methods starts dropping. 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CFE seems to be the worst", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "performing baseline and the validity of the recourses generated by CFE drops very sharply even at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 293, + 265, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 265, + 304 + ], + "score": 1.0, + "content": "small magnitudes of distribution shifts.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 216, + 506, + 304 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 270, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 272, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 272, + 329 + ], + "score": 1.0, + "content": "6 Conclusions & Future Work", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "We proposed a novel framework, RObust Algorithmic Recourse (ROAR), to address the critical but", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "under-explored issue of recourse robustness to model updates. 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We also presented novel theoretical results which demonstrate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "that recourses without accounting for model shifts are likely to be invalidated, underscoring the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "necessity of ROAR. Furthermore, we also showed that the additional cost incurred by robust recourses", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "generated by ROAR are bounded. Extensive experimentation with real world and synthetic datasets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "demonstrated that recourses using ROAR are highly robust to model shifts induced by a range of data", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "distribution shifts. Our work also paves the way for further research into techniques for generating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "robust recourses. For instance, it would be valuable to further analyze the tradeoff between recourse", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 443, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 504, + 456 + ], + "score": 1.0, + "content": "robustness and cost to better understand the impacts to affected individuals. Other interesting future", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "directions include non-linear extensions that leverage novel local linear approximation methods that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 466, + 203, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 203, + 479 + ], + "score": 1.0, + "content": "improve on LIME [33].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 335, + 506, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 488, + 207, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 208, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 208, + 504 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "in part by the NSF awards #IIS-2008461 and #IIS-2040989, and research awards from the Harvard", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "Data Science Institute, Amazon, Bayer, and Google. SJ would like to acknowledge the support of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Center for Research on Computation and Society (CRCS) at the Harvard John A. 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Input:x s.t. fω(x)=O,fw,λ>O,△,learning rate α >0. Initialize x" =x,g =0
repeat = arg maxs∈△ l(fw+8(x"),1)
g =∀[e(fw+8(x"),1)+ λc(x",x)]
x" -=ag
until convergence
Return x"
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Correction ShiftTemporal ShiftGeospatial Shift
Model CostRecourse CFEAvgCost 1.02 ± 0.18MValidity 1.00±0.00MValidity 0.54± 0.27AvgCost 3.57 ± 1.14MValidity 1.00±0.00MValidity 0.31±0.09Avg Cost 8.37±0.73MValidity 0.98±0.03MValidity 0.29±0.09
LRL10.85 ± 0.141.00 ± 0.000.53 ± 0.211.50± 0.281.00 ± 0.000.16 ± 0.065.29 ± 0.281.00 ± 0.000.43 ± 0.14
AR3.14 ± 0.250.99 ± 0.010.98 ±0.0210.88 ± 1.671.00 ± 0.000.67 ± 0.19
ROAR3.13 ± 0.321.00 ± 0.000.94 ± 0.08 0.93 ± 0.07NA
MINT4.73 ± 1.561.00 ± 0.00NANANANANA
ROAR-MINT6.77 ± 0.351.00 ± 0.001.00 ± 0.00NANANANANANA
CFE0.03±0.021.00 ±0.000.56±0.330.24±0.091.00 ± 0.000.26± 0.110.34± 0.041.00±0.000.18 ±0.10
PFCAR0.09 ± 0.021.00 ± 0.000.54 ± 0.270.11 ± 0.021.00 ± 0.000.09 ± 0.050.32 ±0.031.00 ±0.000.24 ± 0.11
ROAR MINT0.36±0.081.00 ± 0.001.00 ± 0.000.44 ± 0.120.99 ± 0.010.98 ± 0.011.20 ± 0.101.00 ± 0.000.91± 0.07
ROAR-MINT1.00 ± 1.151.00 ± 0.00 1.00 ± 0.000.95±0.08 1.00 ± 0.00NANA NANANANANA
L1CFE1.23 ± 0.05 0.55 ±0.101.00± 0.000.47±0.06NA 3.78±0.681.00 ± 0.00NA 0.52±0.09NANANA
AR-LIME0.38 ± 0.150.16 ±0.100.31 ± 0.061.39 ± 0.130.59 ± 0.110.65 ± 0.1710.09± 0.71 9.02 ±1.571.00 ± 0.000.48±0.09 0.83 ±0.10
ROAR-LIME0.76± 0.06
NN1.83 ± 0.190.78 ±0.060.72 ± 0.104.90±0.240.98 ±0.020.97 ±0.0221.05 ± 3.581.00 ± 0.000.97 ±0.03
MINT2.24 ± 1.250.81 ± 0.020.63 ± 0.11NANANANANANA
ROAR-MINT8.59 ± 1.700.90 ±0.030.84 ± 0.04NANANANANANA
CFE0.06±0.021.00± 0.000.51 ± 0.120.19±0.061.00±0.000.50± 0.130.48± 0.061.00±0.000.30±0.14
AR-LIME0.06± 0.030.49 ± 0.110.56± 0.150.11 ± 0.010.54 ± 0.080.62 ± 0.120.78 ± 0.150.84 ± 0.060.82 ± 0.11
PFC ROAR-LIME0.64 ± 0.080.85 ± 0.070.82 ± 0.050.37 ±0.070.99 ± 0.010.99 ±0.01.66 ± 0.211.00 ±0.000.97 ± 0.04
0.60 ± 0.160.82 ±0.070.64 ± 0.15NANANANANA
MINT ROAR-MINT0.60 ±0.070.91± 0.040.81 ± 0.04NANANANANANA NA
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--- /dev/null +++ b/parse/train/HylTBhA5tQ/images/fd21a0bfc81f6dfbdd2d2f30e091f716cbebc45d55bb2aab7e2781590d8269aa.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ffe7028b3589496271eeb67414974bedf6cbef171183f5bd755447e7b64ae59 +size 5777 diff --git a/parse/train/Hyx4knR9Ym/Hyx4knR9Ym.md b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym.md new file mode 100644 index 0000000000000000000000000000000000000000..bbb0ce9c88b938c6dc1cb95ddaa1dc97af7157d0 --- /dev/null +++ b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym.md @@ -0,0 +1,680 @@ +# GENERALIZABLE ADVERSARIAL TRAINING VIA SPECTRAL NORMALIZATION + +Farzan Farnia∗, Jesse M. Zhang∗, David N. Tse Department of Electrical Engineering Stanford University {farnia,jessez,dntse}@stanford.edu + +# ABSTRACT + +Deep neural networks (DNNs) have set benchmarks on a wide array of supervised learning tasks. Trained DNNs, however, often lack robustness to minor adversarial perturbations to the input, which undermines their true practicality. Recent works have increased the robustness of DNNs by fitting networks using adversarially-perturbed training samples, but the improved performance can still be far below the performance seen in non-adversarial settings. A significant portion of this gap can be attributed to the decrease in generalization performance due to adversarial training. In this work, we extend the notion of margin loss to adversarial settings and bound the generalization error for DNNs trained under several well-known gradient-based attack schemes, motivating an effective regularization scheme based on spectral normalization of the DNN’s weight matrices. We also provide a computationally-efficient method for normalizing the spectral norm of convolutional layers with arbitrary stride and padding schemes in deep convolutional networks. We evaluate the power of spectral normalization extensively on combinations of datasets, network architectures, and adversarial training schemes. + +# 1 INTRODUCTION + +Despite their impressive performance on many supervised learning tasks, deep neural networks (DNNs) are often highly susceptible to adversarial perturbations imperceptible to the human eye (Szegedy et al., 2013; Goodfellow et al., 2014b). These “adversarial attacks" have received enormous attention in the machine learning literature over recent years (Goodfellow et al., 2014b; Moosavi Dezfooli et al., 2016; Carlini & Wagner, 2016; Kurakin et al., 2016; Papernot et al., 2016; Carlini & Wagner, 2017; Papernot et al., 2017; Madry et al., 2018; Tramèr et al., 2018). Adversarial attack studies have mainly focused on developing effective attack and defense schemes. While attack schemes attempt to mislead a trained classifier via additive perturbations to the input, defense mechanisms aim to train classifiers robust to these perturbations. Although existing defense methods result in considerably better performance compared to standard training methods, the improved performance can still be far below the performance in non-adversarial settings (Athalye et al., 2018; Schmidt et al., 2018). + +A standard adversarial training scheme involves fitting a classifier using adversarially-perturbed samples (Szegedy et al., 2013; Goodfellow et al., 2014b) with the intention of producing a trained classifier with better robustness to attacks on future (i.e. test) samples. Madry et al. (2018) provides a robust optimization interpretation of the adversarial training approach, demonstrating that this strategy finds the optimal classifier minimizing the average worst-case loss over an adversarial ball centered at each training sample. This minimax interpretation can also be extended to distributionally-robust training methods (Sinha et al., 2018) where the offered robustness is over a Wasserstein-ball around the empirical distribution of training data. + +Recently, Schmidt et al. (2018) have shown that standard adversarial training produces networks that generalize poorly. The performance of adversarially-trained DNNs over test samples can be significantly worse than their training performance, and this gap can be far greater than the generalization gap achieved using standard empirical risk minimization (ERM). This discrepancy suggests that the overall adversarial test performance can be improved by applying effective regularization schemes during adversarial training. + +![](images/ef52baaee3ec3221278850c219c777bc05008a87327e9f87b9f3cb88ff84e222.jpg) +Figure 1: Adversarial training performance with and without spectral normalization (SN) for AlexNet fit on CIFAR10. The gain in the final test accuracies for FGM, PGM, and WRM after spectral normalization are 0.09, 0.11, and 0.04, respectively (see Table 1 in the Appendix). For FGM and PGM, perturbations have $\ell _ { 2 }$ magnitude 2.44. + +In this work, we propose using spectral normalization (SN) (Miyato et al., 2018) as a computationallyefficient and statistically-powerful regularization scheme for adversarial training of DNNs. SN has been successfully implemented and applied for DNNs in the context of generative adversarial networks (GANs) (Goodfellow et al., 2014a), resulting in state-of-the-art deep generative models for several benchmark tasks (Miyato et al., 2018). Moreover, SN (Tsuzuku et al., 2018) and other similar Lipschitz regularization techniques (Cisse et al., 2017) have been successfully applied in non-adversarial training settings to improve the robustness of ERM-trained networks to adversarial attacks. The theoretical results in (Bartlett et al., 2017; Neyshabur et al., 2017a) and empirical results in (Yoshida & Miyato, 2017) also suggest that SN can close the generalization gap for DNNs in non-adversarial ERM setting. + +On the theoretical front, we extend the standard notion of margin loss to adversarial settings. We leverage the PAC-Bayes generalization framework (McAllester, 1999) to prove generalization bounds for spectrally-normalized DNNs in terms of our defined adversarial margin loss. We obtain adversarial generalization error bounds for three well-known gradient-based attack schemes: fast gradient method (FGM) (Goodfellow et al., 2014b), projected gradient method (PGM) (Kurakin et al., 2016), and Wasserstein risk minimization (WRM) (Sinha et al., 2018). Our theoretical analysis shows that the adversarial generalization error will vanish by applying SN to all layers. + +On the empirical front, we show that SN can significantly improve the test performance of adversarially-trained DNNs. We perform numerical experiments over various standard datasets and DNN architectures. In almost all of our experiments, we obtain a better test performance after applying SN. For example, Figure 1 shows the training and validation performance for AlexNet fit on the CIFAR10 dataset using FGM, PGM, and WRM, resulting in adversarial test accuracy improvements of 9, 11, and 4 percent, respectively. To perform our numerical experiments, we develop a computationally-efficient approach for normalizing the spectral norm of convolution layers with arbitrary stride and padding schemes. To summarize, the main contributions of this work are: + +1. Proposing SN as a regularization scheme for adversarial training of DNNs, +2. Extending concepts of margin-based generalization analysis to adversarial settings and proving margin-based generalization bounds for three gradient-based adversarial attack schemes, +3. Developing an efficient method for normalizing the spectral norm of convolutional layers in deep convolution networks, +4. Numerically demonstrating the improved test and generalization performance of DNNs trained with SN. + +# 2 PRELIMINARIES + +In this section, we first review some standard concepts of margin-based generalization analysis in learning theory. We then extend these notions to adversarial training settings. + +2.1 SUPERVISED LEARNING, DEEP NEURAL NETWORKS, GENERALIZATION ERROR + +Consider samples $\left\{ ( \mathbf { x } _ { 1 } , y _ { 1 } ) , \dotsc , ( \mathbf { x } _ { n } , y _ { n } ) \right\}$ drawn i.i.d from underlying distribution $P _ { \mathbf { X } , Y }$ . We suppose $\mathbf x \in \mathcal X$ and $Y \in \{ 1 , 2 , \dots , m \}$ where $m$ represents the number of different labels. Given loss function $\ell$ and function class $\mathcal { F } = \left\{ f _ { \mathbf { w } } , \mathbf { w } \in \mathcal { W } \right\}$ parameterized by w, a supervised learner aims to find the optimal function in $\mathcal { F }$ minimizing the expected loss (risk) averaged over the underlying distribution $P$ . + +We consider $\mathcal { F } _ { \mathrm { n n } }$ as the class of $d$ -layer neural networks with $h$ hidden units per layer and activation functions $\sigma : \mathbb { R } \mathbb { R }$ . Each $f _ { \mathbf { w } } : \mathcal { X } \mathbf { R } ^ { m }$ in $\mathcal { F } _ { \mathrm { n n } }$ maps a data point $\mathbf { x }$ to an $m$ -dimensional vector. Specifically, we can express each $f _ { \mathbf { w } } \in \mathcal { F } _ { \mathrm { n n } }$ as $f _ { \mathbf { w } } ( \mathbf { x } \bar { ) = } { \mathbf { W } _ { d } } \bar { \sigma } ( { \mathbf { W } _ { d - 1 } } \cdot \cdot \cdot \sigma ( { \mathbf { W } _ { 1 } } \mathbf { x } ) \cdot \cdot \cdot ) )$ . We use $\left. \mathbf { W } _ { i } \right. _ { 2 }$ to denote the spectral norm of matrix $\mathbf { W } _ { i }$ , defined as the largest singular value of $\mathbf { W } _ { i }$ , and $\Vert \mathbf { W } _ { i } \Vert _ { F }$ to denote $\mathbf { W } _ { i }$ ’s Frobenius norm. + +A classifier $f _ { \mathbf { w } }$ ’s performance over the true distribution of data can be different from the training performance over the empirical distribution of training samples $\hat { P }$ . The difference between the empirical and true averaged losses, evaluated on respectively training and test samples, is called the generalization error. Similar to Neyshabur et al. (2017a), we evaluate a DNN’s generalization performance using its expected margin loss defined for margin parameter $\gamma > 0$ as + +$$ +L _ { \gamma } ( f _ { \mathbf { w } } ) : = P \bigg ( f _ { \mathbf { w } } ( \mathbf { X } ) [ Y ] \leq \gamma + \operatorname* { m a x } _ { j \neq Y } f _ { \mathbf { w } } ( \mathbf { X } ) [ j ] \bigg ) , +$$ + +where $f _ { \mathbf { w } } ( \mathbf { X } ) [ j ]$ denotes the $j$ th entry of $f _ { \mathbf { w } } ( \mathbf { X } ) \in \mathbb { R } ^ { m }$ . For a given data point $\mathbf { X }$ , we predict the label corresponding to the maximum entry of $f _ { \mathbf { w } } ( \mathbf { X } )$ . Also, we use $\widehat { L } _ { \gamma } ( f _ { \mathbf { w } } )$ to denote the empirical margin loss averaged over the training samples. The goal of margin-based generalization analysis is to provide theoretical comparison between the true and empirical margin risks. + +# 2.2 ADVERSARIAL ATTACKS, ADVERSARIAL TRAINING + +A supervised learner observes only the training samples and hence does not know the true distribution of data. Then, a standard approach to train a classifier is to minimize the empirical expected loss $\ell$ over function class $\mathcal { F } = \{ f _ { \mathbf { w } } : \mathbf { w } \in \mathcal { W } \}$ , which is + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathcal { W } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } _ { i } ) , y _ { i } \big ) . +$$ + +This approach is called empirical risk minimization (ERM). For better optimization performance, the loss function $\ell$ is commonly chosen to be smooth. Hence, 0-1 and margin losses are replaced by smooth surrogate loss functions such as the cross-entropy loss. However, we still use the margin loss as defined in (1) for evaluating the test and generalization performance of DNN classifiers. + +While ERM training usually achieves good performance over DNNs, several recent observations reveal that adding some adversarially-chosen perturbation to each sample can significantly drop the trained DNN’s performance. Given norm function $\| \cdot \|$ and adversarial noise power $\epsilon > 0$ , the adversarial additive noise for sample $\left( \mathbf { x } , y \right)$ and classifier $f _ { \mathbf { w } }$ is defined to be + +$$ +\delta _ { \mathbf w } ^ { \mathrm { a d v } } ( \mathbf x ) : = \operatorname * { a r g m a x } _ { \| \delta \| \leq \epsilon } \ell \big ( f _ { \mathbf w } ( \mathbf x + \pmb \delta ) , y \big ) . +$$ + +To provide adversarial robustness against the above attack scheme, a standard technique, which is called adversarial training, follows ERM training over the adversarially-perturbed samples by solving + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathcal { W } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell \left( f _ { \mathbf { w } } \left( \mathbf { x } _ { i } + \delta _ { \mathbf { w } } ^ { \mathrm { a d v } } ( \mathbf { x } _ { i } ) \right) , y _ { i } \right) : = \operatorname* { m i n } _ { \mathbf { w } \in \mathcal { W } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \operatorname* { m a x } _ { \| \delta _ { i } \| \leq \epsilon } \ell \left( f _ { \mathbf { w } } ( \mathbf { x } _ { i } + \delta _ { i } ) , y _ { i } \right) . +$$ + +However, (3) and (4) are intractable optimization problems. Therefore, several schemes have been proposed in the literature to approximate the optimal solution of (3). In this work, we analyze the generalization performance of the following three gradient-based methods for approximating the solution to (3). We note that several other attack schemes such as DeepFool (Moosavi Dezfooli et al., 2016), CW attacks (Carlini & Wagner, 2017), target and least-likely attacks (Kurakin et al., 2016) have been introduced and examined in the literature, which can lead to interesting future directions for this work. + +1. Fast Gradient Method (FGM) (Goodfellow et al., 2014b): FGM approximates the solution to (3) by considering a linearized DNN loss around a given data point. Hence, FGM perturbs $\left( \mathbf { x } , y \right)$ by adding the following noise vector: + +$$ +\delta _ { \mathbf { w } } ^ { \mathrm { f g m } } ( \mathbf { x } ) : = \underset { \| \delta \| \leq \epsilon } { \mathrm { a r g m a x } } \delta ^ { T } \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) . +$$ + +For the special case of $\ell _ { \infty }$ -norm $\| \cdot \| _ { \infty }$ , the above representation of FGM recovers the fast gradient sign method (FGSM) where each data point $\left( \mathbf { x } , y \right)$ is perturbed by the $\epsilon$ -normalized sign vector of the loss’s gradient. For $\ell _ { 2 }$ -norm $\| \cdot \| _ { 2 }$ , we similarly normalize the loss’s gradient vector to have $\epsilon$ Euclidean norm. + +2. Projected Gradient Method (PGM) (Kurakin et al., 2016): PGM is the iterative version of FGM and applies projected gradient descent to solve (3). PGM follows the following update rules for a given $r$ number of steps: + +$$ +\begin{array} { r l } { \forall 1 \le i \le r : } & { \delta _ { \mathbf { w } } ^ { \mathrm { p g m } , i + 1 } ( \mathbf { x } ) : = \displaystyle \prod _ { \boldsymbol { \epsilon } _ { \epsilon , \parallel } \cdot \parallel ^ { ( 0 ) } } \bigl \{ \delta _ { \mathbf { w } } ^ { \mathrm { p g m } , i } ( \mathbf { x } ) + \alpha \nu _ { \mathbf { w } } ^ { ( i ) } \bigr \} , } \\ & { \nu _ { \mathbf { w } } ^ { ( i ) } : = \underset { \parallel \delta \parallel \le 1 } { \arg \operatorname* { m a x } } \delta ^ { T } \nabla _ { \mathbf { x } } \ell \bigl ( f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { p g m } , i } ( \mathbf { x } ) ) , y \bigr ) . } \end{array} +$$ + +Here, we first find the direction $\nu _ { \mathbf { w } } ^ { ( i ) }$ along which the loss at the $i$ th perturbed point changes the most, and then we move the perturbed point along this direction by stepsize $\alpha$ followed by projecting the resulting perturbation onto the set $\left\{ \delta : \left\| \delta \right\| \leq \epsilon \right\}$ with $\epsilon$ -bounded norm. + +3. Wasserstein Risk Minimization (WRM) (Sinha et al., 2018): WRM solves the following variant of (3) for data-point $\left( \mathbf { x } , y \right)$ where the norm constraint in (3) is replaced by a norm-squared Lagrangian penalty term: + +$$ +\delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) : = \operatorname * { a r g m a x } _ { \delta } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } + \pmb { \delta } ) , y \big ) - \frac { \lambda } { 2 } \| \pmb { \delta } \| ^ { 2 } . +$$ + +As discussed earlier, the optimization problem (3) is generally intractable. However, in the case of Euclidean norm $\| \cdot \| _ { 2 }$ , if we assume $\nabla _ { \mathbf x } \ell ( f _ { \mathbf w } ( \mathbf x ) , y )$ ’s Lipschitz constant is upper-bounded by $\lambda$ , then WRM optimization (7) results in solving a convex optimization problem and can be efficiently solved using gradient methods. + +To obtain efficient adversarial defense schemes, we can substitute $\delta _ { \mathbf { w } } ^ { \mathrm { f g m } }$ , $\delta _ { \mathbf { w } } ^ { \mathrm { p g m } }$ , or $\delta _ { \mathbf { w } } ^ { \mathrm { w r m } }$ for $\delta _ { \mathbf { w } } ^ { \mathrm { a d v } }$ in (4). Instead of fitting the classifier over true adversarial examples, which are NP-hard to obtain, we can instead train the DNN over FGM, PGM, or WRM-adversarially perturbed samples. + +# 2.3 ADVERSARIAL GENERALIZATION ERROR + +The goal of adversarial training is to improve the robustness against adversarial attacks on not only the training samples but also on test samples; however, the adversarial training problem (4) focuses only on the training samples. To evaluate the adversarial generalization performance, we extend the notion of margin loss defined earlier in (1) to adversarial training settings by defining the adversarial margin loss as + +$$ +L _ { \gamma } ^ { \mathrm { a d v } } ( f _ { \mathbf { w } } ) = P \bigg ( f _ { \mathbf { w } } ( \mathbf { X } + \delta _ { \mathbf { w } } ^ { \mathrm { a d v } } ( \mathbf { X } ) ) [ Y ] \leq \gamma + \operatorname* { m a x } _ { j \neq Y } f _ { \mathbf { w } } \big ( \mathbf { X } + \delta _ { \mathbf { w } } ^ { \mathrm { a d v } } ( \mathbf { X } ) \big ) [ j ] \bigg ) . +$$ + +Here, we measure the margin loss over adversarially-perturbed samples, and we use $\widehat { L } _ { \gamma } ^ { \mathrm { a d v } } ( f _ { \mathbf { w } } )$ to denote the empirical adversarial margin loss. We also use $L _ { \gamma } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) , L _ { \gamma } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } )$ , and $L _ { \gamma } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } )$ to denote the adversarial margin losses with FGM (5), PGM (6), and WRM (7) attacks, respectively. + +# 3 MARGIN-BASED ADVERSARIAL GENERALIZATION BOUNDS + +As previously discussed, generalization performance can be different between adversarial and nonadversarial settings. In this section, we provide generalization bounds for DNN classifiers under adversarial attacks in terms of the spectral norms of the trained DNN’s weight matrices. The bounds motivate regularizing these spectral norms in order to limit the DNN’s capacity and improve its generalization performance under adversarial attacks. + +We use the PAC-Bayes framework (McAllester, 1999; 2003) to prove our main results. To derive adversarial generalization error bounds for DNNs with smooth activation functions $\sigma$ , we first extend a recent result on the margin-based generalization bound for the ReLU activation function (Neyshabur et al., 2017a) to general 1-Lipschitz activation functions. + +Theorem 1. Consider $\mathcal { F } _ { n n } = \{ f _ { \mathbf { w } } : \mathbf { w } \in \mathbf { W } \}$ the class of d hidden-layer neural networks with $h$ units per hidden-layer with 1-Lipschitz activation $\sigma$ satisfying $\sigma ( 0 ) = 0$ . Suppose that $\mathcal { X }$ , $\mathbf { X }$ ’s support set, is norm-bounded as $\| \mathbf { x } \| _ { 2 } \leq B$ , $\forall \mathbf { x } \in { \mathcal { X } }$ . Also assume for constant $M \geq 1$ any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ satisfies + +$$ +\forall i : ~ \frac { 1 } { M } \leq \frac { \| \mathbf { W } _ { i } \| _ { 2 } } { \beta _ { \mathbf { w } } } \leq M , \quad \beta _ { \mathbf { w } } : = \big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) ^ { 1 / d } . +$$ + +Here $\beta _ { \mathbf { w } }$ denotes the geometric mean of $f _ { \mathbf { w } }$ ’s spectral norms across all layers. Then, for any $\eta , \gamma > 0$ , with probability at least $1 - \eta$ for any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ we have: + +$$ +L _ { 0 } ( f _ { \mathbf { w } } ) \leq \widehat { L } _ { \gamma } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { B ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi ^ { \mathrm { e r m } } ( f _ { \mathbf { w } } ) + d \log \frac { d n \log M } { \eta } } { \gamma ^ { 2 } n } } \bigg ) , +$$ + +$\begin{array} { r } { \Phi ^ { \mathrm { e r m } } ( f _ { \mathbf { w } } ) : = \left( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } \right) \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } . } \end{array}$ + +Proof. We defer the proof to the Appendix. + +We now generalize this result to adversarial settings where the DNN’s performance is evaluated under adversarial attacks. We prove three separate adversarial generalization error bounds for FGM, PGM, and WRM attacks. + +For the following results, we consider $\mathcal { F } _ { \mathrm { n n } }$ , the class of neural nets defined in Theorem 1. Moreover, we assume that the training loss $\ell ( \hat { y } , y )$ and its first-order derivative are 1-Lipschitz. Similar to Sinha et al. (2018), we assume the activation $\sigma$ is smooth and its derivative $\sigma ^ { \prime }$ is 1-Lipschitz. This class of activations include ELU (Clevert et al., 2015) and tanh functions but not the ReLU function. However, our numerical results in Table 1 from the Appendix suggest similar generalization performance between ELU and ReLU activations. + +Theorem 2. Consider $\mathcal { F } _ { n n }$ , $\mathcal { X }$ in Theorem 1 and training loss function $\ell$ satisfying the assumptions stated above. We consider an FGM attack with noise power $\epsilon$ according to Euclidean norm $\| \cdot \| _ { 2 }$ . For any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ assume $\boldsymbol { \kappa } \leq \| \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) \| _ { 2 }$ holds for constant $\kappa > 0$ , any $y \in \mathcal { V }$ , and any $\mathbf { x } \in B _ { \epsilon , \parallel \cdot \parallel _ { 2 } } ( \mathcal { X } )$ -close to $\mathbf { X }$ ’s support set. Then, for any $\eta , \gamma > 0$ with probability $1 - \eta$ the following bound holds for the FGM margin loss of any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ + +$$ +\begin{array} { r } { L _ { 0 } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) \leq \widehat { L } _ { \gamma } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { ( B + \epsilon ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \epsilon , \kappa } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) + d \log \frac { d n \log M } { \eta } } { \gamma ^ { 2 } n } } \bigg ) , } \end{array} +$$ + +$$ +\begin{array} { r } { \Phi _ { \epsilon , \kappa } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) : = \left\{ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ( 1 + ( \epsilon / \kappa ) ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ) \right\} ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } . } \end{array} +$$ + +Proof. We defer the proof to the Appendix. + +Note that the above theorem assumes that the change rate for the loss function around test samples is at least $\kappa$ , which gives a baseline for measuring the attack power $\epsilon$ . In our numerical experiments, we validate this assumption over standard image recognition tasks. Next, we generalize this result to adversarial settings with PGM attack, i.e. the iterative version of FGM attack. + +Theorem 3. Consider $\mathcal { F } _ { n n } , \mathcal { X }$ and training loss function \` for which the assumptions in Theorem 2 hold. We consider a PGM attack with noise power  given Euclidean norm $\| \cdot \| _ { 2 }$ , $r$ iterations for attack, and stepsize $\alpha$ . Then, for any $\eta , \gamma > 0$ with probability $1 - \eta$ the following bound applies to the PGM margin loss of any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ + +$$ +L _ { 0 } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) \le \widehat { L } _ { \gamma } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { ( B + \epsilon ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \epsilon , \kappa , r , \alpha } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) + d \log \frac { r d n \log M } { \eta } } { \gamma ^ { 2 } n } } \bigg ) . +$$ + +Here we define $\Phi _ { \epsilon , \kappa , r , \alpha } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } )$ as the following expression + +$$ +\left\{ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \left( 1 + ( \alpha / \kappa ) { \frac { 1 - ( 2 \alpha / \kappa ) ^ { r } \varlimsup ( \nabla \ell \circ f _ { \mathbf { w } } ) ^ { r } } { 1 - ( 2 \alpha / \kappa ) \varlimsup ( \nabla \ell \circ f _ { \mathbf { w } } ) } } ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \right) \right\} ^ { 2 } \sum _ { i = 1 } ^ { d } { \frac { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } } +$$ + +where $\begin{array} { r } { \varlimsup ( \nabla \ell \circ f _ { \mathbf { w } } ) : = \bigl ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \bigr ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } } \end{array}$ provides an upper-bound on the Lipschitz constant of $\nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } ) , y )$ . + +Proof. We defer the proof to the Appendix. + +In the above result, notice that if $\overline { { \mathrm { l i p } } } ( \nabla \ell \circ f _ { \mathbf { w } } ) / \kappa < 1 / ( 2 \alpha )$ then for any number of gradient steps the PGM margin-based generalization bound will grow the FGM generalization error bound in Theorem 2 by factor $1 / \big ( 1 - ( \overline { { 2 \alpha / \kappa } } ) \overline { { \mathrm { l i p } } } ( \nabla \ell \circ f _ { \mathbf { w } } ) \big )$ . We next extend our adversarial generalization analysis to WRM attacks. + +Theorem 4. For neural net class $\mathcal { F } _ { n n }$ and training loss $\ell$ satisfying Theorem 2’s assumptions, consider a WRM attack with Lagrangian coefficient $\lambda$ and Euclidean norm $\| \cdot \| _ { 2 }$ . Given parameter $0 < \tau < 1$ , assume $\overline { { \mathrm { l i p } } } ( \nabla \ell \circ f _ { \mathbf { w } } )$ defined in Theorem 3 is upper-bounded by $\lambda ( 1 - \tau )$ for any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ . For any $\eta > 0$ , the following WRM margin-based generalization bound holds with probability $1 - \eta$ for any $f _ { \mathbf { w } } \in \mathcal { F } _ { n n }$ : + +$$ +\begin{array} { r } { \mathbf { \neg _ { v r m } } ( f _ { \mathbf { w } } ) \le \widehat { L } _ { \gamma } ^ { \operatorname { w r m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { ( B + \frac { 1 } { \lambda } \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \lambda } ^ { \operatorname { w r m } } ( f _ { \mathbf { w } } ) + d \log \frac { d n \log M } { \tau \eta } } { \gamma ^ { 2 } n } } \bigg ) } \end{array} +$$ + +where we define + +$$ +\Phi _ { \lambda } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } ) : = \{ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ( 1 + \frac { 1 } { \lambda - \operatorname* { l i m } ( \nabla \ell \circ f _ { \mathbf { w } } ) } ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \} ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } . +$$ + +Proof. We defer the proof to the Appendix. + +As discussed by Sinha et al. (2018), the condition $\mathrm { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) < \lambda$ for the actual Lipschitz constant of $\nabla \ell \circ f _ { \mathbf { w } }$ is in fact required to guarantee WRM’s convergence to the global solution. Notice that the WRM generalization error bound in Theorem 4 is bounded by the product of $\frac { 1 } { \lambda - \mathrm { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) }$ and the FGM generalization bound in Theorem 2. + +# 4 SPECTRAL NORMALIZATION OF CONVOLUTIONAL LAYERS + +To control the Lipschitz constant of our trained network, we need to ensure that the spectral norm associated with each linear operation in the network does not exceed some pre-specified $\beta$ . For fully-connected layers (i.e. regular matrix multiplication), please see Appendix B. For a general class of linear operations including convolution, Tsuzuku et al. (2018) propose to compute the operation’s spectral norm through computing the gradient of the Euclidean norm of the operation’s output. Here, we leverage the deconvolution operation to further simplify and accelerate computing the spectral norm of the convolution operation. Additionally, Sedghi et al. (2018) develop a method for computing all the singular values including the largest one, i.e. the spectral norm. While elegant, the method only applies to convolution filters with stride 1 and zero-padding. However, in practice the normalization factor depends on the stride size and padding scheme governing the convolution operation. Here we develop an efficient approach for computing the maximum singular value, i.e. spectral norm, of convolutional layers with arbitary stride and padding schemes. Note that, as also discussed by Gouk et al. (2018), the ith convolutional layer output feature map $\psi _ { i }$ is a linear operation of the input $X$ : + +$$ +\psi _ { i } ( X ) = \sum _ { j = 1 } ^ { M } F _ { i , j } \star X _ { j } , +$$ + +where $X$ has $M$ feature maps, $F _ { i , j }$ is a filter, and $\star$ denotes the convolution operation (which also encapsulates stride size and padding scheme). For simplicity, we ignore the additive bias terms here. By vectorizing $X$ and letting $V _ { i , j }$ represent the overall linear operation associated with $F _ { i , j }$ , we see that + +$$ +\psi _ { i } ( X ) = [ V _ { 1 , 1 } \quad \cdot \cdot \cdot \quad V _ { 1 , M } ] X , +$$ + +and therefore the overall convolution operation can be described using + +$$ +\psi ( X ) = \left[ \begin{array} { c c c } { V _ { 1 , 1 } } & { \ldots } & { V _ { 1 , M } } \\ { \vdots } & { \ddots } & { \vdots } \\ { V _ { N , 1 } } & { \ldots } & { V _ { N , M } } \end{array} \right] X = W X . +$$ + +While explicitly reconstructing $W$ is expensive, we can still compute $\sigma ( W )$ , the spectral norm of $W$ , by leveraging the convolution transpose operation implemented by several modern-day deep learning packages. This allows us to efficiently performs matrix multiplication with $W ^ { T }$ without explicitly constructing $W$ . Therefore we can approximate $\sigma ( W )$ using a modified version of power iteration (Algorithm 1), wrapping the appropriate stride size and padding arguments into the convolution and convolution transpose operations. After obtaining $\sigma ( W )$ , we compute $W _ { \mathrm { S N } }$ in the same manner as for the fully-connected layers. Like Miyato et al., we exploit the fact that SGD only makes small updates to $W$ from training step to training step, reusing the same $\tilde { \mathbf { u } }$ and running only one iteration per step. Unlike Miyato et al., rather than enforcing $\sigma ( W ) = \beta$ , we instead enforce the looser constraint $\sigma ( W ) \leq \beta$ : + +$$ +W _ { \mathrm { S N } } = W / \operatorname* { m a x } ( 1 , \sigma ( W ) / \beta ) , +$$ + +which we observe to result in faster training for supervised learning tasks. + +
Algorithm1 Convolutional power iteration
Initialize ü with a random vector matching the shape of the convolution input
for t = 0,...,T-1do
ν ← conv(W,u)/llconv(W,u)ll2
ü ← conv_transpose(W,v)/llconv_transpose(W,v)ll2
end for
σ ←v· conv(W,u)
+ +# 5 NUMERICAL EXPERIMENTS + +In this section we provide an array of empirical experiments to validate both the bounds we derived in Section 3 and our implementation of spectral normalization described in section 4. We show that spectral normalization improves both test accuracy and generalization for a variety of adversarial training schemes, datasets, and network architectures. + +All experiments are implemented in TensorFlow (Abadi et al., 2016). For each experiment, we cross validate 4 to 6 values of $\beta$ (see (9)) using a fixed validation set of 500 samples. For PGM, we used $r = 1 5$ iterations and $\alpha = 2 \epsilon / r$ . Additionally, for FGM and PGM we used $\ell _ { 2 }$ -type attacks (unless specified) with magnitude $\epsilon = 0 . 0 5 \mathbb { E } _ { \hat { P } } [ \| \mathbf { X } \| _ { 2 } ]$ (this value was approximately 2.44 for CIFAR10). For WRM, we implemented gradient ascent as discussed by Sinha et al. (2018). Additionally, for WRM training we used a Lagrangian coefficient of $0 . 0 0 2 \mathbb { E } _ { \hat { P } } [ \| \mathbf { X } \| _ { 2 } ]$ for CIFAR10 and SVHN and a Lagrangian coefficient of $0 . \bar { 0 4 } \mathbb { E } _ { \hat { P } } [ \| \mathbf { X } \| _ { 2 } ]$ for MNIST in a similar manner to Sinha et al. (2018). The code will be made readily available. + +We first demonstrate the effect of the proposed spectral normalization approach on the final DNN weights by comparing the $\ell _ { 2 }$ norm of the input $\mathbf { x }$ to that of the output $f _ { \mathbf { w } } ( \mathbf { x } )$ . As shown in Figure 2(a), without spectral normalization ( $\beta = \infty$ in (9)), the norm gain can be large. Additionally, because we are using cross-entropy loss, the weights (and therefore the norm gain) can grow arbitrarily high if we continue training as reported by Neyshabur et al. (2017b). As we decrease $\beta$ , however, we produce more constrained networks, resulting in a decrease in norm gain. At $\beta = 1$ , the gain of the network cannot be greater than 1, which is consistent with what we observe. Additionally, we provide a comparison of our method to that of Miyato et al. (2018) in Appendix A.1, empirically demonstrating that Miyato et al.’s method does not properly control the spectral norm of convolutional layers, resulting in worse generalization performance. + +Figure 2(b) shows that the $\ell _ { 2 }$ norms of the gradients with respect to the training samples are nicely distributed after spectral normalization. Additionally, this figure suggests that the minimum gradient $\ell _ { 2 }$ -norm assumption (the $\kappa$ condition in Theorems 2 and 3) holds for spectrally-normalized networks. + +The first column of Figure 3 shows that, as observed by Bartlett et al. (2017), AlexNet trained using ERM generates similar margin distributions for both random and true labels on CIFAR10 unless we normalize the margins appropriately. We see that even without further correction, ERM training with SN allows AlexNet to have distinguishable performance between the two datasets. This observation suggests that SN as a regularization scheme enforces the generalization error bounds shown for spectrally-normalized DNNs by Bartlett et al. (2017) and Neyshabur et al. (2017a). Additionally, the margin normalization factor (the capacity norm $\Phi$ in Theorems 1-4) is much smaller for networks trained with SN. As demonstrated by the other columns in Figure 3, a smaller normalization factor results in larger normalized margin values and much tighter margin-based generalization bounds (a factor of $1 0 ^ { 2 }$ for ERM and a factor of $1 0 ^ { 5 }$ for FGM and PGM) (see Theorems 1-4). + +(a) $\ell _ { 2 }$ norm gain due to the network $f$ trained using ERM. + +![](images/2d10a59f5082d7d00e28922000204f0b02852c3f5bbd768c93322e043f96d3e2.jpg) +(b) Distributions of the $\ell _ { 2 }$ norms of the gradients with respect to training samples. Training regularized with SN. + +![](images/5e7d4faf907f521308a2e89fec6b112cbfa9f7b7dbda6618248bffd9196a9578.jpg) +Figure 2: Validation of SN implementation and distribution of the gradient norms using AlexNet trained on CIFAR10. + +5.2 SPECTRAL NORMALIZATION IMPROVES GENERALIZATION AND ADVERSARIAL ROBUSTNESS + +The phenomenon of overfitting random labels described by Zhang et al. (2016) can be observed even for adversarial training methods. Figure 4 shows how the FGM, PGM, or WRM adversarial training schemes only slightly delay the rate at which AlexNet fits random labels on CIFAR10, and therefore the generalization gap can be quite large without proper regularization. After introducing spectral normalization, however, we see that the network has a much harder time fitting both the random and true labels. With the proper amount of SN (chosen via cross validation), we can obtain networks that struggle to fit random labels while still obtaining the same or better test performance on true labels. + +We also observe that training schemes regularized with SN result in networks more robust to adversarial attacks. Figure 5 shows that even without adversarial training, AlexNet with SN becomes more robust to FGM, PGM, and WRM attacks. Adversarial training improves adversarial robustness more than SN by itself; however we see that we can further improve the robustness of the trained networks significantly by combining SN with adversarial training. + +![](images/5abc549a3964c1047d4e227f84672de7c547b6a42071166acdc50926bfe7c32a.jpg) +Figure 3: Effect of SN on distributions of unnormalized (leftmost column) and normalized (other three columns) margins for AlexNet fit on CIFAR10. The normalization factor is described by the capacity norm $\Phi$ reported in Theorems 1-4. + +![](images/06e6f1a8a28a2c03851b9b32128613974a14b5a3ef44a3dd2b6e434f5392405a.jpg) +Figure 4: Fitting random and true labels on CIFAR10 with AlexNet using adversarial training. + +![](images/c09a5b4382e86db70c17af5cad57f0b16c5aa7c6ba79c2f41bb4c07e45bcb080.jpg) +Figure 5: Robustness of AlexNet trained on CIFAR10 to various adversarial attacks. + +# 5.3 OTHER DATASETS AND ARCHITECTURES + +We demonstrate the power of regularization via SN on several combinations of datasets, network architectures, and adversarial training schemes. The datasets we evaluate are CIFAR10, MNIST, and SVHN. We fit CIFAR10 using the AlexNet and Inception networks described by Zhang et al. (2016), 1-hidden-layer and 2-hidden-layer multi layer perceptrons (MLPs) with ELU activation and 512 hidden nodes in each layer, and the ResNet architecture (He et al. (2016)) provided in TensorFlow for fitting CIFAR10. We fit MNIST using the ELU network described by Sinha et al. (2018) and the 1-hidden-layer and 2-hidden-layer MLPs. Finally, we fit SVHN using the same AlexNet architecture we used to fit CIFAR10. Our implementations do not use any additional regularization schemes including weight decay, dropout (Srivastava et al., 2014), and batch normalization (Ioffe & Szegedy, 2015) as these approaches are not motivated by the theory developed in this work; however, we provide numerical experiments comparing the proposed approach with weight decay, dropout, and batch normalization in Appendix A.2. + +Table 1 in the Appendix reports the pre and post-SN test accuracies for all 42 combinations evaluated. Figure 1 in the Introduction and Figures 7-9 in the Appendix show examples of training and validation curves on some of these combinations. We see that the validation curve generally improves after regularization with SN, and the observed improvements in validation accuracy are confirmed by the test accuracies reported in Table 1. Figure 6 visually summarizes Table 1, showing how SN can often significantly improve the test accuracy (and therefore decrease the generalization gap) for several of the combinations. We also provide Table 2 in the Appendix which shows the proportional increase in training time after introducing SN with our TensorFlow implementation. + +![](images/369c35ed7e831dc332f3c540e68035bbfacc5e5bdb1f8405ad7d01ddfec1fa0d.jpg) +Figure 6: Test accuracy improvement after SN for various datasets and network architectures. + +# 6 RELATED WORKS + +Providing theoretical guarantees for adversarial robustness of various classifiers has been studied in multiple works. Wang et al. (2017) targets analyzing the adversarial robustness of the nearest neighbor approach. Gilmer et al. (2018) studies the effect of the complexity of the data-generating manifold on the final adversarial robustness for a specific trained model. Fawzi et al. (2018) proves lower-bounds for the complexity of robust learning in adversarial settings, targeting the population distribution of data. Xu et al. (2009) shows that the regularized support vector machine (SVM) can be interpreted via robust optimization. Fawzi et al. (2016) analyzes the robustness of a fixed classifier to random and adversarial perturbations of the input data. While all of these works seek to understand the robustness properties of different classification function classes, unlike our work they do not focus on the generalization aspects of learning over DNNs under adversarial attacks. + +Concerning the generalization aspect of adversarial training, Sinha et al. (2018) provides optimization and generalization guarantees for WRM under the assumptions discussed after Theorem 4. However, their generalization guarantee only applies to the Wasserstein cost function, which is different from the 0-1 or margin loss and does not explicitly suggest a regularization scheme. In a recent related work, Schmidt et al. (2018) numerically shows the wide generalization gap in PGM adversarial training and theoretically establishes lower-bounds on the sample complexity of linear classifiers in Gaussian settings. While our work does not provide sample complexity lower-bounds, we study the broader function class of DNNs where we provide upper-bounds on adversarial generalization error and suggest an explicit regularization scheme for adversarial training over DNNs. + +Generalization in deep learning has been a topic of great interest in machine learning (Zhang et al., 2016). In addition to margin-based bounds (Bartlett et al., 2017; Neyshabur et al., 2017a), various other tools including VC dimension (Anthony & Bartlett, 2009), norm-based capacity scores (Bartlett & Mendelson, 2002; Neyshabur et al., 2015), and flatness of local minima (Keskar et al., 2016; Neyshabur et al., 2017b) have been used to analyze generalization properties of DNNs. Recently, Arora et al. (2018) introduced a compression approach to further improve the margin-based bounds presented by Bartlett et al. (2017); Neyshabur et al. (2017a). The PAC-Bayes bound has also been considered and computed by Dziugaite & Roy (2017), resulting in non-vacuous bounds for the MNIST dataset. + +# ACKNOWLEDGMENTS + +We are grateful for support under the National Science Foundation grant under CCF-1563098, and the Center for Science of Information (CSoI), an NSF Science and Technology Center under grant agreement CCF-0939370. + +# REFERENCES + +Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. +Martin Anthony and Peter L Bartlett. Neural network learning: Theoretical foundations. cambridge university press, 2009. +Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. 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Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. + +# Appendices + +A FURTHER EXPERIMENTAL RESULTS + +Table 1: Train and test accuracies before and after spectral normalization for various datasets, network architectures, and training schemes. The amount of spectral normalization was selected from 4-6 values of $\beta$ via cross validation on 500 samples. For each row, the greater test accuracy is bolded (both are bolded in the event of a tie). $\ell _ { \infty }$ adversarial training was performed with magnitude 0.1. + +
DatasetArchitectureTrainingTrain accTest accTrain acc (SN)Test acc (SN)
CIFAR10AlexNetERM1.000.791.000.79
CIFAR10AlexNetFGM l20.980.540.930.63
CIFAR10AlexNetFGM lo1.000.510.670.56
CIFAR10AlexNetPGM l20.990.500.920.62
CIFAR10AlexNetPGMlo0.990.440.860.54
CIFAR10AlexNetWRM1.000.610.760.65
CIFAR10ELU-AlexNetERM1.000.791.000.79
CIFAR10ELU-AlexNetFGM l20.970.520.680.60
CIFAR10ELU-AlexNetPGMl20.980.530.880.61
CIFAR10ELU-AlexNetWRM1.000.601.000.60
CIFAR10InceptionERM1.000.851.000.86
CIFAR10InceptionPGM l20.990.531.000.58
CIFAR10InceptionPGM lo0.980.480.620.56
CIFAR10InceptionWRM1.000.661.000.67
CIFAR101-layer MLPERM0.980.490.680.53
CIFAR101-layer MLPFGM l20.600.360.600.46
CIFAR101-layer MLPPGM l20.570.360.550.46
CIFAR101-layer MLPWRM0.600.410.620.50
CIFAR102-layer MLPERM0.990.510.790.56
CIFAR102-layer MLPFGM l20.570.360.660.49
CIFAR102-layer MLPPGM l20.930.350.660.48
CIFAR102-layer MLPWRM0.870.350.730.52
CIFAR10ResNetERM1.000.801.000.83
CIFAR10ResNetPGM l20.990.491.000.55
CIFAR10ResNetPGM lo0.980.440.720.53
CIFAR10ResNetWRM1.000.631.000.66
MNISTELU-NetERM1.000.991.000.99*
MNISTELU-NetFGM l20.980.971.000.97
MNISTELU-NetPGM l20.990.971.000.97
MNISTELU-NetWRM0.950.920.950.93
MNIST1-layer MLPERM1.000.981.000.98*
MNIST1-layer MLPFGM l20.880.881.000.96
MNIST1-layer MLPPGM l21.000.961.000.96
MNIST1-layer MLPWRM0.920.880.920.88
MNIST2-layer MLPERM1.000.981.000.98
MNIST2-layer MLPFGM l20.970.911.000.96
MNIST2-layer MLPPGM l21.000.961.000.97
MNIST2-layer MLPWRM0.970.880.980.90
SVHNAlexNetERM1.000.931.000.93*
SVHNAlexNetFGM l20.970.760.950.83
SVHNAlexNetPGM l21.000.780.850.81
SVHNAlexNetWRM1.000.830.870.84
+ +\* $\overline { { \beta = \infty } }$ (i.e. no spectral normalization) achieved the highest validation accuracy. + +Table 2: Runtime increase after introducing spectral normalization for various datasets, network architectures, and training schemes. These ratios were obtained by running the experiments on one NVIDIA Titan $\mathrm { X p }$ GPU for 40 epochs. + +
DatasetArchitectureTrainingno SN runtime SN runtimeratio
CIFAR10AlexNetERM229 s283 s1.24
CIFAR10AlexNetFGM l2407 s463 s1.14
CIFAR10AlexNetFGM loo408 s465 s1.14
CIFAR10AlexNetPGM l22917 s3077 s1.05
CIFAR10AlexNetPGM lo2896 s3048 s1.05
CIFAR10AlexNetWRM3076 s3151 s1.02
CIFAR10ELU-AlexNetERM231 s283 s1.23
CIFAR10ELU-AlexNetFGM l2410 s466 s1.14
CIFAR10ELU-AlexNetPGM l22939 s3093 s1.05
CIFAR10ELU-AlexNetWRM3094 s3150 s1.02
CIFAR10InceptionERM632 s734 s1.16
CIFAR10InceptionPGM l29994 s6082 s0.61
CIFAR10InceptionPGM lo9948 s6063 s0.61
CIFAR10InceptionWRM10247 s6356 s0.62
CIFAR101-layer MLPERM22 s31 s1.42
CIFAR101-layer MLPFGM l225 s35s1.43
CIFAR101-layer MLPPGM l279 s93 s1.18
CIFAR101-layer MLPWRM73 s86 s1.18
CIFAR102-layer MLPERM23 s37 s1.59
CIFAR102-layer MLPFGM l227 s41 s1.51
CIFAR102-layer MLPPGMl291 s108 s1.19
CIFAR102-layer MLPWRM85 s103 s1.21
CIFAR10ResNetERM315 s547 s1.73
CIFAR10ResNetPGM l22994 s3300 s1.10
CIFAR10ResNetPGM lo2980 s3300 s1.11
CIFAR10ResNetWRM3187 s3457 s1.08
MNISTELU-NetERM55 s97s1.76
MNISTELU-NetFGM l291s136 s1.49
MNISTELU-NetPGM l2614 s676 s1.10
MNISTELU-NetWRM635 s670 s1.06
MNIST1-layer MLPERM15 s24 s1.60
MNIST1-layer MLPFGM l217 s27s1.57
MNIST1-layer MLPPGM l257s71 s1.24
MNIST1-layer MLPWRM51 s63 s1.24
MNIST2-layer MLPERM17 s31 s1.84
MNIST2-layer MLPFGM l220 s35 s1.77
MNIST2-layer MLPPGM l267 s89 s1.32
MNIST2-layer MLPWRM62 s81 s1.30
SVHNAlexNetERM334 s412 s1.23
SVHNAlexNetFGM l2596 s676 s1.13
SVHNAlexNetPGM l24270 s4495 s1.05
SVHNAlexNetWRM4501 s4572 s1.02
+ +![](images/aad3b3b45c18f07506ccaffc3664feb697a9ac5ac91974f7442ee3d4bc8902d0.jpg) +Figure 7: Adversarial training performance with and without spectral normalization for AlexNet fit on CIFAR10. + +![](images/db9080bcbeb5283fc0d6b1c8dc283eaf945003257db19d46a41039c072336cef.jpg) +Figure 8: Adversarial training performance with and without spectral normalization for Inception and ResNet fit on CIFAR10. + +![](images/36ce01c99f6af2bf5a31795012fb7704c427a47cebb8a0901fe0558ed4f8a1ec.jpg) +Figure 9: Adversarial training performance with and without spectral normalization for AlexNet with ELU activation functions fit on CIFAR10. + +# A.1 COMPARISON OF PROPOSED METHOD TO MIYATO ET AL. (2018)’S METHOD + +For the optimal $\beta$ chosen when fitting AlexNet to CIFAR10 with PGM, we repeat the experiment using the spectral normalization approach suggested by Miyato et al. (2018). This approach performs spectral normalization on convolutional layers by scaling the convolution kernel by the spectral norm of the kernel rather than the spectral norm of the overall convolution operation. Because it does not account for how the kernel can amplify perturbations in a single pixel multiple times (see Section 4), it does not properly control the spectral norm. + +In Figure 10, we see that for the optimal $\beta$ reported in the main text, using Miyato et al. (2018)’s SN method results in worse generalization performance. This is because although we specified that $\beta \ : = \ : 1 . 6$ , the actual $\beta$ obtained using Miyato et al. (2018)’s method can be much greater for convolutional layers, resulting in overfitting (hence the training curve quickly approaches 1.0 accuracy). The AlexNet architecture used has two convolutional layers. For the proposed method, the final spectral norms of the convolutional layers were both 1.60; for Miyato et al. (2018)’s method, the final spectral norms of the convolutional layers were 7.72 and 7.45 despite the corresponding convolution kernels having spectral norms of 1.60. + +Our proposed method is less computationally efficient in comparison to Miyato et al. (2018)’s approach because each power iteration step requires a convolution operation rather than a division operation. As shown in Table 3, the proposed approach is not significantly less efficient with our TensorFlow implementation. + +![](images/ecf0f675b93c30caffac318e0a783a77d6f0fb6a507e2dc92bf11229ad28d266.jpg) +Figure 10: Adversarial training performance with proposed SN versus Miyato et al. (2018)’s SN for AlexNet fit on CIFAR10 using PGM. The final train and validation accuracies for the proposed method are 0.92 and 0.60. The final train and validation accuracies for Miyato et al. (2018)’s are 1.00 and 0.55. + +Table 3: Runtime increase of the proposed spectral normalization approach compared to Miyato et al. (2018)’s approach for CIFAR10 and various network architectures and training schemes. These ratios were obtained by running the experiments on one NVIDIA Titan $\mathrm { X p }$ GPU for 40 epochs. + +
DatasetArchitectureTrainingproposed SN runtime Miyato SN runtime
CIFAR10AlexNetERM1.11
CIFAR10AlexNetFGM l21.06
CIFAR10AlexNetFGMloo1.11
CIFAR10AlexNetPGM l21.01
CIFAR10AlexNetPGM lo1.11
CIFAR10AlexNetWRM1.02
CIFAR10InceptionERM0.98
CIFAR10InceptionPGM l21.04
CIFAR10InceptionPGM loo1.06
CIFAR10InceptionWRM1.03
+ +![](images/2aa56c0998b21e60c302c7532ab72c986434aecdb5230fcbe7694c08766a94f1.jpg) +Figure 11: Adversarial training performance with proposed SN versus batch normalization, weight decay, and dropout for AlexNet fit on CIFAR10 using PGM. The dropout rate was 0.8, and the amount of weight decay was 5e-4 for all weights. The leftmost plot is from Figure 1 and compares final performance of no regularization (train accuracy 1.00, validation accuracy 0.48) to that of SN (train accuracy 0.92, validation accuracy 0.60). The final train and validation accuracies for batch normalization are 1.00 and 0.54; the final train and validation accuracies for weight decay are 0.84 and 0.55; and the final train and validation accuracies for dropout are 0.99 and 0.52. + +# B SPECTRAL NORMALIZATION OF FULLY-CONNECTED LAYERS + +For fully-connected layers, we approximate the spectral norm of a given matrix $W$ using the approach described by Miyato et al. (2018): the power iteration method. For each $W$ , we randomly initialize a vector $\tilde { \mathbf { u } }$ and approximate both the left and right singular vectors by iterating the update rules + +$$ +\begin{array} { r l } & { \tilde { \mathbf { v } } W \tilde { \mathbf { u } } / \| W \tilde { \mathbf { u } } \| _ { 2 } } \\ & { \tilde { \mathbf { u } } W ^ { T } \tilde { \mathbf { v } } / \| W ^ { T } \tilde { \mathbf { v } } \| _ { 2 } . } \end{array} +$$ + +The final singular value can be approximated with $\sigma ( W ) \approx \tilde { \mathbf { v } } ^ { T } W \tilde { \mathbf { u } }$ . Like Miyato et al., we exploit the fact that SGD only makes small updates to $W$ from training step to training step, reusing the same $\tilde { \mathbf { u } }$ and running only one iteration per step. Unlike Miyato et al., rather than enforcing $\sigma ( W ) = \beta$ , we instead enforce the looser constraint $\sigma ( W ) \le \beta$ as described by Gouk et al. (2018): + +$$ +W _ { \mathrm { S N } } = W / \operatorname* { m a x } ( 1 , \sigma ( W ) / \beta ) , +$$ + +which we observe to result in faster training in practice for supervised learning tasks. + +# C PROOFS + +# C.1 PROOF OF THEOREM 1 + +First let us quote the following two lemmas from (Neyshabur et al., 2017a). + +Lemma 1 (Neyshabur et al. (2017a)). Consider $\mathcal { F } _ { n n } = \{ f _ { \mathbf { w } } : \mathbf { w } \in \mathcal { W } \}$ as the class of neural nets parameterized by w where each $f _ { \mathbf { w } }$ maps input $\mathbf { x } \in \mathcal { X }$ to $\mathbb { R } ^ { m }$ . Let $Q$ be a distribution on parameter vector chosen independently from the n training samples. Then, for each $\eta > 0$ with probability at least $1 - \eta$ for any w and any random perturbation u satisfying $\operatorname* { P r } _ { \mathbf { u } } \left( \operatorname* { m a x } _ { \mathbf { x } \in \mathcal { X } } \| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } ) - \right.$ $\begin{array} { r } { f _ { \mathbf { w } } ( \mathbf { x } ) \lVert _ { \infty } \leq \frac { \gamma } { 4 } ) \geq \frac { 1 } { 2 } } \end{array}$ we have + +$$ +L _ { 0 } ( f _ { \mathbf { w } } ) \leq \widehat { L } _ { \gamma } ( f _ { \mathbf { w } } ) + 4 \sqrt { \frac { K L ( P _ { \mathbf { w } + \mathbf { u } } \| Q ) + \log \frac { 6 n } { \eta } } { n - 1 } } . +$$ + +Lemma 2 (Neyshabur et al. (2017a)). Consider a $d$ -layer neural net $f _ { \mathbf { w } }$ with 1-Lipschitz activation function $\sigma$ where $\sigma ( 0 ) = 0$ . Then for any norm-bounded input $\| \mathbf { x } \| _ { 2 } \leq B$ and weight perturbation $\begin{array} { r } { \mathbf { \dot { u } } : \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ , we have the following perturbation bound: + +$$ +\| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } ) - f _ { \mathbf { w } } ( \mathbf { x } ) \| _ { 2 } \leq e B \left( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \right) \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } . +$$ + +To prove Theorem 1, consider $f _ { \widetilde { \mathbf { w } } }$ with weights $\widetilde { \mathbf { w } }$ . Since $( 1 + \textstyle { \frac { 1 } { d } } ) ^ { d } \leq e$ and $\begin{array} { r } { \frac { 1 } { e } \leq ( 1 - \frac { 1 } { d } ) ^ { d - 1 } } \end{array}$ , for any weight vector w such that $\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ for every $i$ we have: + +$$ +( 1 / e ) ^ { \frac { d } { d - 1 } } \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \leq \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \leq e \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } . +$$ + +We apply Lemma 1, choosing $Q$ to be a zero-mean multivariate Gaussian distribution with diagonal covariance matrix, where each entry of the $i$ th layer $\mathbf { U } _ { i }$ has standard deviation $\begin{array} { r } { \xi _ { i } = \frac { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } { \beta _ { \widetilde { \mathbf { w } } } } \xi } \end{array}$ kWfik2 ξ with $\xi$ chosen later in the proof. Note that $\beta _ { \mathbf { w } }$ edefined earlier in the theorem is the geometric average of spectral norms across all layers. Then for the ith layer’s random perturbation vector $\mathbf { u } _ { i } \sim \mathcal { N } ( 0 , \xi _ { i } ^ { 2 } I )$ , we get the following bound from (Tropp, 2012) with $h$ representing the width of the ith hidden layer: + +$$ +\mathrm { P r } \big ( \beta _ { \widetilde { \mathbf { w } } } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } > t \big ) \le 2 h \exp ( - \frac { t ^ { 2 } } { 2 h \xi ^ { 2 } } ) . +$$ + +We now use a union bound over all layers for a maximum union probability of $1 / 2$ , which implies the normalized w kUik2 for each layer is upper-bounded by ξp2h log(4hd). Then for any w satisfying $\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ for all $i$ ’s + +$$ +\begin{array} { r l } { \displaystyle \operatorname* { m a x } _ { \| \mathbf x \| _ { 2 } \leq B } \| f _ { \mathbf w + \mathbf u } ( \mathbf x ) - f _ { \mathbf w } ( \mathbf x ) \| _ { 2 } \leq e B \left( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \right) \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } } & { } \\ { \displaystyle } & { \overset { ( a ) } \leq e ^ { 2 } B \left( \displaystyle \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \right) \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } } \\ & { = e ^ { 2 } B \beta _ { \widetilde { \mathbf { w } } } ^ { d - 1 } \displaystyle \sum _ { i = 1 } ^ { d } \beta _ { \widetilde { \mathbf { w } } } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } } \\ & { \leq e ^ { 2 } d B \beta _ { \widetilde { \mathbf { w } } } ^ { d - 1 } \xi \sqrt { 2 h \log ( 4 h d ) } . } \end{array} +$$ + +Here (a) holds, since $\begin{array} { r } { \frac { 1 } { \| \mathbf { W } _ { j } \| } \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \leq \frac { e } { \| \widetilde { \mathbf { W } } _ { j } \| } \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ is true for each $j$ . Hence we choose $\begin{array} { r } { \xi = \frac { \gamma } { 3 0 d B \beta _ { \widetilde { \mathbf { w } } } ^ { d - 1 } \sqrt { h \log ( 4 h d ) } } } \end{array}$ for which the perturbation vector satisfies the assumptions of Lemma 2. eThen, we bound the KL-divergence term in Lemma 1 as + +$$ +\begin{array} { r l r } { { K L ( P _ { \mathbf { w } + \mathbf { u } } | | Q ) \leq \displaystyle \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { 2 \xi _ { i } ^ { 2 } } } } \\ & { } & { = \frac { 3 0 ^ { 2 } d ^ { 2 } B ^ { 2 } \beta _ { \mathbf { w } } ^ { 2 } \| \mathbf { b } \log ( 4 h d ) } { 2 \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \widehat { \mathbf { W } } _ { i } \| _ { 2 } ^ { 2 } } } \\ & { } & { \overset { ( b ) } { \leq } \frac { 3 0 ^ { 2 } e ^ { 2 } d ^ { 2 } B ^ { 2 } \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } h \log ( 4 h d ) } { 2 \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } } \\ & { } & { = \mathcal { O } \Big ( d ^ { 2 } B ^ { 2 } h \log ( h d ) \frac { \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } { \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } \Big ) . } \end{array} +$$ + +Note that (b) holds, because we assume $\begin{array} { r l r } { | \| { \bf W } _ { i } \| _ { 2 } ~ - ~ \| \widetilde { { \bf W } } _ { i } \| _ { 2 } \Big | } & { { } \le ~ } & { \frac { 1 } { d } \| \widetilde { { \bf W } } _ { i } \| _ { 2 } } \end{array}$ implying 1kW k Qdi=1 kWfik2 ≤ (1 − 1d )−(d−1) 1kWjk $\begin{array} { r } { \frac { 1 } { \| \widetilde { \mathbf { W } } _ { i } \| } \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \leq ( 1 - \frac { 1 } { d } ) ^ { - ( d - 1 ) } \frac { 1 } { \| \mathbf { W } _ { j } \| } \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \leq \frac { e } { \| \mathbf { W } _ { j } \| } \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ for each $j$ . Therefore, Lemma 1 implies with probability $1 - \eta$ we have the following bound hold for any w satisfying + +$\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ for all $i$ ’s, + +$$ +L _ { 0 } ( f _ { \mathbf { w } } ) \leq \widehat { L } _ { \gamma } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { B ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi ^ { \mathrm { e r m } } ( f _ { \mathbf { w } } ) + \log \frac { n } { \widetilde { \eta } } } { \gamma ^ { 2 } n } } \bigg ) . +$$ + +Then, we can give an upper-bound over all the functions in $\mathcal { F } _ { \mathrm { n n } }$ by finding the covering number of the set of We o $\widetilde { \mathbf { w } }$ ’s where for each feasible w need to form the bound for $\begin{array} { r } { \big ( \frac { \gamma } { 2 B } \big ) ^ { 1 / d } \leq \beta _ { \mathbf { w } } \leq \big ( \frac { \gamma \sqrt { n } } { 2 B } \big ) ^ { 1 / d } } \end{array}$ ition satisfied for at least one of we ’s.which can be covered using a cover of size $d n ^ { 1 / 2 d }$ as discussed in (Neyshabur et al., 2017a). Then, from the theorem’s assumption we know each $\Vert \mathbf { W } _ { i } \Vert _ { 2 }$ will be in the interval $[ \frac { 1 } { M } \beta _ { \mathbf { w } } , M \beta _ { \mathbf { w } } ]$ which we want to cover such that for any $\beta$ in the interval there exists a $\widetilde { \beta }$ satisfying $| \beta - \widetilde { \beta } | \leq \widetilde { \beta } / d$ . For this purpose we can use a cover of size $2 \log _ { 1 + 1 / d } M \leq 2 ( d + 1 ) \log M ,$ 1 which combined for all $i$ ’s gives a cover with size ${ \mathcal { O } } ( ( d \log M ) ^ { d } )$ whose logarithm is growing as $d \log ( d \log M )$ . This together with (15) completes the proof. + +# C.2 PROOF OF THEOREM 2 + +We start by proving the following lemmas providing perturbation bound for FGM attacks. + +Lemma 3. Consider a $d$ -layer neural net $f _ { \mathbf { w } }$ with 1-Lipschitz and 1-smooth (1-Lipschitz derivative) activation $\sigma$ where $\sigma ( 0 ) = 0$ . Let training loss $\ell : ( \mathbb { R } ^ { m } , \mathcal { Y } ) \to \mathbb { R }$ also be 1-Lipschitz and 1-smooth for any fixed label $y \in \mathcal { D }$ . Then, for any input $\mathbf { x }$ , label $y$ , and perturbation vector u satisfying $\begin{array} { r } { \forall i : \| \dot { \mathbf { U } } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ we have + +$$ +\begin{array} { r l } & { \quad \left\| \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } ) , y ) - \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) \right\| _ { 2 } } \\ & { \leq e ^ { 2 } ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \displaystyle \sum _ { i = 1 } ^ { d } \biggl [ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + \| \mathbf { x } \| _ { 2 } ( \displaystyle \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ) \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \biggr ] . } \end{array} +$$ + +Proof. Since for a fixed $y \ell$ satisfies the same Lipschitzness and smoothness properties as $\sigma$ , then $\| \nabla _ { \mathbf { z } } \ell ( \mathbf { z } , y ) \| _ { 2 } \leq 1$ and applying the chain rule implies: + +$$ +\begin{array} { r l } & { \quad \| \nabla _ { \mathbf x } \ell ( \mathbf r _ { \infty + \mathbf n } ( \mathbf x ) , y ) - \nabla _ { \mathbf x } \ell ( f _ { \infty } ( \mathbf x ) , y ) \| _ { 2 } } \\ & { = \| ( \nabla _ { \mathbf x } f _ { \infty + \mathbf n } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) - ( \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty } ( \mathbf x ) , y ) \| _ { 2 } } \\ & { \le \| ( \nabla _ { \mathbf x } f _ { \infty + \mathbf n } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) - ( \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) \| _ { 2 } } \\ & { \quad + \| ( \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) - ( \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) ) ( \nabla \ell ) ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) \| _ { 2 } } \\ & { \le \| \nabla _ { \mathbf x } f _ { \infty + \mathbf n } ( \mathbf x ) - \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) \| _ { 2 } + ( \prod _ { i } \| \mathbf x _ { i } \| _ { 2 } ) \| ( \| \nabla \ell ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) - ( \nabla \ell ) ( f _ { \infty } ( \mathbf x ) , y ) \| _ { 2 } } \\ & { \le \| \nabla _ { \mathbf x } f _ { \infty + \mathbf n } ( \mathbf x ) - \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) \| _ { 2 } + ( \prod _ { i = 1 } ^ { d } \| \nabla _ { \mathbf x } ) \| ( \nabla \ell ( f _ { \infty + \mathbf n } ( \mathbf x ) , y ) - ( \nabla \ell ) ( f _ { \infty } ( \mathbf x ) , y ) \| _ { 2 } } \\ & \le \| \nabla _ { \mathbf x } f _ { \infty + \mathbf n } ( \mathbf x ) - \nabla _ { \mathbf x } f _ { \infty } ( \mathbf x ) \| \end{array} +$$ + +The above result is a conclusion of Lemma 2 and the lemma’s assumptions implying $\left\| \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } ) \right\| _ { 2 } \leq$ $\begin{array} { r } { \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ for every x. Now, we define $\Delta _ { k } = \left. \nabla _ { \mathbf { x } } f _ { \mathbf { w } + \mathbf { u } } ^ { ( k ) } ( \mathbf { x } ) - \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \right. _ { 2 }$ where $f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) : =$ $\mathbf { W } _ { k } \sigma ( \mathbf { W } _ { k - 1 } \cdot \cdot \cdot \sigma ( \mathbf { W } _ { 1 } \mathbf { x } ) ) \cdot \cdot \cdot )$ denotes the DNN’s output at layer $k$ . With (17) in mind, we complete this lemma’s proof by showing the following inequality via induction: + +$$ +\Delta _ { k } \leq e ( 1 + \frac { 1 } { d } ) ^ { k } \big ( \prod _ { i = 1 } ^ { k } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { k } \left[ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + \| \mathbf { x } \| _ { 2 } \big ( \prod _ { j = 1 } ^ { i - 1 } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \sum _ { j = 1 } ^ { i - 1 } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \right] . +$$ + +1Note that log 1x ≥ 1 − x implying log 11− 1d+1 $\textstyle { \frac { 1 } { 1 - { \frac { 1 } { d + 1 } } } } \geq { \frac { 1 } { d + 1 } }$ and hence $( \log ( 1 + 1 / d ) ) ^ { - 1 } \leq d + 1$ . + +The above equation will prove the lemma because for $k \leq d$ we have $\begin{array} { r } { ( 1 + \frac { 1 } { d } ) ^ { k } \le ( 1 + \frac { 1 } { d } ) ^ { d } \le e } \end{array}$ . For $k = 0$ , $\Delta _ { 0 } = 0$ since $f _ { \mathbf { w } } ^ { ( 0 ) } ( \mathbf { x } ) = \mathbf { x }$ and does not change with w. Given that (18) holds for $k$ we have + +$$ +\begin{array} { r l } { \mathbf { \Phi } } & { = | \nabla _ { x } \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} } \\ & { - \nabla _ { x } \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} } \\ & - \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \} \{ \mathbf { S } _ { 0 } ^ { \varepsilon } \ \end{array} +$$ + +Therefore, combining (17) and (18) the lemma’s proof is complete + +Before presenting the perturbation bound for FGM attacks, we first prove the following simple lemma. + +Lemma 4. Consider vectors $\mathbf { z } _ { 1 } , \mathbf { z } _ { 2 }$ and norm function $\| \cdot \|$ $| . \ I f \operatorname* { m a x } \{ \| \mathbf { z } _ { 1 } \| , \| \mathbf { z } _ { 2 } \| \} \geq \kappa ,$ , then + +$$ +\Bigl \| \frac { \epsilon } { \| \mathbf { z } _ { 1 } \| } \mathbf { z } _ { 1 } - \frac { \epsilon } { \| \mathbf { z } _ { 2 } \| } \mathbf { z } _ { 2 } \Bigr \| \leq \frac { 2 \epsilon } { \kappa } \| \mathbf { z } _ { 1 } - \mathbf { z } _ { 2 } \| . +$$ + +Proof. Without loss of generality suppose $\| \mathbf { z } _ { 2 } \| \leq \| \mathbf { z } _ { 1 } \|$ and therefore $\kappa \leq \| \mathbf { z } _ { 1 } \|$ . Then, + +$$ +\begin{array} { r l } & { \displaystyle \| \frac { \epsilon } { \| { \bf z } _ { 1 } \| } { \bf z } _ { 1 } - \frac { \epsilon } { \| { \bf z } _ { 2 } \| } { \bf z } _ { 2 } \| = \epsilon \big \| \frac { 1 } { \| { \bf z } _ { 1 } \| } \big ( { \bf z } _ { 1 } - { \bf z } _ { 2 } \big ) - \frac { \| { \bf z } _ { 1 } \| - \| { \bf z } _ { 2 } \| } { \| { \bf z } _ { 1 } \| } \frac { 1 } { \| { \bf z } _ { 2 } \| } \big \| } \\ & { \displaystyle \quad \quad \quad \leq \frac { \epsilon } { \| { \bf z } _ { 1 } \| } \| { \bf z } _ { 1 } - { \bf z } _ { 2 } \| + \frac { \epsilon } { \| { \bf z } _ { 1 } \| } \big | \| { \bf z } _ { 1 } \| - \| { \bf z } _ { 2 } \| \big | } \\ & { \displaystyle \quad \quad \leq \frac { 2 \epsilon } { \| { \bf z } _ { 1 } \| } \| { \bf z } _ { 1 } - { \bf z } _ { 2 } \| } \\ & { \displaystyle \quad \quad \quad \leq \frac { 2 \epsilon } { \kappa } \| { \bf z } _ { 1 } - { \bf z } _ { 2 } \| . } \end{array} +$$ + +Lemma 5. Consider a $d$ -layer neural network function $f _ { \mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\sigma$ where $\sigma ( 0 ) = 0$ . Consider FGM attacks with noise power  according to Euclidean norm $| | \cdot | | _ { 2 }$ . Suppose $\kappa \leq \| \nabla _ { \mathbf x } \ell ( f _ { \mathbf w } ( \mathbf x ) , y ) \| _ { 2 }$ holds over the $\epsilon$ -ball around the support set $\mathcal { X }$ . Then, for any norm-bounded perturbation vector u such that $\begin{array} { r } { \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } . \forall i } \end{array}$ , we have + +$$ +\big \| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { f g m } } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { f g m } } ( \mathbf { x } ) \big \| _ { 2 } \leq \frac { 2 e ^ { 2 } \epsilon } { \kappa } \big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { d } \left[ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + \| \mathbf { x } \| _ { 2 } \big ( \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \right] . +$$ + +Proof. The FGM attack according to Euclidean norm is simply the DNN loss’s gradient normalized to have $\epsilon$ -Euclidean norm. The lemma is hence a direct result of combining Lemmas 3 and 4. + +To prove Theorem 2, we apply Lemma 1 together with the result in Lemma 5. Similar to the proof for Theorem 1, given weights $\widetilde { \mathbf { w } }$ we consider a zero-mean multivariate Gaussian perturbation vector u with diagonal covariance matrix where each element in the ith layer $\mathbf { u } _ { i }$ varies with the scaled standard deviation kWfik2βw ξ with ξ properly chosen later in the proof. Consider weights w for which + +$$ +\forall i : \ | \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \leq \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } . +$$ + +Since $\mathbf { u } _ { i } \sim \mathcal { N } ( 0 , \xi _ { i } ^ { 2 } I )$ , (Tropp, 2012) shows the following bound holds + +$$ +\mathrm { P r } \big ( \beta _ { \widetilde { \mathbf { w } } } \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } > t \big ) \le 2 h \exp ( - \frac { t ^ { 2 } } { 2 h \xi ^ { 2 } } ) . +$$ + +Then we apply a union bound over all layers for a maximum union probability of $1 / 2$ implying the normalized kUik2 for each layer is upper-bounded by ξp2h log(4hd). Now, if the assumptions of Lemma 5 hold for perturbation vector $\mathbf { u }$ given the choice of $\xi$ , for the FGM attack with noise power $\epsilon$ according to Euclidean norm $\| \cdot \| _ { 2 }$ we have + +$$ +\begin{array} { r l } & { \quad \| f _ { w + \mathbf { n } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) - f _ { \mathbf { w } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) \| _ { 2 } } \\ & { \leq \| f _ { w + \mathbf { n } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) - f _ { \mathbf { w } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) \| _ { 2 } + \| f _ { w } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) - f _ { w } ( \mathbf x + \delta _ { w } ^ { \mathrm { t e m } } ( \mathbf x ) ) \| _ { 2 } } \\ & { \leq \| f _ { \mathbf { w + \mathbf { n } } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) - f _ { \mathbf { w } } ( \mathbf x + \delta _ { w + \mathbf { n } } ^ { \mathrm { t e m } } ( \mathbf x ) ) \| _ { 2 } + ( \underbrace { d } _ { \mathbf { w - \mathbf { n } } } ( \mathbf x ) \| _ { 2 } ) \| \delta _ { \mathbf { w + \mathbf { n } } } ^ { \mathrm { t e m } } ( \mathbf x ) - \delta _ { w } ^ { \mathrm { t e m } } ( \mathbf x ) \| _ { 2 } } \\ & { \leq e ( B + e ) \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { i } \| _ { 2 } \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { i } \| _ { 2 } + 2 e ^ { 2 } \frac { d } { \kappa } \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } + B ( \frac { 1 } { \sqrt { 1 + 1 } } W _ { i } ) \| _ { 2 } \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { i } \| _ { 2 } \displaystyle \frac { d } { i - 1 } \| \mathbf { W } _ { j } \| } \\ & \leq e ^ { 2 } ( B + e ) \displaystyle \sum _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \displaystyle \sum _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ \end{array} +$$ + +Hence we choose + +$$ +\begin{array} { r } { \dot { \mathbf { \xi } } = \frac { \gamma } { 8 e ^ { 5 } d ( B + \epsilon ) \sqrt { 2 h \log ( 4 h d ) } \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big ( 1 + \frac { \epsilon } { \kappa } \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } ( 1 / B + \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \widetilde { \mathbf { W } } _ { j } \| _ { 2 } ) \big ) } , } \end{array} +$$ + +for which the assumptions of Lemmas 1 and 5 hold. Assuming $B \geq 1$ , similar to Theorem 1’s proof we can show for any w such that $\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ we have + +$$ +\begin{array} { l l } { \displaystyle K L ( P _ { \mathbf { w } + \mathbf { u } } | | Q ) \leq \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { 2 \xi _ { i } ^ { 2 } } } \\ { \displaystyle = \mathcal { O } \bigg ( d ^ { 2 } ( B + \epsilon ) ^ { 2 } h \log ( h d ) \frac { \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } ^ { 2 } \{ 1 + \frac { \epsilon } { \kappa } \big ( \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \widetilde { \mathbf { W } } _ { j } \| _ { 2 } \big ) \} ^ { 2 } } { \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } ^ { 2 } } } \\ { \leq \mathcal { O } \bigg ( d ^ { 2 } ( B + \epsilon ) ^ { 2 } h \log ( h d ) \frac { \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } \{ 1 + \frac { \epsilon } { \kappa } \big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \} ^ { 2 } } { \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } } \end{array} +$$ + +Then, applying Lemma 1 reveals that given any $\eta > 0$ with probability at least $1 - \eta$ for any w such that $\begin{array} { r } { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ we have + +$$ +L _ { 0 } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) \le \widehat { L } _ { \gamma } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \biggl ( \sqrt { \frac { ( B + \epsilon ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \epsilon , \kappa } ^ { \mathrm { f g m } } ( f _ { \mathbf { w } } ) + \log \frac { n } { \eta } } { \gamma ^ { 2 } n } } \biggr ) +$$ + +where $\begin{array} { r } { \Phi _ { \epsilon , \kappa } ^ { \mathrm { f o m } } ( f _ { \mathbf { w } } ) : = \left\{ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ( 1 + \frac { \epsilon } { \kappa } \{ \left( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \right) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ) \} ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } \right\} . } \end{array}$ Note that similar to our proof for Theorem 1 we can find a cover of size $O ( ( d \log M ) ^ { d } d n ^ { 1 / 2 d } )$ for the spectral norms of the weights feasible set, where for any $\Vert \mathbf { W } _ { i } \Vert _ { 2 }$ we have $a _ { i }$ such that $\left| \| \mathbf { W } _ { i } \| _ { 2 } - a _ { i } \right| \leq a _ { i } / d$ . Applying this covering number bound to (24) completes the proof. + +# C.3 PROOF OF THEOREM 3 + +We use the following two lemmas to extend the proof of Theorem 2 for FGM attacks to show Theorem 3 for PGM attacks. + +Lemma 6. Consider a $d$ -layer neural network function $f _ { \mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\sigma$ where $\sigma ( 0 ) = 0$ . We consider PGM attacks with noise power  according to Euclidean norm $| | \cdot | | _ { 2 } , r$ iterations and stepsize $\alpha$ . Suppose $\boldsymbol { \kappa } \leq \| \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) \| _ { 2 }$ holds over the $\epsilon$ -ball around the support set $\mathcal { X }$ . Then for any perturbation vector u such that $\begin{array} { r } { \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ for every $i$ we have + +$$ +\begin{array} { r l } & { \| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { p g m } , r } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { p g m } , r } ( \mathbf { x } ) \| _ { 2 } \leq e ^ { 2 } ( 2 \alpha / \kappa ) \frac { 1 - ( 2 \alpha / \kappa ) ^ { r } \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) ^ { r } } { 1 - ( 2 \alpha / \kappa ) \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) } } \\ & { \times \big ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \displaystyle \sum _ { i = 1 } ^ { d } \biggl [ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + ( \| \mathbf { x } \| _ { 2 } + \epsilon ) \big ( \displaystyle \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \displaystyle \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \biggr ] . } \end{array} +$$ + +Here $\operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } )$ denotes the actual Lipschitz constant of $\nabla _ { \mathbf x } \ell ( f _ { \mathbf w } ( \mathbf x ) , y )$ . + +Proof. We use induction to show this lemma for different $r$ values. The result for case $r = 1$ is a direct consequence of Lemma 5. Suppose that the result is true for $r = k$ . Then, Lemmas 3 and 4 imply + +$$ +\begin{array} { r l } & { | | E _ { \frac { d } { d } } ^ { \mathrm { L C } } ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) | \leq x _ { 0 } ^ { 3 } , } \\ & { \leq \frac { \varepsilon ^ { 2 } } { \varepsilon } \log ( x _ { 0 } ^ { 4 } + \varepsilon \varepsilon ( \frac { 1 } { \varepsilon } ) ^ { 2 } + \varepsilon ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) , } \\ & { \leq \frac { \varepsilon ^ { 2 } } { \varepsilon } \log ( x _ { 0 } ^ { 4 } + \varepsilon ( \frac { 1 } { \varepsilon } ) ^ { 2 } + \varepsilon ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \ \mathbf { S } ^ { \varepsilon } \ \ \exp ( \frac { 1 } { \varepsilon } ) \cdot \mathbf { S } ^ { \varepsilon } \ } \\ & { \quad + \ \frac { \varepsilon ^ { 2 } } { \varepsilon } \log ( x _ { 0 } ^ { 4 } + \varepsilon ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \ \mathbf { S } ^ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } \ \mathbf { S } ^ { \varepsilon } \ \exp ( \frac { 1 } { \varepsilon } ) \cdot \mathbf { S } ^ { \varepsilon } \ ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) } \\ & \leq \frac { \varepsilon ^ { 2 } } { \varepsilon } \log ( x _ { 0 } ^ { 4 } + \varepsilon ( \frac { 1 } { \varepsilon } ) ^ { 2 } ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot ( \mathbf { S } _ { \varepsilon } \cdot \mathbf { S } ^ { \varepsilon } ) \cdot ( \mathbf { S } _ \end{array} +$$ + +where the last line follows from the equality $\textstyle \sum _ { i = 0 } ^ { k } s ^ { i } = { \frac { 1 - s ^ { k + 1 } } { 1 - s } }$ . Therefore, by induction the lemma holds for every value $r \geq 1$ . + +Lemma 7. Consider a $d$ -layer neural network function $f _ { \mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\sigma$ where $\sigma ( 0 ) = 0$ . Also, assume that training loss $\ell$ is 1-Lipschitz and 1-smooth. Then, + +$$ +\operatorname* { l i p } \bigl ( \nabla _ { \mathbf { x } } \ell \bigl ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \bigr ) \bigr ) \leq \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) : = \bigl ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \bigr ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } . +$$ + +Proof. First of all note that according to the chain rule + +$$ +\begin{array} { r l } & { \displaystyle \operatorname* { l i p } \bigg ( \nabla _ { \mathbf { x } } \ell \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) \bigg ) = \operatorname* { l i p } \bigg ( \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } ) ( \nabla \ell ) \big ( f _ { \mathbf { w } } ( \mathbf { x } ) , y \big ) \bigg ) } \\ & { \quad \quad \quad \leq \operatorname* { l i p } \big ( \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } ) \big ) + \operatorname* { l i p } ( f _ { \mathbf { w } } ) ^ { 2 } } \\ & { \quad \quad \quad \leq \operatorname* { l i p } \big ( \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } ) \big ) + \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } . } \end{array} +$$ + +Considering the above result, we complete the proof by inductively proving $\begin{array} { r } { \operatorname* { l i p } \bigl ( \nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } ) \bigr ) \ \leq } \end{array}$ $( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i - 1 } \| \mathbf { W } _ { j } \| _ { 2 }$ . For $d = 1$ , $\nabla _ { \mathbf { x } } f _ { \mathbf { w } } ( \mathbf { x } )$ is constant and hence the result holds. Assume the statement holds for $d = k$ . Due to the chain rule, + +$$ +\nabla _ { \mathbf { x } } { f } _ { \mathbf { w } } ^ { ( k + 1 ) } ( \mathbf { x } ) = \nabla _ { \mathbf { x } } \mathbf { W } _ { k + 1 } \sigma \big ( { f } _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \big ) = \nabla _ { \mathbf { x } } { f } _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \sigma ^ { \prime } \big ( { f } _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \big ) \mathbf { W } _ { k + 1 } ^ { T } +$$ + +and therefore for any $\mathbf { x }$ and $\mathbf { v }$ + +$$ +\begin{array} { r l } & { \quad \| \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) - \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \| _ { 2 } } \\ & { \leq \| \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) \nabla _ { x + 1 } ^ { F } - \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) \mathbf { W } _ { k + 1 } ^ { T } \| _ { 2 } } \\ & { \leq \| \mathbf { W } _ { k + 1 } \| _ { 2 } \big \| \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) - \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \big ) \big \| _ { 2 } } \\ & { \leq \| \mathbf { W } _ { k + 1 } \| _ { 2 } \big \| \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) - \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) \big \| _ { 2 } } \\ & \quad + \| \mathbf { W } _ { k + 1 } \| _ { 2 } \big \| \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \sigma ^ { \prime } \big ( f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } + \mathbf { v } ) \big ) - \nabla _ { x } f _ { \mathbf { w } } ^ { ( k ) } ( \mathbf { x } ) \sigma ^ { \prime } \big ( f _ \mathbf w \end{array} +$$ + +which shows the statement holds for $d = k + 1$ and therefore completes the proof via induction. + +In order to prove Theorem 3, we note that for any norm-bounded $\| \mathbf { x } \| _ { 2 } \leq B$ and perturbation vector $\mathbf { u }$ such that $\forall i$ , $\begin{array} { r } { \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ we have + +$$ +\begin{array} { r l } & { \quad \| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathbf { w } \times \mathbf { n } } , \mathbf { r } ( \mathbf { x } ) ) \| _ { 2 } } \\ & { \leq \| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathbf { w } \mathbf { n } , \mathbf { r } } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathbf { w } \mathbf { n } , \mathbf { r } } ( \mathbf { x } ) ) \| _ { 2 } } \\ & { \quad + \| f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathbf { w } \mathbf { n } , \mathbf { r } } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathbf { w } \mathbf { n } , \mathbf { r } } ( \mathbf { x } ) ) \| _ { 2 } } \\ & { \leq e ( B + \epsilon ) ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { w } _ { i } \| _ { 2 } ) \displaystyle \sum _ { i = 1 } ^ { d } \| \frac { \| \mathbf { u } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + e ^ { 2 } ( 2 \alpha / \kappa ) \frac { 1 - ( 2 \alpha / \kappa ) ^ { r } \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) ^ { r } } { 1 - ( 2 \alpha / \kappa ) \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) } } \\ & { \qquad \times \displaystyle ( \prod _ { i = 1 } ^ { d } \| \mathbf { w } _ { i } \| _ { 2 } ^ { 2 } ) \displaystyle \sum _ { i = 1 } ^ { d } \Big [ \| \mathbf { u } _ { i } \| _ { 2 } } \\ & \qquad \leq e ( B + \epsilon ) ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { w } _ { i } \| _ { 2 } ) \displaystyle \sum _ { i = 1 } ^ { d } \| \mathbf { Z } _ { i } \| _ { 2 } + ( B + \epsilon ) ( \displaystyle \prod _ { j = 1 } ^ { i } \end{array} +$$ + +$$ +\times \left( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } \right) \sum _ { i = 1 } ^ { d } \left[ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + ( B + \epsilon ) ( \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ) \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \right] +$$ + +The last inequality holds since as shown in Lemma $7 \ : \operatorname* { l i p } \bigl ( \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } ) , y ) \bigr ) \leq \ : \varlimsup _ { \mathbf { \theta } } ( \nabla \ell \circ f _ { \mathbf { w } } ) : =$ $\begin{array} { r } { \left( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \right) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } } \end{array}$ . Here the upper-bound $\overline { { \mathrm { l i p } } } ( \nabla \ell \circ f _ { \widetilde { \mathbf { w } } } )$ for $\widetilde { \mathbf { w } }$ changes by a factor at most $e ^ { 2 / r }$ for w such that $| | \mathbf { W } _ { i } | | _ { 2 } - | | \widetilde { \mathbf { W } } _ { i } | | _ { 2 } \Big | \leq \frac { 1 } { r _ { i } ^ { d } } | | \widetilde { \mathbf { W } } _ { i } | | _ { 2 }$ . Therefore, given $\widetilde { \mathbf { w } }$ if similar to the proof for Theorem 2 we choose a zero-mean multivariate Gaussian distribution $Q$ efor $\mathbf { u }$ with the ith layer $\mathbf { u } _ { i }$ ’s standard deviation to be $\begin{array} { r } { \xi _ { i } = \frac { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } { \beta _ { \widetilde { \mathbf { w } } } } \xi } \end{array}$ kWfik2 ξ where + +$$ +\begin{array} { r } { \overline { { 8 d ( B + \epsilon ) \sqrt { 2 h \log ( 4 h d ) } } } \epsilon ^ { 4 } ( \alpha / \kappa ) \frac { \gamma } { 1 - e ^ { 2 \left( 2 \alpha / \kappa \right) \sqrt { \operatorname* { l i p } } ( \nabla \ell \circ f _ { \widetilde { \infty } } ) ^ { r } } } ( \prod _ { i = 1 } ^ { d } \Vert \widetilde { \mathbf { W } } _ { i } \Vert _ { 2 } ) \big ( 1 + \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \Vert \widetilde { \mathbf { W } } _ { j } \Vert _ { 2 } ) . } \end{array} +$$ + +Then for any w satisfying $\begin{array} { r } { \left| \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \right| \le \frac { 1 } { r d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ , applying union bound shows that the assumptiholds for of Lemma 1 , and further $\begin{array} { r } { \operatorname* { P r } _ { \mathbf { u } } \bigl ( \operatorname* { m a x } _ { \mathbf { x } \in \mathcal { X } } \| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { p g m } , r } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { p g m } , r } ( \mathbf { x } ) ) \| _ { \infty } \leq \frac { \gamma } { 4 } \bigr ) \geq \frac { 1 } { 2 } } \end{array}$ $Q$ + +$$ +\begin{array} { r l } & { K L ( P _ { \mathbf { w } + \mathbf { u } } | | Q ) \leq \displaystyle \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { \mathcal { E } } ^ { 2 } } { 2 \xi _ { i } ^ { 2 } } } \\ & { = \mathcal { O } \Big ( d ^ { 2 } ( B + c ) ^ { 2 } \widehat { l } \mathrm { l o g } ( i a ) \Big ) \times } \\ & { \displaystyle \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 \frac { 1 - e ^ { 2 } ( 2 a \rho \gamma ) ^ { \top } \widehat { \mathbb { H } } ( \nabla \xi \circ f _ { \varphi _ { i } } ) } { 1 - e ^ { 2 } \gamma ( 2 a \rho \gamma ) \widehat { l } \mathrm { l o g } ( \widehat { \mathbf { W } } \xi \circ f _ { \varphi _ { i } } ) } } ^ { 2 } \{ 1 + \frac { \alpha } { \kappa } \Big ( \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \Big ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \widetilde { \mathbf { W } } _ { j } \| _ { 2 } \Big \} ^ { 2 } \displaystyle \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { \mathcal { E } } ^ { 2 } } { | \widetilde { \mathbf { W } } _ { i } \| _ { 2 } ^ { 2 } } \Big ) } \\ & { \leq \mathcal { O } \Big ( d ^ { 2 } ( B + c ) ^ { 2 } \widehat { l } \mathrm { l o g } ( k a ) \Big ) \times } \\ & \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 \frac { 1 - ( 2 \sigma \gamma ) \widehat { l } \mathrm { l o g } ( \gamma \circ f _ { \varphi _ { i } } ) r } { 1 - ( 2 \sigma \gamma ) \widehat { l } \mathrm { l o g } ( \widehat { \mathbf { W } } \xi \circ f _ { \varphi _ { i } } ) } ^ { 2 } \Big \{ 1 + \frac { \alpha } { \kappa } \Big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \Big ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { \mathcal { F } } ^ { 2 } } | \end{array} +$$ + +Applying the above bound to Lemma 1 shows that for any $\eta > 0$ the following holds with probability $1 - \eta$ for any w where $\begin{array} { r } { \left| \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \right| \le \frac { 1 } { r d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ : + +$$ +L _ { 0 } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) \le \widehat { L } _ { \gamma } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { ( B + \epsilon ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \epsilon , \kappa , r , \alpha } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } ) + \log \frac { n } { \eta } } { \gamma ^ { 2 } n } } \bigg ) , +$$ + +where we consider $\Phi _ { \epsilon , \kappa , r , \alpha } ^ { \mathrm { p g m } } ( f _ { \mathbf { w } } )$ as the following expression + +$$ +\Big [ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ( 1 + ( \alpha / \kappa ) \frac { 1 - ( 2 \alpha / \kappa ) ^ { r } \| \mathbf { \overline { { \log } } } ( \nabla \ell \circ f _ { \mathbf { w } } ) ^ { r } } { 1 - ( 2 \alpha / \kappa ) \| \mathbf { \overline { { \log } } } ( \nabla \ell \circ f _ { \mathbf { w } } ) } \big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \Big \} ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } . +$$ + +Using a similar argument to our proof of Theorem 2, we can properly cover the spectral norms for each $\mathbf { W } _ { i }$ with $2 r d \log M$ points, such that for any feasible $\Vert \mathbf { W } _ { i } \Vert _ { 2 }$ value, satisfying the assumptions, we have value $a _ { i }$ in our cover where $\begin{array} { r } { | \| \mathbf { W } _ { i } \| _ { 2 } - a _ { i } | \le \frac { 1 } { r d } \overset { \cdot \cdot } { a } _ { i } } \end{array}$ . Therefore, we can cover all feasible combinations of spectral norms with $( 2 r d \log M ) ^ { d } d n ^ { 1 / 2 d }$ , which combined with the above discussion completes the proof. + +# C.4 PROOF OF THEOREM 4 + +We first show the following lemma providing a perturbation bound for WRM attacks. + +Lemma 8. Consider a $d$ -layer neural net $f _ { \mathbf { w } }$ satisfying the assumptions of Lemma 3. Then, for any weight perturbation u such that $\begin{array} { r } { \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ we have + +$$ +\| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { w r m } } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) \| _ { 2 } \leq \frac { e ^ { 2 } } { \lambda - \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) } +$$ + +$$ +\times \big ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ) \sum _ { i = 1 } ^ { d } \bigg [ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + \big ( \| \mathbf { x } \| _ { 2 } + \frac { \prod _ { j = 1 } ^ { d } \| \mathbf { W } _ { j } \| _ { 2 } } { \lambda } \big ) \big ( \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \bigg ] . +$$ + +In the above inequality, $\operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } )$ denotes the Lipschitz constant of $\nabla _ { \mathbf x } \ell ( f _ { \mathbf w } ( \mathbf x ) , y )$ + +Proof. First of all note that for any $\mathbf { x }$ we have $\begin{array} { r } { \| \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) \| _ { 2 } \leq ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) / \lambda } \end{array}$ , because we assume $\mathrm { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) < \lambda$ implying WRM’s optimization is a convex optimization problem with the global solution $\delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } )$ satisfying $\begin{array} { r } { \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) = \frac { 1 } { \lambda } \nabla \ell \circ f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) ) } \end{array}$ which is norm-bounded by $\begin{array} { r } { \frac { \operatorname* { l i p } ( \ell \circ f _ { \mathbf { w } } ) } { \lambda } \leq ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) / \lambda } \end{array}$ . Moreover, applying Lemma 3 we have + +$$ +\begin{array} { r l } & { \quad \left\| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { s r m } } ( \mathbf { x } ) \right\| _ { 2 } } \\ & { = \big \| \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) - \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) ) \big \| _ { 2 } } \\ & { \leq \big \| \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) - \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) ) \big \| _ { 2 } } \\ & { \quad \quad + \big \| \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) - \frac { 1 } { \lambda } \nabla _ { \mathbf { x } } \ell ( f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { s r m } } ( \mathbf { x } ) ) ) \big \| _ { 2 } } \\ & { \leq \frac { \epsilon ^ { 2 } } { \lambda } ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \Big [ \big \| \mathbf { U } _ { i } \big \| _ { 2 } } \\ & { \quad \quad + \frac { \operatorname* { l i p } ( \nabla \ell \cup f _ { \mathbf { w } } ) } { \lambda } \big \| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s r m } } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { s r m } } ( \mathbf { x } ) \big \| _ { 2 } } \end{array} +$$ + +which shows the following inequality and hence completes the proof: + +$$ +\begin{array} { r l } & { \quad \bigl ( 1 - \frac { \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) } { \lambda } \bigr ) \left\| \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { w r m } } ( \mathbf { x } ) - \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) \right\| _ { 2 } } \\ & { \leq \frac { e ^ { 2 } } { \lambda } ( \displaystyle \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \displaystyle \sum _ { i = 1 } ^ { d } \left[ \frac { \| \mathbf { U } _ { i } \| _ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } } + \bigl ( \| \mathbf { x } \| _ { 2 } + \frac { \prod _ { j = 1 } ^ { d } \| \mathbf { W } _ { j } \| _ { 2 } } { \lambda } \bigr ) ( \displaystyle \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } ) \sum _ { j = 1 } ^ { i } \frac { \| \mathbf { U } _ { j } \| _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \right] . } \end{array} +$$ + +Combining the above lemma with Lemma 2, for any norm-bounded $\| \mathbf { x } \| _ { 2 } \leq B$ and perturbation vector $\mathbf { u }$ where $\begin{array} { r } { \| \mathbf { U } _ { i } \| _ { 2 } \leq \frac { 1 } { d } \| \mathbf { W } _ { i } \| _ { 2 } } \end{array}$ , + +$$ +\begin{array} { r l } & { \quad \left\| \int _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { m a x } } ( \mathbf { x } ) ) - \int _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s e m } } ( \mathbf { x } ) ) \right\| _ { 2 } } \\ & { \leq \Big \| \int _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { m a x } } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { s e m } } ( \mathbf { x } ) ) \Big \| _ { 2 } + \Big \| f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { m a x } } ( \mathbf { x } ) ) - f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { m m } } ( \mathbf { x } ) ) \Big \| _ { 2 } } \\ & { \leq e ( B + \frac { \prod _ { j = 1 } ^ { d } | \mathbf { w } _ { j } | } { \lambda } ) ( \underset { \mathrm { L } ^ { - 1 } = 1 } { \overset { d } { \prod } } | \mathbb { W } _ { 1 } | _ { 2 } ) \underset { i = 1 } { \overset { d } { \sum } } \Big \| \mathbb { W } _ { 1 } | _ { 2 } + \Big ( \underset { i = 1 } { \overset { d } { \prod } } | \mathbb { W } _ { 1 } | _ { 2 } ) } \\ & { \qquad \times \frac { e ^ { 2 } } { \lambda - \operatorname { l i p } ( \nabla \xi \ell \int _ { \mathbf { w } } ) } \underset { i = 1 } { \overset { d } { \sum } } \bigg [ \frac { \| \mathbf { U } _ { 1 } | _ { 2 } } { \| \mathbf { W } _ { 1 } \| _ { 2 } } + \big ( B + \frac { \prod _ { j = 1 } ^ { d } \| \mathbf { W } _ { 2 } \| _ { 2 } } { \lambda } \big ) ( \underset { j = 1 } { \overset { i } { \prod } } | \mathbf { W } _ { j } | | _ { 2 } ) \underset { j = 1 } { \overset { i } { \sum } } \frac { \| \mathbf { U } _ { j } | _ { 2 } } { \| \mathbf { W } _ { j } \| _ { 2 } } \bigg ] } \\ & \leq e \end{array} +$$ + +Similar to the proofs of Theorems 2,3, given $\widetilde { \mathbf { w } }$ we choose a zero-mean multivariate Gaussian distribution $Q$ e with diagonal covariance matrix for random perturbation u, with the $i$ th layer $\mathbf { u } _ { i }$ ’s standard deviation parameter $\begin{array} { r } { \xi _ { i } = \frac { \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } { \beta _ { \widetilde { \mathbf { w } } } } \xi } \end{array}$ kWfik2β ξ where + +$$ +\begin{array} { r } { = \frac { \gamma } { 8 e ^ { 5 } d \sqrt { 2 h \log ( 4 h d ) } ( B + \prod _ { j = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { j } \| _ { 2 } / \lambda ) \left( \prod _ { i = 1 } ^ { d } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \right) \left( 1 + \frac { 1 } { \lambda - \operatorname { l i p } ( \nabla \ell \circ f _ { \infty } ) } \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \widetilde { \mathbf { W } } _ { j } \| _ { 2 } \right) } } \end{array} +$$ + +Using a union bound suggests the assumption of Lemma $\begin{array} { r } { \operatorname* { P r } _ { \mathbf { u } } \left( \operatorname* { m a x } _ { \mathbf { x } \in \mathcal { X } } \| f _ { \mathbf { w } + \mathbf { u } } ( \mathbf { x } + \delta _ { \mathbf { w } + \mathbf { u } } ^ { \mathrm { w r m } } ( \mathbf { x } ) ) - \right. } \end{array}$ $\begin{array} { r } { f _ { \mathbf { w } } ( \mathbf { x } + \delta _ { \mathbf { w } } ^ { \mathrm { w r m } } ( \mathbf { x } ) ) \| _ { \infty } \leq \frac { \gamma } { 4 } \big ) \geq \frac { 1 } { 2 } } \end{array}$ holds for $Q$ . Then, for any w satisfying $\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \leq } \end{array}$ $\frac { 1 } { 4 d / \tau } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 }$ we have li $\begin{array} { r } { \overline { { \mathsf { p } } } ( \ell \circ f _ { \mathbf { w } } ) \leq ( e ^ { \tau / 4 } ) ^ { 2 } \overline { { \operatorname { l i p } } } ( \ell \circ f _ { \widetilde { \mathbf { w } } } ) \leq ( e ^ { \tau / 4 } ) ^ { 2 } \lambda ( 1 - \tau ) \leq \frac { 1 - \tau } { 1 - \tau / 2 } \lambda \leq ( 1 - \frac { \tau } { 2 } ) \lambda } \end{array}$ which implies the guard-band $\tau$ for $\widetilde { \mathbf { w } }$ eapplies to w after being modified by a factor 2. Hence, + +$$ +\begin{array} { r l } & { K L ( P _ { \Psi + \Psi } | | Q ) \le \displaystyle \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { \mathcal { F } } ^ { 2 } } { 2 \xi _ { i } ^ { 2 } } } \\ & { \le \mathcal { O } \Big ( d ^ { 2 } ( B + \displaystyle \prod _ { j = 1 } ^ { d } | | \widehat { \mathbf { W } } _ { i j } | | / \lambda ) ^ { 2 } h \log ( \bar { h } d ) } \\ & { \times \displaystyle \frac { ( | \mathbf { I } _ { i - 1 } ^ { d } | | \widehat { \mathbf { W } } _ { i j } | | ^ { 2 } ) \big ( 1 + \frac { 1 } { \lambda - \log ( \nabla \xi _ { \mathcal { F } _ { \Psi } } ) } \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } | \widehat { \mathbf { W } } _ { i j } | \big ) ^ { 2 } } { \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { | \mathbf { W } _ { i } | | _ { \mathcal { F } } ^ { 2 } } { | \widehat { \mathbf { W } } _ { i j } | | _ { 2 } ^ { 2 } } \Big ) } \\ & { \le \mathcal { O } \Big ( d ^ { 2 } ( B + \displaystyle \prod _ { j = 1 } ^ { d } | | \mathbf { W } _ { i j } | | _ { \mathcal { F } } | ) ^ { 2 } ) ^ { 2 } h \log ( \bar { h } d ) } \\ & { \times \displaystyle \frac { ( | \mathbf { I } _ { i - 1 } ^ { d } | | \mathbf { W } _ { i } | | _ { 2 } ^ { 2 } ) \big ( 1 + \frac { 1 } { \lambda - \log ( \nabla \xi _ { \Psi } ) } \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { d } | \mathbf { W } _ { j } | | _ { 2 } \big ) ^ { 2 } } { \gamma ^ { 2 } } \sum _ { i = 1 } ^ { d } \frac { | \mathbf { W } _ { i } | | _ { \mathcal { F } } ^ { 2 } } { | \widehat { \mathbf { W } } _ { i j } | | _ { 2 } ^ { 2 } } , } \end{array} +$$ + +Using this bound in Lemma 1 implies that for any $\eta > 0$ the following bound will hold with probability $1 - \eta$ for any w where $\begin{array} { r } { \| \mathbf { W } _ { i } \| _ { 2 } - \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } \big | \le \frac { 1 } { 4 d / \tau } \| \widetilde { \mathbf { W } } _ { i } \| _ { 2 } } \end{array}$ : + +$$ +L _ { 0 } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } ) \leq \widehat { L } _ { \gamma } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } ) + \mathcal { O } \bigg ( \sqrt { \frac { \big ( B + \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } / \lambda \big ) ^ { 2 } d ^ { 2 } h \log ( d h ) \Phi _ { \lambda } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } ) + d \log \frac { n } { \eta } } { \gamma ^ { 2 } n } } \bigg ) . +$$ + +where we define $\Phi _ { \lambda } ^ { \mathrm { w r m } } ( f _ { \mathbf { w } } )$ to be + +$$ +\displaystyle \{ \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } \big ( 1 + \frac { 1 } { \lambda - \operatorname* { l i p } ( \nabla \ell \circ f _ { \mathbf { w } } ) } ( \prod _ { i = 1 } ^ { d } \| \mathbf { W } _ { i } \| _ { 2 } ) \sum _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { i } \| \mathbf { W } _ { j } \| _ { 2 } \big ) \big \} ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \| \mathbf { W } _ { i } \| _ { F } ^ { 2 } } { \| \mathbf { W } _ { i } \| _ { 2 } ^ { 2 } } . +$$ + +Using a similar argument to our proofs of Theorems 2 and 3, we can cover the possible spectral norms for each $\mathbf { W } _ { i }$ with $O ( ( 8 d / \tau ) \log M )$ points, such that for any feasible $\Vert \mathbf { W } _ { i } \Vert _ { 2 }$ value satisfying the theorem’s assumptions, we have value $a _ { i }$ in our cover where $\begin{array} { r } { \left| \| \mathbf { W } _ { i } \right\| _ { 2 } - a _ { i } \Big | \le \frac { 1 } { 4 d / \tau } a _ { i } } \end{array}$ . Therefore, we can cover all feasible combinations of spectral norms with $O ( ( ( 8 d / \tau ) \log M ) ^ { d } d n ^ { 1 / 2 d } )$ , which combined with the above discussion finishes the proof. \ No newline at end of file diff --git a/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_content_list.json b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6507e371a0ccc2e52a35cd2b9351d7185f19fdd4 --- /dev/null +++ b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_content_list.json @@ -0,0 +1,3099 @@ +[ + { + "type": "text", + "text": "GENERALIZABLE ADVERSARIAL TRAINING VIA SPECTRAL NORMALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 99, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Farzan Farnia∗, Jesse M. Zhang∗, David N. Tse Department of Electrical Engineering Stanford University {farnia,jessez,dntse}@stanford.edu ", + "bbox": [ + 184, + 169, + 517, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (DNNs) have set benchmarks on a wide array of supervised learning tasks. Trained DNNs, however, often lack robustness to minor adversarial perturbations to the input, which undermines their true practicality. Recent works have increased the robustness of DNNs by fitting networks using adversarially-perturbed training samples, but the improved performance can still be far below the performance seen in non-adversarial settings. A significant portion of this gap can be attributed to the decrease in generalization performance due to adversarial training. In this work, we extend the notion of margin loss to adversarial settings and bound the generalization error for DNNs trained under several well-known gradient-based attack schemes, motivating an effective regularization scheme based on spectral normalization of the DNN’s weight matrices. We also provide a computationally-efficient method for normalizing the spectral norm of convolutional layers with arbitrary stride and padding schemes in deep convolutional networks. We evaluate the power of spectral normalization extensively on combinations of datasets, network architectures, and adversarial training schemes. ", + "bbox": [ + 233, + 295, + 766, + 502 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 532, + 336, + 547 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite their impressive performance on many supervised learning tasks, deep neural networks (DNNs) are often highly susceptible to adversarial perturbations imperceptible to the human eye (Szegedy et al., 2013; Goodfellow et al., 2014b). These “adversarial attacks\" have received enormous attention in the machine learning literature over recent years (Goodfellow et al., 2014b; Moosavi Dezfooli et al., 2016; Carlini & Wagner, 2016; Kurakin et al., 2016; Papernot et al., 2016; Carlini & Wagner, 2017; Papernot et al., 2017; Madry et al., 2018; Tramèr et al., 2018). Adversarial attack studies have mainly focused on developing effective attack and defense schemes. While attack schemes attempt to mislead a trained classifier via additive perturbations to the input, defense mechanisms aim to train classifiers robust to these perturbations. Although existing defense methods result in considerably better performance compared to standard training methods, the improved performance can still be far below the performance in non-adversarial settings (Athalye et al., 2018; Schmidt et al., 2018). ", + "bbox": [ + 173, + 565, + 825, + 731 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A standard adversarial training scheme involves fitting a classifier using adversarially-perturbed samples (Szegedy et al., 2013; Goodfellow et al., 2014b) with the intention of producing a trained classifier with better robustness to attacks on future (i.e. test) samples. Madry et al. (2018) provides a robust optimization interpretation of the adversarial training approach, demonstrating that this strategy finds the optimal classifier minimizing the average worst-case loss over an adversarial ball centered at each training sample. This minimax interpretation can also be extended to distributionally-robust training methods (Sinha et al., 2018) where the offered robustness is over a Wasserstein-ball around the empirical distribution of training data. ", + "bbox": [ + 174, + 738, + 825, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, Schmidt et al. (2018) have shown that standard adversarial training produces networks that generalize poorly. The performance of adversarially-trained DNNs over test samples can be significantly worse than their training performance, and this gap can be far greater than the generalization gap achieved using standard empirical risk minimization (ERM). This discrepancy suggests that the overall adversarial test performance can be improved by applying effective regularization schemes during adversarial training. ", + "bbox": [ + 176, + 857, + 825, + 898 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/ef52baaee3ec3221278850c219c777bc05008a87327e9f87b9f3cb88ff84e222.jpg", + "image_caption": [ + "Figure 1: Adversarial training performance with and without spectral normalization (SN) for AlexNet fit on CIFAR10. The gain in the final test accuracies for FGM, PGM, and WRM after spectral normalization are 0.09, 0.11, and 0.04, respectively (see Table 1 in the Appendix). For FGM and PGM, perturbations have $\\ell _ { 2 }$ magnitude 2.44. " + ], + "image_footnote": [], + "bbox": [ + 183, + 103, + 812, + 268 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 372, + 825, + 415 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we propose using spectral normalization (SN) (Miyato et al., 2018) as a computationallyefficient and statistically-powerful regularization scheme for adversarial training of DNNs. SN has been successfully implemented and applied for DNNs in the context of generative adversarial networks (GANs) (Goodfellow et al., 2014a), resulting in state-of-the-art deep generative models for several benchmark tasks (Miyato et al., 2018). Moreover, SN (Tsuzuku et al., 2018) and other similar Lipschitz regularization techniques (Cisse et al., 2017) have been successfully applied in non-adversarial training settings to improve the robustness of ERM-trained networks to adversarial attacks. The theoretical results in (Bartlett et al., 2017; Neyshabur et al., 2017a) and empirical results in (Yoshida & Miyato, 2017) also suggest that SN can close the generalization gap for DNNs in non-adversarial ERM setting. ", + "bbox": [ + 174, + 421, + 825, + 560 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "On the theoretical front, we extend the standard notion of margin loss to adversarial settings. We leverage the PAC-Bayes generalization framework (McAllester, 1999) to prove generalization bounds for spectrally-normalized DNNs in terms of our defined adversarial margin loss. We obtain adversarial generalization error bounds for three well-known gradient-based attack schemes: fast gradient method (FGM) (Goodfellow et al., 2014b), projected gradient method (PGM) (Kurakin et al., 2016), and Wasserstein risk minimization (WRM) (Sinha et al., 2018). Our theoretical analysis shows that the adversarial generalization error will vanish by applying SN to all layers. ", + "bbox": [ + 174, + 568, + 825, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "On the empirical front, we show that SN can significantly improve the test performance of adversarially-trained DNNs. We perform numerical experiments over various standard datasets and DNN architectures. In almost all of our experiments, we obtain a better test performance after applying SN. For example, Figure 1 shows the training and validation performance for AlexNet fit on the CIFAR10 dataset using FGM, PGM, and WRM, resulting in adversarial test accuracy improvements of 9, 11, and 4 percent, respectively. To perform our numerical experiments, we develop a computationally-efficient approach for normalizing the spectral norm of convolution layers with arbitrary stride and padding schemes. To summarize, the main contributions of this work are: ", + "bbox": [ + 174, + 671, + 825, + 784 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. Proposing SN as a regularization scheme for adversarial training of DNNs, \n2. Extending concepts of margin-based generalization analysis to adversarial settings and proving margin-based generalization bounds for three gradient-based adversarial attack schemes, \n3. Developing an efficient method for normalizing the spectral norm of convolutional layers in deep convolution networks, \n4. Numerically demonstrating the improved test and generalization performance of DNNs trained with SN. ", + "bbox": [ + 210, + 795, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 339, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we first review some standard concepts of margin-based generalization analysis in learning theory. We then extend these notions to adversarial training settings. ", + "bbox": [ + 174, + 132, + 825, + 161 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 SUPERVISED LEARNING, DEEP NEURAL NETWORKS, GENERALIZATION ERROR ", + "bbox": [ + 173, + 178, + 758, + 193 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider samples $\\left\\{ ( \\mathbf { x } _ { 1 } , y _ { 1 } ) , \\dotsc , ( \\mathbf { x } _ { n } , y _ { n } ) \\right\\}$ drawn i.i.d from underlying distribution $P _ { \\mathbf { X } , Y }$ . We suppose $\\mathbf x \\in \\mathcal X$ and $Y \\in \\{ 1 , 2 , \\dots , m \\}$ where $m$ represents the number of different labels. Given loss function $\\ell$ and function class $\\mathcal { F } = \\left\\{ f _ { \\mathbf { w } } , \\mathbf { w } \\in \\mathcal { W } \\right\\}$ parameterized by w, a supervised learner aims to find the optimal function in $\\mathcal { F }$ minimizing the expected loss (risk) averaged over the underlying distribution $P$ . ", + "bbox": [ + 173, + 202, + 825, + 273 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider $\\mathcal { F } _ { \\mathrm { n n } }$ as the class of $d$ -layer neural networks with $h$ hidden units per layer and activation functions $\\sigma : \\mathbb { R } \\mathbb { R }$ . Each $f _ { \\mathbf { w } } : \\mathcal { X } \\mathbf { R } ^ { m }$ in $\\mathcal { F } _ { \\mathrm { n n } }$ maps a data point $\\mathbf { x }$ to an $m$ -dimensional vector. Specifically, we can express each $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { \\mathrm { n n } }$ as $f _ { \\mathbf { w } } ( \\mathbf { x } \\bar { ) = } { \\mathbf { W } _ { d } } \\bar { \\sigma } ( { \\mathbf { W } _ { d - 1 } } \\cdot \\cdot \\cdot \\sigma ( { \\mathbf { W } _ { 1 } } \\mathbf { x } ) \\cdot \\cdot \\cdot ) )$ . We use $\\left. \\mathbf { W } _ { i } \\right. _ { 2 }$ to denote the spectral norm of matrix $\\mathbf { W } _ { i }$ , defined as the largest singular value of $\\mathbf { W } _ { i }$ , and $\\Vert \\mathbf { W } _ { i } \\Vert _ { F }$ to denote $\\mathbf { W } _ { i }$ ’s Frobenius norm. ", + "bbox": [ + 173, + 280, + 825, + 351 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A classifier $f _ { \\mathbf { w } }$ ’s performance over the true distribution of data can be different from the training performance over the empirical distribution of training samples $\\hat { P }$ . The difference between the empirical and true averaged losses, evaluated on respectively training and test samples, is called the generalization error. Similar to Neyshabur et al. (2017a), we evaluate a DNN’s generalization performance using its expected margin loss defined for margin parameter $\\gamma > 0$ as ", + "bbox": [ + 173, + 357, + 825, + 429 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a628ba20a03d1f96572de5b8f76fd2200a4e39d6eca919f74c20ccd2cfcbde74.jpg", + "text": "$$\nL _ { \\gamma } ( f _ { \\mathbf { w } } ) : = P \\bigg ( f _ { \\mathbf { w } } ( \\mathbf { X } ) [ Y ] \\leq \\gamma + \\operatorname* { m a x } _ { j \\neq Y } f _ { \\mathbf { w } } ( \\mathbf { X } ) [ j ] \\bigg ) ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 433, + 666, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $f _ { \\mathbf { w } } ( \\mathbf { X } ) [ j ]$ denotes the $j$ th entry of $f _ { \\mathbf { w } } ( \\mathbf { X } ) \\in \\mathbb { R } ^ { m }$ . For a given data point $\\mathbf { X }$ , we predict the label corresponding to the maximum entry of $f _ { \\mathbf { w } } ( \\mathbf { X } )$ . Also, we use $\\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } )$ to denote the empirical margin loss averaged over the training samples. The goal of margin-based generalization analysis is to provide theoretical comparison between the true and empirical margin risks. ", + "bbox": [ + 173, + 470, + 825, + 530 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 ADVERSARIAL ATTACKS, ADVERSARIAL TRAINING ", + "text_level": 1, + "bbox": [ + 173, + 546, + 570, + 561 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A supervised learner observes only the training samples and hence does not know the true distribution of data. Then, a standard approach to train a classifier is to minimize the empirical expected loss $\\ell$ over function class $\\mathcal { F } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathcal { W } \\}$ , which is ", + "bbox": [ + 173, + 571, + 825, + 614 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c8d00f90a14fe2bb5837f15560fad6b8ab432d251b342e057e55573e51a93d2d.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathcal { W } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) , y _ { i } \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 618, + 589, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This approach is called empirical risk minimization (ERM). For better optimization performance, the loss function $\\ell$ is commonly chosen to be smooth. Hence, 0-1 and margin losses are replaced by smooth surrogate loss functions such as the cross-entropy loss. However, we still use the margin loss as defined in (1) for evaluating the test and generalization performance of DNN classifiers. ", + "bbox": [ + 173, + 662, + 825, + 719 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While ERM training usually achieves good performance over DNNs, several recent observations reveal that adding some adversarially-chosen perturbation to each sample can significantly drop the trained DNN’s performance. Given norm function $\\| \\cdot \\|$ and adversarial noise power $\\epsilon > 0$ , the adversarial additive noise for sample $\\left( \\mathbf { x } , y \\right)$ and classifier $f _ { \\mathbf { w } }$ is defined to be ", + "bbox": [ + 173, + 726, + 825, + 782 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e436fb0d7d7b2445b4dd44a4739193aaec85d8b390822261bb760b0cb52cca15.jpg", + "text": "$$\n\\delta _ { \\mathbf w } ^ { \\mathrm { a d v } } ( \\mathbf x ) : = \\operatorname * { a r g m a x } _ { \\| \\delta \\| \\leq \\epsilon } \\ell \\big ( f _ { \\mathbf w } ( \\mathbf x + \\pmb \\delta ) , y \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 786, + 629, + 815 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To provide adversarial robustness against the above attack scheme, a standard technique, which is called adversarial training, follows ERM training over the adversarially-perturbed samples by solving ", + "bbox": [ + 174, + 819, + 825, + 848 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0ef5ff3deff2585f04b6f6a366234cb82ddf00a783b310abb55d68c66342465b.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathcal { W } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell \\left( f _ { \\mathbf { w } } \\left( \\mathbf { x } _ { i } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { x } _ { i } ) \\right) , y _ { i } \\right) : = \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathcal { W } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\| \\delta _ { i } \\| \\leq \\epsilon } \\ell \\left( f _ { \\mathbf { w } } ( \\mathbf { x } _ { i } + \\delta _ { i } ) , y _ { i } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 222, + 851, + 777, + 892 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "However, (3) and (4) are intractable optimization problems. Therefore, several schemes have been proposed in the literature to approximate the optimal solution of (3). In this work, we analyze the generalization performance of the following three gradient-based methods for approximating the solution to (3). We note that several other attack schemes such as DeepFool (Moosavi Dezfooli et al., 2016), CW attacks (Carlini & Wagner, 2017), target and least-likely attacks (Kurakin et al., 2016) have been introduced and examined in the literature, which can lead to interesting future directions for this work. ", + "bbox": [ + 173, + 895, + 826, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 174 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Fast Gradient Method (FGM) (Goodfellow et al., 2014b): FGM approximates the solution to (3) by considering a linearized DNN loss around a given data point. Hence, FGM perturbs $\\left( \\mathbf { x } , y \\right)$ by adding the following noise vector: ", + "bbox": [ + 173, + 184, + 825, + 228 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a8103fe9da87a8078dd9c58670c3b37f7687231941d0b675721b45084e38b8d8.jpg", + "text": "$$\n\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) : = \\underset { \\| \\delta \\| \\leq \\epsilon } { \\mathrm { a r g m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 234, + 635, + 265 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the special case of $\\ell _ { \\infty }$ -norm $\\| \\cdot \\| _ { \\infty }$ , the above representation of FGM recovers the fast gradient sign method (FGSM) where each data point $\\left( \\mathbf { x } , y \\right)$ is perturbed by the $\\epsilon$ -normalized sign vector of the loss’s gradient. For $\\ell _ { 2 }$ -norm $\\| \\cdot \\| _ { 2 }$ , we similarly normalize the loss’s gradient vector to have $\\epsilon$ Euclidean norm. ", + "bbox": [ + 173, + 272, + 825, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2. Projected Gradient Method (PGM) (Kurakin et al., 2016): PGM is the iterative version of FGM and applies projected gradient descent to solve (3). PGM follows the following update rules for a given $r$ number of steps: ", + "bbox": [ + 174, + 333, + 825, + 377 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6b7e6cd6e53ed6b5fe1283c39d11ed9b1865b5c19487a5861e666b531ebf7034.jpg", + "text": "$$\n\\begin{array} { r l } { \\forall 1 \\le i \\le r : } & { \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i + 1 } ( \\mathbf { x } ) : = \\displaystyle \\prod _ { \\boldsymbol { \\epsilon } _ { \\epsilon , \\parallel } \\cdot \\parallel ^ { ( 0 ) } } \\bigl \\{ \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) + \\alpha \\nu _ { \\mathbf { w } } ^ { ( i ) } \\bigr \\} , } \\\\ & { \\nu _ { \\mathbf { w } } ^ { ( i ) } : = \\underset { \\parallel \\delta \\parallel \\le 1 } { \\arg \\operatorname* { m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) ) , y \\bigr ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 382, + 738, + 454 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, we first find the direction $\\nu _ { \\mathbf { w } } ^ { ( i ) }$ along which the loss at the $i$ th perturbed point changes the most, and then we move the perturbed point along this direction by stepsize $\\alpha$ followed by projecting the resulting perturbation onto the set $\\left\\{ \\delta : \\left\\| \\delta \\right\\| \\leq \\epsilon \\right\\}$ with $\\epsilon$ -bounded norm. ", + "bbox": [ + 176, + 462, + 823, + 507 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3. Wasserstein Risk Minimization (WRM) (Sinha et al., 2018): WRM solves the following variant of (3) for data-point $\\left( \\mathbf { x } , y \\right)$ where the norm constraint in (3) is replaced by a norm-squared Lagrangian penalty term: ", + "bbox": [ + 173, + 511, + 825, + 555 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f669b3f84fa5632204c604cc43c989f5b73c08aa68e89307525ded9698c0c27f.jpg", + "text": "$$\n\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) : = \\operatorname * { a r g m a x } _ { \\delta } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\pmb { \\delta } ) , y \\big ) - \\frac { \\lambda } { 2 } \\| \\pmb { \\delta } \\| ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 561, + 661, + 594 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As discussed earlier, the optimization problem (3) is generally intractable. However, in the case of Euclidean norm $\\| \\cdot \\| _ { 2 }$ , if we assume $\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )$ ’s Lipschitz constant is upper-bounded by $\\lambda$ , then WRM optimization (7) results in solving a convex optimization problem and can be efficiently solved using gradient methods. ", + "bbox": [ + 173, + 601, + 825, + 657 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To obtain efficient adversarial defense schemes, we can substitute $\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } }$ , $\\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } }$ , or $\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } }$ for $\\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } }$ in (4). Instead of fitting the classifier over true adversarial examples, which are NP-hard to obtain, we can instead train the DNN over FGM, PGM, or WRM-adversarially perturbed samples. ", + "bbox": [ + 173, + 669, + 826, + 713 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 ADVERSARIAL GENERALIZATION ERROR ", + "text_level": 1, + "bbox": [ + 176, + 731, + 495, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The goal of adversarial training is to improve the robustness against adversarial attacks on not only the training samples but also on test samples; however, the adversarial training problem (4) focuses only on the training samples. To evaluate the adversarial generalization performance, we extend the notion of margin loss defined earlier in (1) to adversarial training settings by defining the adversarial margin loss as ", + "bbox": [ + 173, + 756, + 825, + 827 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fba65fc355814dd336ebccebd4538ef1d48d9e8e914b9f6b75fded54737a2eba.jpg", + "text": "$$\nL _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } ) = P \\bigg ( f _ { \\mathbf { w } } ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) ) [ Y ] \\leq \\gamma + \\operatorname* { m a x } _ { j \\neq Y } f _ { \\mathbf { w } } \\big ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) \\big ) [ j ] \\bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 241, + 833, + 754, + 868 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, we measure the margin loss over adversarially-perturbed samples, and we use $\\widehat { L } _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } )$ to denote the empirical adversarial margin loss. We also use $L _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) , L _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )$ , and $L _ { \\gamma } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } )$ to denote the adversarial margin losses with FGM (5), PGM (6), and WRM (7) attacks, respectively. ", + "bbox": [ + 174, + 877, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 MARGIN-BASED ADVERSARIAL GENERALIZATION BOUNDS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 697, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As previously discussed, generalization performance can be different between adversarial and nonadversarial settings. In this section, we provide generalization bounds for DNN classifiers under adversarial attacks in terms of the spectral norms of the trained DNN’s weight matrices. The bounds motivate regularizing these spectral norms in order to limit the DNN’s capacity and improve its generalization performance under adversarial attacks. ", + "bbox": [ + 174, + 133, + 825, + 204 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use the PAC-Bayes framework (McAllester, 1999; 2003) to prove our main results. To derive adversarial generalization error bounds for DNNs with smooth activation functions $\\sigma$ , we first extend a recent result on the margin-based generalization bound for the ReLU activation function (Neyshabur et al., 2017a) to general 1-Lipschitz activation functions. ", + "bbox": [ + 174, + 209, + 825, + 267 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1. Consider $\\mathcal { F } _ { n n } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathbf { W } \\}$ the class of d hidden-layer neural networks with $h$ units per hidden-layer with 1-Lipschitz activation $\\sigma$ satisfying $\\sigma ( 0 ) = 0$ . Suppose that $\\mathcal { X }$ , $\\mathbf { X }$ ’s support set, is norm-bounded as $\\| \\mathbf { x } \\| _ { 2 } \\leq B$ , $\\forall \\mathbf { x } \\in { \\mathcal { X } }$ . Also assume for constant $M \\geq 1$ any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ satisfies ", + "bbox": [ + 173, + 270, + 825, + 327 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/65149fb37096cc18401f4f83d66787664661b763120128178de050f90d13c0c9.jpg", + "text": "$$\n\\forall i : ~ \\frac { 1 } { M } \\leq \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } } { \\beta _ { \\mathbf { w } } } \\leq M , \\quad \\beta _ { \\mathbf { w } } : = \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) ^ { 1 / d } .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 325, + 678, + 369 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here $\\beta _ { \\mathbf { w } }$ denotes the geometric mean of $f _ { \\mathbf { w } }$ ’s spectral norms across all layers. Then, for any $\\eta , \\gamma > 0$ , with probability at least $1 - \\eta$ for any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ we have: ", + "bbox": [ + 173, + 372, + 825, + 401 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2c3f8efecae8bdf8059492f399508d55a04cd934671cf4808d438540b9a69797.jpg", + "text": "$$\nL _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { B ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) ,\n$$", + "text_format": "latex", + "bbox": [ + 256, + 407, + 740, + 450 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) : = \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}$ ", + "bbox": [ + 173, + 459, + 689, + 481 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. We defer the proof to the Appendix. ", + "bbox": [ + 174, + 497, + 460, + 512 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now generalize this result to adversarial settings where the DNN’s performance is evaluated under adversarial attacks. We prove three separate adversarial generalization error bounds for FGM, PGM, and WRM attacks. ", + "bbox": [ + 176, + 529, + 825, + 571 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For the following results, we consider $\\mathcal { F } _ { \\mathrm { n n } }$ , the class of neural nets defined in Theorem 1. Moreover, we assume that the training loss $\\ell ( \\hat { y } , y )$ and its first-order derivative are 1-Lipschitz. Similar to Sinha et al. (2018), we assume the activation $\\sigma$ is smooth and its derivative $\\sigma ^ { \\prime }$ is 1-Lipschitz. This class of activations include ELU (Clevert et al., 2015) and tanh functions but not the ReLU function. However, our numerical results in Table 1 from the Appendix suggest similar generalization performance between ELU and ReLU activations. ", + "bbox": [ + 173, + 577, + 825, + 661 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 2. Consider $\\mathcal { F } _ { n n }$ , $\\mathcal { X }$ in Theorem 1 and training loss function $\\ell$ satisfying the assumptions stated above. We consider an FGM attack with noise power $\\epsilon$ according to Euclidean norm $\\| \\cdot \\| _ { 2 }$ . For any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ assume $\\boldsymbol { \\kappa } \\leq \\| \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\| _ { 2 }$ holds for constant $\\kappa > 0$ , any $y \\in \\mathcal { V }$ , and any $\\mathbf { x } \\in B _ { \\epsilon , \\parallel \\cdot \\parallel _ { 2 } } ( \\mathcal { X } )$ \u000f-close to $\\mathbf { X }$ ’s support set. Then, for any $\\eta , \\gamma > 0$ with probability $1 - \\eta$ the following bound holds for the FGM margin loss of any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ ", + "bbox": [ + 174, + 665, + 825, + 739 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4dfa8e7000269871e712b2d591a10c764c3edb7ad81fad4be3ed77e7f6821266.jpg", + "text": "$$\n\\begin{array} { r } { L _ { 0 } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 746, + 777, + 789 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d13a266f750f4645802622beec38ccec9b17630427cfc888afb53d947d477897.jpg", + "text": "$$\n\\begin{array} { r } { \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) : = \\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + ( \\epsilon / \\kappa ) ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\right\\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 795, + 825, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. We defer the proof to the Appendix. ", + "bbox": [ + 174, + 835, + 460, + 851 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that the above theorem assumes that the change rate for the loss function around test samples is at least $\\kappa$ , which gives a baseline for measuring the attack power $\\epsilon$ . In our numerical experiments, we validate this assumption over standard image recognition tasks. Next, we generalize this result to adversarial settings with PGM attack, i.e. the iterative version of FGM attack. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 3. Consider $\\mathcal { F } _ { n n } , \\mathcal { X }$ and training loss function \\` for which the assumptions in Theorem 2 hold. We consider a PGM attack with noise power \u000f given Euclidean norm $\\| \\cdot \\| _ { 2 }$ , $r$ iterations for attack, and stepsize $\\alpha$ . Then, for any $\\eta , \\gamma > 0$ with probability $1 - \\eta$ the following bound applies to the PGM margin loss of any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ ", + "bbox": [ + 174, + 102, + 825, + 161 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d306a56826d0e7ad984fa7002dccb30221035d090d09bcf4a30f9a96f4a1c5bf.jpg", + "text": "$$\nL _ { 0 } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { r d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 202, + 166, + 794, + 209 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here we define $\\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )$ as the following expression ", + "bbox": [ + 173, + 215, + 542, + 232 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0ff186e4f785a9950d0f414c3d2f8a72571c1056f6477a93831335f5c9ee6a6c.jpg", + "text": "$$\n\\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\left( 1 + ( \\alpha / \\kappa ) { \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\right) \\right\\} ^ { 2 } \\sum _ { i = 1 } ^ { d } { \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } }\n$$", + "text_format": "latex", + "bbox": [ + 169, + 239, + 816, + 284 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : = \\bigl ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\bigr ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\end{array}$ provides an upper-bound on the Lipschitz constant of $\\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y )$ . ", + "bbox": [ + 174, + 291, + 825, + 328 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof. We defer the proof to the Appendix. ", + "bbox": [ + 174, + 342, + 460, + 356 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the above result, notice that if $\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) / \\kappa < 1 / ( 2 \\alpha )$ then for any number of gradient steps the PGM margin-based generalization bound will grow the FGM generalization error bound in Theorem 2 by factor $1 / \\big ( 1 - ( \\overline { { 2 \\alpha / \\kappa } } ) \\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) \\big )$ . We next extend our adversarial generalization analysis to WRM attacks. ", + "bbox": [ + 173, + 372, + 825, + 431 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4. For neural net class $\\mathcal { F } _ { n n }$ and training loss $\\ell$ satisfying Theorem 2’s assumptions, consider a WRM attack with Lagrangian coefficient $\\lambda$ and Euclidean norm $\\| \\cdot \\| _ { 2 }$ . Given parameter $0 < \\tau < 1$ , assume $\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )$ defined in Theorem 3 is upper-bounded by $\\lambda ( 1 - \\tau )$ for any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ . For any $\\eta > 0$ , the following WRM margin-based generalization bound holds with probability $1 - \\eta$ for any $f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }$ : ", + "bbox": [ + 173, + 434, + 825, + 508 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0bb541ac528bf72e5af822360659726a00fd8b2e0d2e882a08df010a717a2f66.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { \\neg _ { v r m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\operatorname { w r m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\frac { 1 } { \\lambda } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\lambda } ^ { \\operatorname { w r m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\tau \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 513, + 828, + 558 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where we define ", + "bbox": [ + 173, + 563, + 281, + 577 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/060b3e7e1a0b340c8e6abdcbef6054054392615d5e9f190fa8791738add1a866.jpg", + "text": "$$\n\\Phi _ { \\lambda } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } ) : = \\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ( 1 + \\frac { 1 } { \\lambda - \\operatorname* { l i m } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 583, + 820, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof. We defer the proof to the Appendix. ", + "bbox": [ + 176, + 642, + 460, + 657 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As discussed by Sinha et al. (2018), the condition $\\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) < \\lambda$ for the actual Lipschitz constant of $\\nabla \\ell \\circ f _ { \\mathbf { w } }$ is in fact required to guarantee WRM’s convergence to the global solution. Notice that the WRM generalization error bound in Theorem 4 is bounded by the product of $\\frac { 1 } { \\lambda - \\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) }$ and the FGM generalization bound in Theorem 2. ", + "bbox": [ + 173, + 671, + 825, + 733 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 SPECTRAL NORMALIZATION OF CONVOLUTIONAL LAYERS ", + "text_level": 1, + "bbox": [ + 174, + 753, + 689, + 770 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To control the Lipschitz constant of our trained network, we need to ensure that the spectral norm associated with each linear operation in the network does not exceed some pre-specified $\\beta$ . For fully-connected layers (i.e. regular matrix multiplication), please see Appendix B. For a general class of linear operations including convolution, Tsuzuku et al. (2018) propose to compute the operation’s spectral norm through computing the gradient of the Euclidean norm of the operation’s output. Here, we leverage the deconvolution operation to further simplify and accelerate computing the spectral norm of the convolution operation. Additionally, Sedghi et al. (2018) develop a method for computing all the singular values including the largest one, i.e. the spectral norm. While elegant, the method only applies to convolution filters with stride 1 and zero-padding. However, in practice the normalization factor depends on the stride size and padding scheme governing the convolution operation. Here we develop an efficient approach for computing the maximum singular value, i.e. spectral norm, of convolutional layers with arbitary stride and padding schemes. Note that, as also discussed by Gouk et al. (2018), the ith convolutional layer output feature map $\\psi _ { i }$ is a linear operation of the input $X$ : ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 826, + 146 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/a7b109fb87215c86eee6b6f6e0de886f2b623d07c6e10e13662d630b1404d38a.jpg", + "text": "$$\n\\psi _ { i } ( X ) = \\sum _ { j = 1 } ^ { M } F _ { i , j } \\star X _ { j } ,\n$$", + "text_format": "latex", + "bbox": [ + 418, + 154, + 578, + 198 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $X$ has $M$ feature maps, $F _ { i , j }$ is a filter, and $\\star$ denotes the convolution operation (which also encapsulates stride size and padding scheme). For simplicity, we ignore the additive bias terms here. By vectorizing $X$ and letting $V _ { i , j }$ represent the overall linear operation associated with $F _ { i , j }$ , we see that ", + "bbox": [ + 173, + 205, + 826, + 261 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/7bd6c684e0e41d9a398f8f6c95fe98ad0d67a8e28cac55e123617acd6920f3ab.jpg", + "text": "$$\n\\psi _ { i } ( X ) = [ V _ { 1 , 1 } \\quad \\cdot \\cdot \\cdot \\quad V _ { 1 , M } ] X ,\n$$", + "text_format": "latex", + "bbox": [ + 392, + 260, + 604, + 279 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "and therefore the overall convolution operation can be described using ", + "bbox": [ + 173, + 281, + 635, + 296 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/6f25d869d5ec816a388331ca815af54063973249a89cdd3e821245987c4d202f.jpg", + "text": "$$\n\\psi ( X ) = \\left[ \\begin{array} { c c c } { V _ { 1 , 1 } } & { \\ldots } & { V _ { 1 , M } } \\\\ { \\vdots } & { \\ddots } & { \\vdots } \\\\ { V _ { N , 1 } } & { \\ldots } & { V _ { N , M } } \\end{array} \\right] X = W X .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 303, + 640, + 362 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "While explicitly reconstructing $W$ is expensive, we can still compute $\\sigma ( W )$ , the spectral norm of $W$ , by leveraging the convolution transpose operation implemented by several modern-day deep learning packages. This allows us to efficiently performs matrix multiplication with $W ^ { T }$ without explicitly constructing $W$ . Therefore we can approximate $\\sigma ( W )$ using a modified version of power iteration (Algorithm 1), wrapping the appropriate stride size and padding arguments into the convolution and convolution transpose operations. After obtaining $\\sigma ( W )$ , we compute $W _ { \\mathrm { S N } }$ in the same manner as for the fully-connected layers. Like Miyato et al., we exploit the fact that SGD only makes small updates to $W$ from training step to training step, reusing the same $\\tilde { \\mathbf { u } }$ and running only one iteration per step. Unlike Miyato et al., rather than enforcing $\\sigma ( W ) = \\beta$ , we instead enforce the looser constraint $\\sigma ( W ) \\leq \\beta$ : ", + "bbox": [ + 173, + 375, + 826, + 516 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ca14e0c3edc816fcdfe37e0c0e2d28bc7f91cb9625a4ca1d31e7edb2b8d6840b.jpg", + "text": "$$\nW _ { \\mathrm { S N } } = W / \\operatorname* { m a x } ( 1 , \\sigma ( W ) / \\beta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 395, + 523, + 601, + 541 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "which we observe to result in faster training for supervised learning tasks. ", + "bbox": [ + 174, + 547, + 655, + 563 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/4d4baebdd3d4ef5e60a5f873e769aee730ab59700e34e386d4d6f3c2c9e1c516.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm1 Convolutional power iteration
Initialize ü with a random vector matching the shape of the convolution input
for t = 0,...,T-1do
ν ← conv(W,u)/llconv(W,u)ll2
ü ← conv_transpose(W,v)/llconv_transpose(W,v)ll2
end for
σ ←v· conv(W,u)
", + "bbox": [ + 174, + 575, + 825, + 689 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 NUMERICAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 717, + 436, + 733 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section we provide an array of empirical experiments to validate both the bounds we derived in Section 3 and our implementation of spectral normalization described in section 4. We show that spectral normalization improves both test accuracy and generalization for a variety of adversarial training schemes, datasets, and network architectures. ", + "bbox": [ + 173, + 750, + 825, + 805 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "All experiments are implemented in TensorFlow (Abadi et al., 2016). For each experiment, we cross validate 4 to 6 values of $\\beta$ (see (9)) using a fixed validation set of 500 samples. For PGM, we used $r = 1 5$ iterations and $\\alpha = 2 \\epsilon / r$ . Additionally, for FGM and PGM we used $\\ell _ { 2 }$ -type attacks (unless specified) with magnitude $\\epsilon = 0 . 0 5 \\mathbb { E } _ { \\hat { P } } [ \\| \\mathbf { X } \\| _ { 2 } ]$ (this value was approximately 2.44 for CIFAR10). For WRM, we implemented gradient ascent as discussed by Sinha et al. (2018). Additionally, for WRM training we used a Lagrangian coefficient of $0 . 0 0 2 \\mathbb { E } _ { \\hat { P } } [ \\| \\mathbf { X } \\| _ { 2 } ]$ for CIFAR10 and SVHN and a Lagrangian coefficient of $0 . \\bar { 0 4 } \\mathbb { E } _ { \\hat { P } } [ \\| \\mathbf { X } \\| _ { 2 } ]$ for MNIST in a similar manner to Sinha et al. (2018). The code will be made readily available. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first demonstrate the effect of the proposed spectral normalization approach on the final DNN weights by comparing the $\\ell _ { 2 }$ norm of the input $\\mathbf { x }$ to that of the output $f _ { \\mathbf { w } } ( \\mathbf { x } )$ . As shown in Figure 2(a), without spectral normalization ( $\\beta = \\infty$ in (9)), the norm gain can be large. Additionally, because we are using cross-entropy loss, the weights (and therefore the norm gain) can grow arbitrarily high if we continue training as reported by Neyshabur et al. (2017b). As we decrease $\\beta$ , however, we produce more constrained networks, resulting in a decrease in norm gain. At $\\beta = 1$ , the gain of the network cannot be greater than 1, which is consistent with what we observe. Additionally, we provide a comparison of our method to that of Miyato et al. (2018) in Appendix A.1, empirically demonstrating that Miyato et al.’s method does not properly control the spectral norm of convolutional layers, resulting in worse generalization performance. ", + "bbox": [ + 173, + 131, + 825, + 270 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 2(b) shows that the $\\ell _ { 2 }$ norms of the gradients with respect to the training samples are nicely distributed after spectral normalization. Additionally, this figure suggests that the minimum gradient $\\ell _ { 2 }$ -norm assumption (the $\\kappa$ condition in Theorems 2 and 3) holds for spectrally-normalized networks. ", + "bbox": [ + 176, + 277, + 823, + 319 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The first column of Figure 3 shows that, as observed by Bartlett et al. (2017), AlexNet trained using ERM generates similar margin distributions for both random and true labels on CIFAR10 unless we normalize the margins appropriately. We see that even without further correction, ERM training with SN allows AlexNet to have distinguishable performance between the two datasets. This observation suggests that SN as a regularization scheme enforces the generalization error bounds shown for spectrally-normalized DNNs by Bartlett et al. (2017) and Neyshabur et al. (2017a). Additionally, the margin normalization factor (the capacity norm $\\Phi$ in Theorems 1-4) is much smaller for networks trained with SN. As demonstrated by the other columns in Figure 3, a smaller normalization factor results in larger normalized margin values and much tighter margin-based generalization bounds (a factor of $1 0 ^ { 2 }$ for ERM and a factor of $1 0 ^ { 5 }$ for FGM and PGM) (see Theorems 1-4). ", + "bbox": [ + 173, + 325, + 825, + 465 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(a) $\\ell _ { 2 }$ norm gain due to the network $f$ trained using ERM. ", + "bbox": [ + 205, + 643, + 467, + 669 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/2d10a59f5082d7d00e28922000204f0b02852c3f5bbd768c93322e043f96d3e2.jpg", + "image_caption": [ + "(b) Distributions of the $\\ell _ { 2 }$ norms of the gradients with respect to training samples. Training regularized with SN. " + ], + "image_footnote": [], + "bbox": [ + 534, + 484, + 781, + 625 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5e7d4faf907f521308a2e89fec6b112cbfa9f7b7dbda6618248bffd9196a9578.jpg", + "image_caption": [ + "Figure 2: Validation of SN implementation and distribution of the gradient norms using AlexNet trained on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 222, + 488, + 449, + 633 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 SPECTRAL NORMALIZATION IMPROVES GENERALIZATION AND ADVERSARIAL ROBUSTNESS ", + "bbox": [ + 176, + 737, + 751, + 763 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The phenomenon of overfitting random labels described by Zhang et al. (2016) can be observed even for adversarial training methods. Figure 4 shows how the FGM, PGM, or WRM adversarial training schemes only slightly delay the rate at which AlexNet fits random labels on CIFAR10, and therefore the generalization gap can be quite large without proper regularization. After introducing spectral normalization, however, we see that the network has a much harder time fitting both the random and true labels. With the proper amount of SN (chosen via cross validation), we can obtain networks that struggle to fit random labels while still obtaining the same or better test performance on true labels. ", + "bbox": [ + 174, + 776, + 825, + 875 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also observe that training schemes regularized with SN result in networks more robust to adversarial attacks. Figure 5 shows that even without adversarial training, AlexNet with SN becomes more robust to FGM, PGM, and WRM attacks. Adversarial training improves adversarial robustness more than SN by itself; however we see that we can further improve the robustness of the trained networks significantly by combining SN with adversarial training. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5abc549a3964c1047d4e227f84672de7c547b6a42071166acdc50926bfe7c32a.jpg", + "image_caption": [ + "Figure 3: Effect of SN on distributions of unnormalized (leftmost column) and normalized (other three columns) margins for AlexNet fit on CIFAR10. The normalization factor is described by the capacity norm $\\Phi$ reported in Theorems 1-4. " + ], + "image_footnote": [], + "bbox": [ + 240, + 101, + 758, + 258 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 345, + 821, + 375 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/06e6f1a8a28a2c03851b9b32128613974a14b5a3ef44a3dd2b6e434f5392405a.jpg", + "image_caption": [ + "Figure 4: Fitting random and true labels on CIFAR10 with AlexNet using adversarial training. " + ], + "image_footnote": [], + "bbox": [ + 253, + 387, + 743, + 517 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/c09a5b4382e86db70c17af5cad57f0b16c5aa7c6ba79c2f41bb4c07e45bcb080.jpg", + "image_caption": [ + "Figure 5: Robustness of AlexNet trained on CIFAR10 to various adversarial attacks. " + ], + "image_footnote": [], + "bbox": [ + 261, + 571, + 735, + 787 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 OTHER DATASETS AND ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 173, + 829, + 495, + 843 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We demonstrate the power of regularization via SN on several combinations of datasets, network architectures, and adversarial training schemes. The datasets we evaluate are CIFAR10, MNIST, and SVHN. We fit CIFAR10 using the AlexNet and Inception networks described by Zhang et al. (2016), 1-hidden-layer and 2-hidden-layer multi layer perceptrons (MLPs) with ELU activation and 512 hidden nodes in each layer, and the ResNet architecture (He et al. (2016)) provided in TensorFlow for fitting CIFAR10. We fit MNIST using the ELU network described by Sinha et al. (2018) and the 1-hidden-layer and 2-hidden-layer MLPs. Finally, we fit SVHN using the same AlexNet architecture we used to fit CIFAR10. Our implementations do not use any additional regularization schemes including weight decay, dropout (Srivastava et al., 2014), and batch normalization (Ioffe & Szegedy, 2015) as these approaches are not motivated by the theory developed in this work; however, we provide numerical experiments comparing the proposed approach with weight decay, dropout, and batch normalization in Appendix A.2. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Table 1 in the Appendix reports the pre and post-SN test accuracies for all 42 combinations evaluated. Figure 1 in the Introduction and Figures 7-9 in the Appendix show examples of training and validation curves on some of these combinations. We see that the validation curve generally improves after regularization with SN, and the observed improvements in validation accuracy are confirmed by the test accuracies reported in Table 1. Figure 6 visually summarizes Table 1, showing how SN can often significantly improve the test accuracy (and therefore decrease the generalization gap) for several of the combinations. We also provide Table 2 in the Appendix which shows the proportional increase in training time after introducing SN with our TensorFlow implementation. ", + "bbox": [ + 174, + 208, + 825, + 319 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/369c35ed7e831dc332f3c540e68035bbfacc5e5bdb1f8405ad7d01ddfec1fa0d.jpg", + "image_caption": [ + "Figure 6: Test accuracy improvement after SN for various datasets and network architectures. " + ], + "image_footnote": [], + "bbox": [ + 238, + 334, + 758, + 443 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 174, + 488, + 354, + 505 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Providing theoretical guarantees for adversarial robustness of various classifiers has been studied in multiple works. Wang et al. (2017) targets analyzing the adversarial robustness of the nearest neighbor approach. Gilmer et al. (2018) studies the effect of the complexity of the data-generating manifold on the final adversarial robustness for a specific trained model. Fawzi et al. (2018) proves lower-bounds for the complexity of robust learning in adversarial settings, targeting the population distribution of data. Xu et al. (2009) shows that the regularized support vector machine (SVM) can be interpreted via robust optimization. Fawzi et al. (2016) analyzes the robustness of a fixed classifier to random and adversarial perturbations of the input data. While all of these works seek to understand the robustness properties of different classification function classes, unlike our work they do not focus on the generalization aspects of learning over DNNs under adversarial attacks. ", + "bbox": [ + 174, + 520, + 825, + 660 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Concerning the generalization aspect of adversarial training, Sinha et al. (2018) provides optimization and generalization guarantees for WRM under the assumptions discussed after Theorem 4. However, their generalization guarantee only applies to the Wasserstein cost function, which is different from the 0-1 or margin loss and does not explicitly suggest a regularization scheme. In a recent related work, Schmidt et al. (2018) numerically shows the wide generalization gap in PGM adversarial training and theoretically establishes lower-bounds on the sample complexity of linear classifiers in Gaussian settings. While our work does not provide sample complexity lower-bounds, we study the broader function class of DNNs where we provide upper-bounds on adversarial generalization error and suggest an explicit regularization scheme for adversarial training over DNNs. ", + "bbox": [ + 173, + 666, + 825, + 791 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Generalization in deep learning has been a topic of great interest in machine learning (Zhang et al., 2016). In addition to margin-based bounds (Bartlett et al., 2017; Neyshabur et al., 2017a), various other tools including VC dimension (Anthony & Bartlett, 2009), norm-based capacity scores (Bartlett & Mendelson, 2002; Neyshabur et al., 2015), and flatness of local minima (Keskar et al., 2016; Neyshabur et al., 2017b) have been used to analyze generalization properties of DNNs. Recently, Arora et al. (2018) introduced a compression approach to further improve the margin-based bounds presented by Bartlett et al. (2017); Neyshabur et al. (2017a). The PAC-Bayes bound has also been considered and computed by Dziugaite & Roy (2017), resulting in non-vacuous bounds for the MNIST dataset. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 356, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We are grateful for support under the National Science Foundation grant under CCF-1563098, and the Center for Science of Information (CSoI), an NSF Science and Technology Center under grant agreement CCF-0939370. ", + "bbox": [ + 174, + 133, + 825, + 175 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 196, + 287, + 212 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. \nMartin Anthony and Peter L Bartlett. 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", + "bbox": [ + 176, + 174, + 823, + 202 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Appendices ", + "text_level": 1, + "bbox": [ + 174, + 98, + 341, + 127 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A FURTHER EXPERIMENTAL RESULTS ", + "bbox": [ + 174, + 150, + 503, + 166 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/5c56193725552d961043513d727239883cf551d4eb5a0daefd29c624f78bd3e7.jpg", + "table_caption": [ + "Table 1: Train and test accuracies before and after spectral normalization for various datasets, network architectures, and training schemes. The amount of spectral normalization was selected from 4-6 values of $\\beta$ via cross validation on 500 samples. For each row, the greater test accuracy is bolded (both are bolded in the event of a tie). $\\ell _ { \\infty }$ adversarial training was performed with magnitude 0.1. " + ], + "table_footnote": [ + "\\* $\\overline { { \\beta = \\infty } }$ (i.e. no spectral normalization) achieved the highest validation accuracy. " + ], + "table_body": "
DatasetArchitectureTrainingTrain accTest accTrain acc (SN)Test acc (SN)
CIFAR10AlexNetERM1.000.791.000.79
CIFAR10AlexNetFGM l20.980.540.930.63
CIFAR10AlexNetFGM lo1.000.510.670.56
CIFAR10AlexNetPGM l20.990.500.920.62
CIFAR10AlexNetPGMlo0.990.440.860.54
CIFAR10AlexNetWRM1.000.610.760.65
CIFAR10ELU-AlexNetERM1.000.791.000.79
CIFAR10ELU-AlexNetFGM l20.970.520.680.60
CIFAR10ELU-AlexNetPGMl20.980.530.880.61
CIFAR10ELU-AlexNetWRM1.000.601.000.60
CIFAR10InceptionERM1.000.851.000.86
CIFAR10InceptionPGM l20.990.531.000.58
CIFAR10InceptionPGM lo0.980.480.620.56
CIFAR10InceptionWRM1.000.661.000.67
CIFAR101-layer MLPERM0.980.490.680.53
CIFAR101-layer MLPFGM l20.600.360.600.46
CIFAR101-layer MLPPGM l20.570.360.550.46
CIFAR101-layer MLPWRM0.600.410.620.50
CIFAR102-layer MLPERM0.990.510.790.56
CIFAR102-layer MLPFGM l20.570.360.660.49
CIFAR102-layer MLPPGM l20.930.350.660.48
CIFAR102-layer MLPWRM0.870.350.730.52
CIFAR10ResNetERM1.000.801.000.83
CIFAR10ResNetPGM l20.990.491.000.55
CIFAR10ResNetPGM lo0.980.440.720.53
CIFAR10ResNetWRM1.000.631.000.66
MNISTELU-NetERM1.000.991.000.99*
MNISTELU-NetFGM l20.980.971.000.97
MNISTELU-NetPGM l20.990.971.000.97
MNISTELU-NetWRM0.950.920.950.93
MNIST1-layer MLPERM1.000.981.000.98*
MNIST1-layer MLPFGM l20.880.881.000.96
MNIST1-layer MLPPGM l21.000.961.000.96
MNIST1-layer MLPWRM0.920.880.920.88
MNIST2-layer MLPERM1.000.981.000.98
MNIST2-layer MLPFGM l20.970.911.000.96
MNIST2-layer MLPPGM l21.000.961.000.97
MNIST2-layer MLPWRM0.970.880.980.90
SVHNAlexNetERM1.000.931.000.93*
SVHNAlexNetFGM l20.970.760.950.83
SVHNAlexNetPGM l21.000.780.850.81
SVHNAlexNetWRM1.000.830.870.84
", + "bbox": [ + 181, + 246, + 823, + 864 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/de5d326fb415fc4e25fc11066dd65e760acdc2c84caa2df43041d267608adec4.jpg", + "table_caption": [ + "Table 2: Runtime increase after introducing spectral normalization for various datasets, network architectures, and training schemes. These ratios were obtained by running the experiments on one NVIDIA Titan $\\mathrm { X p }$ GPU for 40 epochs. " + ], + "table_footnote": [], + "table_body": "
DatasetArchitectureTrainingno SN runtime SN runtimeratio
CIFAR10AlexNetERM229 s283 s1.24
CIFAR10AlexNetFGM l2407 s463 s1.14
CIFAR10AlexNetFGM loo408 s465 s1.14
CIFAR10AlexNetPGM l22917 s3077 s1.05
CIFAR10AlexNetPGM lo2896 s3048 s1.05
CIFAR10AlexNetWRM3076 s3151 s1.02
CIFAR10ELU-AlexNetERM231 s283 s1.23
CIFAR10ELU-AlexNetFGM l2410 s466 s1.14
CIFAR10ELU-AlexNetPGM l22939 s3093 s1.05
CIFAR10ELU-AlexNetWRM3094 s3150 s1.02
CIFAR10InceptionERM632 s734 s1.16
CIFAR10InceptionPGM l29994 s6082 s0.61
CIFAR10InceptionPGM lo9948 s6063 s0.61
CIFAR10InceptionWRM10247 s6356 s0.62
CIFAR101-layer MLPERM22 s31 s1.42
CIFAR101-layer MLPFGM l225 s35s1.43
CIFAR101-layer MLPPGM l279 s93 s1.18
CIFAR101-layer MLPWRM73 s86 s1.18
CIFAR102-layer MLPERM23 s37 s1.59
CIFAR102-layer MLPFGM l227 s41 s1.51
CIFAR102-layer MLPPGMl291 s108 s1.19
CIFAR102-layer MLPWRM85 s103 s1.21
CIFAR10ResNetERM315 s547 s1.73
CIFAR10ResNetPGM l22994 s3300 s1.10
CIFAR10ResNetPGM lo2980 s3300 s1.11
CIFAR10ResNetWRM3187 s3457 s1.08
MNISTELU-NetERM55 s97s1.76
MNISTELU-NetFGM l291s136 s1.49
MNISTELU-NetPGM l2614 s676 s1.10
MNISTELU-NetWRM635 s670 s1.06
MNIST1-layer MLPERM15 s24 s1.60
MNIST1-layer MLPFGM l217 s27s1.57
MNIST1-layer MLPPGM l257s71 s1.24
MNIST1-layer MLPWRM51 s63 s1.24
MNIST2-layer MLPERM17 s31 s1.84
MNIST2-layer MLPFGM l220 s35 s1.77
MNIST2-layer MLPPGM l267 s89 s1.32
MNIST2-layer MLPWRM62 s81 s1.30
SVHNAlexNetERM334 s412 s1.23
SVHNAlexNetFGM l2596 s676 s1.13
SVHNAlexNetPGM l24270 s4495 s1.05
SVHNAlexNetWRM4501 s4572 s1.02
", + "bbox": [ + 236, + 146, + 761, + 773 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/aad3b3b45c18f07506ccaffc3664feb697a9ac5ac91974f7442ee3d4bc8902d0.jpg", + "image_caption": [ + "Figure 7: Adversarial training performance with and without spectral normalization for AlexNet fit on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 184, + 102, + 812, + 268 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/db9080bcbeb5283fc0d6b1c8dc283eaf945003257db19d46a41039c072336cef.jpg", + "image_caption": [ + "Figure 8: Adversarial training performance with and without spectral normalization for Inception and ResNet fit on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 183, + 337, + 813, + 505 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/36ce01c99f6af2bf5a31795012fb7704c427a47cebb8a0901fe0558ed4f8a1ec.jpg", + "image_caption": [ + "Figure 9: Adversarial training performance with and without spectral normalization for AlexNet with ELU activation functions fit on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 183, + 573, + 813, + 739 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.1 COMPARISON OF PROPOSED METHOD TO MIYATO ET AL. (2018)’S METHOD ", + "text_level": 1, + "bbox": [ + 174, + 814, + 740, + 829 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For the optimal $\\beta$ chosen when fitting AlexNet to CIFAR10 with PGM, we repeat the experiment using the spectral normalization approach suggested by Miyato et al. (2018). This approach performs spectral normalization on convolutional layers by scaling the convolution kernel by the spectral norm of the kernel rather than the spectral norm of the overall convolution operation. Because it does not account for how the kernel can amplify perturbations in a single pixel multiple times (see Section 4), it does not properly control the spectral norm. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Figure 10, we see that for the optimal $\\beta$ reported in the main text, using Miyato et al. (2018)’s SN method results in worse generalization performance. This is because although we specified that $\\beta \\ : = \\ : 1 . 6$ , the actual $\\beta$ obtained using Miyato et al. (2018)’s method can be much greater for convolutional layers, resulting in overfitting (hence the training curve quickly approaches 1.0 accuracy). The AlexNet architecture used has two convolutional layers. For the proposed method, the final spectral norms of the convolutional layers were both 1.60; for Miyato et al. (2018)’s method, the final spectral norms of the convolutional layers were 7.72 and 7.45 despite the corresponding convolution kernels having spectral norms of 1.60. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Our proposed method is less computationally efficient in comparison to Miyato et al. (2018)’s approach because each power iteration step requires a convolution operation rather than a division operation. As shown in Table 3, the proposed approach is not significantly less efficient with our TensorFlow implementation. ", + "bbox": [ + 174, + 222, + 825, + 279 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/ecf0f675b93c30caffac318e0a783a77d6f0fb6a507e2dc92bf11229ad28d266.jpg", + "image_caption": [ + "Figure 10: Adversarial training performance with proposed SN versus Miyato et al. (2018)’s SN for AlexNet fit on CIFAR10 using PGM. The final train and validation accuracies for the proposed method are 0.92 and 0.60. The final train and validation accuracies for Miyato et al. (2018)’s are 1.00 and 0.55. " + ], + "image_footnote": [], + "bbox": [ + 228, + 294, + 769, + 463 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/1164eb8c45c370478ddf6d65d41f48ddcc98ff94fc8279d8dff399fea1bb8da9.jpg", + "table_caption": [ + "Table 3: Runtime increase of the proposed spectral normalization approach compared to Miyato et al. (2018)’s approach for CIFAR10 and various network architectures and training schemes. These ratios were obtained by running the experiments on one NVIDIA Titan $\\mathrm { X p }$ GPU for 40 epochs. " + ], + "table_footnote": [], + "table_body": "
DatasetArchitectureTrainingproposed SN runtime Miyato SN runtime
CIFAR10AlexNetERM1.11
CIFAR10AlexNetFGM l21.06
CIFAR10AlexNetFGMloo1.11
CIFAR10AlexNetPGM l21.01
CIFAR10AlexNetPGM lo1.11
CIFAR10AlexNetWRM1.02
CIFAR10InceptionERM0.98
CIFAR10InceptionPGM l21.04
CIFAR10InceptionPGM loo1.06
CIFAR10InceptionWRM1.03
", + "bbox": [ + 305, + 616, + 692, + 789 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/2aa56c0998b21e60c302c7532ab72c986434aecdb5230fcbe7694c08766a94f1.jpg", + "image_caption": [ + "Figure 11: Adversarial training performance with proposed SN versus batch normalization, weight decay, and dropout for AlexNet fit on CIFAR10 using PGM. The dropout rate was 0.8, and the amount of weight decay was 5e-4 for all weights. The leftmost plot is from Figure 1 and compares final performance of no regularization (train accuracy 1.00, validation accuracy 0.48) to that of SN (train accuracy 0.92, validation accuracy 0.60). The final train and validation accuracies for batch normalization are 1.00 and 0.54; the final train and validation accuracies for weight decay are 0.84 and 0.55; and the final train and validation accuracies for dropout are 0.99 and 0.52. " + ], + "image_footnote": [], + "bbox": [ + 183, + 150, + 813, + 315 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B SPECTRAL NORMALIZATION OF FULLY-CONNECTED LAYERS ", + "text_level": 1, + "bbox": [ + 173, + 464, + 707, + 481 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For fully-connected layers, we approximate the spectral norm of a given matrix $W$ using the approach described by Miyato et al. (2018): the power iteration method. For each $W$ , we randomly initialize a vector $\\tilde { \\mathbf { u } }$ and approximate both the left and right singular vectors by iterating the update rules ", + "bbox": [ + 174, + 496, + 826, + 537 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/3bdfe0bd367ad47a4229eb9fcec7567c84cee3aa594a03aa0337345dc664d933.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathbf { v } } W \\tilde { \\mathbf { u } } / \\| W \\tilde { \\mathbf { u } } \\| _ { 2 } } \\\\ & { \\tilde { \\mathbf { u } } W ^ { T } \\tilde { \\mathbf { v } } / \\| W ^ { T } \\tilde { \\mathbf { v } } \\| _ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 421, + 544, + 575, + 585 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The final singular value can be approximated with $\\sigma ( W ) \\approx \\tilde { \\mathbf { v } } ^ { T } W \\tilde { \\mathbf { u } }$ . Like Miyato et al., we exploit the fact that SGD only makes small updates to $W$ from training step to training step, reusing the same $\\tilde { \\mathbf { u } }$ and running only one iteration per step. Unlike Miyato et al., rather than enforcing $\\sigma ( W ) = \\beta$ , we instead enforce the looser constraint $\\sigma ( W ) \\le \\beta$ as described by Gouk et al. (2018): ", + "bbox": [ + 173, + 598, + 825, + 655 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/fa45b9adad6d632c2f957c14da0851003d412fba397a6bd2ce7a27d051bb6e26.jpg", + "text": "$$\nW _ { \\mathrm { S N } } = W / \\operatorname* { m a x } ( 1 , \\sigma ( W ) / \\beta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 395, + 661, + 599, + 679 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "which we observe to result in faster training in practice for supervised learning tasks. ", + "bbox": [ + 174, + 693, + 728, + 709 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 728, + 276, + 744 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.1 PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 174, + 758, + 374, + 775 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "First let us quote the following two lemmas from (Neyshabur et al., 2017a). ", + "bbox": [ + 174, + 785, + 668, + 801 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 1 (Neyshabur et al. (2017a)). Consider $\\mathcal { F } _ { n n } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathcal { W } \\}$ as the class of neural nets parameterized by w where each $f _ { \\mathbf { w } }$ maps input $\\mathbf { x } \\in \\mathcal { X }$ to $\\mathbb { R } ^ { m }$ . Let $Q$ be a distribution on parameter vector chosen independently from the n training samples. Then, for each $\\eta > 0$ with probability at least $1 - \\eta$ for any w and any random perturbation u satisfying $\\operatorname* { P r } _ { \\mathbf { u } } \\left( \\operatorname* { m a x } _ { \\mathbf { x } \\in \\mathcal { X } } \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) - \\right.$ $\\begin{array} { r } { f _ { \\mathbf { w } } ( \\mathbf { x } ) \\lVert _ { \\infty } \\leq \\frac { \\gamma } { 4 } ) \\geq \\frac { 1 } { 2 } } \\end{array}$ we have ", + "bbox": [ + 173, + 804, + 825, + 880 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2032f981072cad7dea010ab4eb3a7554675ba428a604ffc9b987dc40106ff5a7.jpg", + "text": "$$\nL _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + 4 \\sqrt { \\frac { K L ( P _ { \\mathbf { w } + \\mathbf { u } } \\| Q ) + \\log \\frac { 6 n } { \\eta } } { n - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 886, + 669, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 2 (Neyshabur et al. (2017a)). Consider a $d$ -layer neural net $f _ { \\mathbf { w } }$ with 1-Lipschitz activation function $\\sigma$ where $\\sigma ( 0 ) = 0$ . Then for any norm-bounded input $\\| \\mathbf { x } \\| _ { 2 } \\leq B$ and weight perturbation $\\begin{array} { r } { \\mathbf { \\dot { u } } : \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ , we have the following perturbation bound: ", + "bbox": [ + 171, + 102, + 825, + 147 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/87b4d53b6814707767de659c98397b67d7e33d71416fb1722cd8b32231162e97.jpg", + "text": "$$\n\\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) - f _ { \\mathbf { w } } ( \\mathbf { x } ) \\| _ { 2 } \\leq e B \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 151, + 686, + 194 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To prove Theorem 1, consider $f _ { \\widetilde { \\mathbf { w } } }$ with weights $\\widetilde { \\mathbf { w } }$ . Since $( 1 + \\textstyle { \\frac { 1 } { d } } ) ^ { d } \\leq e$ and $\\begin{array} { r } { \\frac { 1 } { e } \\leq ( 1 - \\frac { 1 } { d } ) ^ { d - 1 } } \\end{array}$ , for any weight vector w such that $\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ for every $i$ we have: ", + "bbox": [ + 173, + 204, + 825, + 242 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/fad54310d0374c9f4833c0c3598de2616f71377b0620a9acc8b76c2f36ce6bb1.jpg", + "text": "$$\n( 1 / e ) ^ { \\frac { d } { d - 1 } } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq e \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 244, + 671, + 287 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We apply Lemma 1, choosing $Q$ to be a zero-mean multivariate Gaussian distribution with diagonal covariance matrix, where each entry of the $i$ th layer $\\mathbf { U } _ { i }$ has standard deviation $\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}$ kWfik2 ξ with $\\xi$ chosen later in the proof. Note that $\\beta _ { \\mathbf { w } }$ edefined earlier in the theorem is the geometric average of spectral norms across all layers. Then for the ith layer’s random perturbation vector $\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )$ , we get the following bound from (Tropp, 2012) with $h$ representing the width of the ith hidden layer: ", + "bbox": [ + 173, + 290, + 826, + 368 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/506da5167bfdd899767c975aa46e9bfba65fc3a47e9f7d01b83bac8d4dac1384.jpg", + "text": "$$\n\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 371, + 635, + 409 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We now use a union bound over all layers for a maximum union probability of $1 / 2$ , which implies the normalized w kUik2 for each layer is upper-bounded by ξp2h log(4hd). Then for any w satisfying $\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ for all $i$ ’s ", + "bbox": [ + 173, + 411, + 825, + 468 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/be569788850b7e55151611a430c41a1d3e886b24248d085fab4b1ade54c03fa9.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\operatorname* { m a x } _ { \\| \\mathbf x \\| _ { 2 } \\leq B } \\| f _ { \\mathbf w + \\mathbf u } ( \\mathbf x ) - f _ { \\mathbf w } ( \\mathbf x ) \\| _ { 2 } \\leq e B \\left( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } } & { } \\\\ { \\displaystyle } & { \\overset { ( a ) } \\leq e ^ { 2 } B \\left( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } } \\\\ & { = e ^ { 2 } B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } } \\\\ & { \\leq e ^ { 2 } d B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\xi \\sqrt { 2 h \\log ( 4 h d ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 281, + 472, + 715, + 623 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here (a) holds, since $\\begin{array} { r } { \\frac { 1 } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq \\frac { e } { \\| \\widetilde { \\mathbf { W } } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ is true for each $j$ . Hence we choose $\\begin{array} { r } { \\xi = \\frac { \\gamma } { 3 0 d B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\sqrt { h \\log ( 4 h d ) } } } \\end{array}$ for which the perturbation vector satisfies the assumptions of Lemma 2. eThen, we bound the KL-divergence term in Lemma 1 as ", + "bbox": [ + 173, + 627, + 826, + 686 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/410d4e9bc8a295499f4d3f96be2a810cec09e76be2ebd041b970549d6a3a3982.jpg", + "text": "$$\n\\begin{array} { r l r } { { K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } } \\\\ & { } & { = \\frac { 3 0 ^ { 2 } d ^ { 2 } B ^ { 2 } \\beta _ { \\mathbf { w } } ^ { 2 } \\| \\mathbf { b } \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\widehat { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\overset { ( b ) } { \\leq } \\frac { 3 0 ^ { 2 } e ^ { 2 } d ^ { 2 } B ^ { 2 } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } h \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { = \\mathcal { O } \\Big ( d ^ { 2 } B ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } \\Big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 689, + 727, + 866 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Note that (b) holds, because we assume $\\begin{array} { r l r } { | \\| { \\bf W } _ { i } \\| _ { 2 } ~ - ~ \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } \\Big | } & { { } \\le ~ } & { \\frac { 1 } { d } \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } } \\end{array}$ implying 1kW k Qdi=1 kWfik2 ≤ (1 − 1d )−(d−1) 1kWjk $\\begin{array} { r } { \\frac { 1 } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq ( 1 - \\frac { 1 } { d } ) ^ { - ( d - 1 ) } \\frac { 1 } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq \\frac { e } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ for each $j$ . Therefore, Lemma 1 implies with probability $1 - \\eta$ we have the following bound hold for any w satisfying ", + "bbox": [ + 173, + 869, + 826, + 925 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ for all $i$ ’s, ", + "bbox": [ + 174, + 99, + 452, + 121 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e0cae9a451c5154192abf400017cb2d5e33ce8c64cf91b80fa87faf93c5c9802.jpg", + "text": "$$\nL _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { B ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) + \\log \\frac { n } { \\widetilde { \\eta } } } { \\gamma ^ { 2 } n } } \\bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 128, + 715, + 171 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Then, we can give an upper-bound over all the functions in $\\mathcal { F } _ { \\mathrm { n n } }$ by finding the covering number of the set of We o $\\widetilde { \\mathbf { w } }$ ’s where for each feasible w need to form the bound for $\\begin{array} { r } { \\big ( \\frac { \\gamma } { 2 B } \\big ) ^ { 1 / d } \\leq \\beta _ { \\mathbf { w } } \\leq \\big ( \\frac { \\gamma \\sqrt { n } } { 2 B } \\big ) ^ { 1 / d } } \\end{array}$ ition satisfied for at least one of we ’s.which can be covered using a cover of size $d n ^ { 1 / 2 d }$ as discussed in (Neyshabur et al., 2017a). Then, from the theorem’s assumption we know each $\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }$ will be in the interval $[ \\frac { 1 } { M } \\beta _ { \\mathbf { w } } , M \\beta _ { \\mathbf { w } } ]$ which we want to cover such that for any $\\beta$ in the interval there exists a $\\widetilde { \\beta }$ satisfying $| \\beta - \\widetilde { \\beta } | \\leq \\widetilde { \\beta } / d$ . For this purpose we can use a cover of size $2 \\log _ { 1 + 1 / d } M \\leq 2 ( d + 1 ) \\log M ,$ 1 which combined for all $i$ ’s gives a cover with size ${ \\mathcal { O } } ( ( d \\log M ) ^ { d } )$ whose logarithm is growing as $d \\log ( d \\log M )$ . This together with (15) completes the proof. ", + "bbox": [ + 173, + 178, + 826, + 300 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2 PROOF OF THEOREM 2 ", + "text_level": 1, + "bbox": [ + 174, + 318, + 375, + 332 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We start by proving the following lemmas providing perturbation bound for FGM attacks. ", + "bbox": [ + 169, + 343, + 761, + 359 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 3. Consider a $d$ -layer neural net $f _ { \\mathbf { w } }$ with 1-Lipschitz and 1-smooth (1-Lipschitz derivative) activation $\\sigma$ where $\\sigma ( 0 ) = 0$ . Let training loss $\\ell : ( \\mathbb { R } ^ { m } , \\mathcal { Y } ) \\to \\mathbb { R }$ also be 1-Lipschitz and 1-smooth for any fixed label $y \\in \\mathcal { D }$ . Then, for any input $\\mathbf { x }$ , label $y$ , and perturbation vector u satisfying $\\begin{array} { r } { \\forall i : \\| \\dot { \\mathbf { U } } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ we have ", + "bbox": [ + 173, + 364, + 825, + 421 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/1f2fb2faad31d4e398abc0624ebb831984923c939c7a9872c789a79dd9123fbc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left\\| \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) , y ) - \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\right\\| _ { 2 } } \\\\ & { \\leq e ^ { 2 } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\biggl [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\| \\mathbf { x } \\| _ { 2 } ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\biggr ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 429, + 723, + 497 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. Since for a fixed $y \\ell$ satisfies the same Lipschitzness and smoothness properties as $\\sigma$ , then $\\| \\nabla _ { \\mathbf { z } } \\ell ( \\mathbf { z } , y ) \\| _ { 2 } \\leq 1$ and applying the chain rule implies: ", + "bbox": [ + 171, + 515, + 823, + 545 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/7f3ade952da620e1f0ddfd6b6cef205bc1187f12c0e1f0f6bcb9ca2671ec19ff.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\mathbf x } \\ell ( \\mathbf r _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - \\nabla _ { \\mathbf x } \\ell ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { = \\| ( \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| ( \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\quad + \\| ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| _ { 2 } + ( \\prod _ { i } \\| \\mathbf x _ { i } \\| _ { 2 } ) \\| ( \\| \\nabla \\ell ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| _ { 2 } + ( \\prod _ { i = 1 } ^ { d } \\| \\nabla _ { \\mathbf x } ) \\| ( \\nabla \\ell ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 551, + 797, + 770 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The above result is a conclusion of Lemma 2 and the lemma’s assumptions implying $\\left\\| \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\right\\| _ { 2 } \\leq$ $\\begin{array} { r } { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ for every x. Now, we define $\\Delta _ { k } = \\left. \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } + \\mathbf { u } } ^ { ( k ) } ( \\mathbf { x } ) - \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\right. _ { 2 }$ where $f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) : =$ $\\mathbf { W } _ { k } \\sigma ( \\mathbf { W } _ { k - 1 } \\cdot \\cdot \\cdot \\sigma ( \\mathbf { W } _ { 1 } \\mathbf { x } ) ) \\cdot \\cdot \\cdot )$ denotes the DNN’s output at layer $k$ . With (17) in mind, we complete this lemma’s proof by showing the following inequality via induction: ", + "bbox": [ + 173, + 776, + 825, + 842 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/c9200bcc6c904b97cfecd1c24a74e527c09706625c3d87ff8b92938e04023675.jpg", + "text": "$$\n\\Delta _ { k } \\leq e ( 1 + \\frac { 1 } { d } ) ^ { k } \\big ( \\prod _ { i = 1 } ^ { k } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { k } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\| \\mathbf { x } \\| _ { 2 } \\big ( \\prod _ { j = 1 } ^ { i - 1 } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 233, + 849, + 764, + 893 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1Note that log 1x ≥ 1 − x implying log 11− 1d+1 $\\textstyle { \\frac { 1 } { 1 - { \\frac { 1 } { d + 1 } } } } \\geq { \\frac { 1 } { d + 1 } }$ and hence $( \\log ( 1 + 1 / d ) ) ^ { - 1 } \\leq d + 1$ . ", + "bbox": [ + 187, + 905, + 751, + 928 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The above equation will prove the lemma because for $k \\leq d$ we have $\\begin{array} { r } { ( 1 + \\frac { 1 } { d } ) ^ { k } \\le ( 1 + \\frac { 1 } { d } ) ^ { d } \\le e } \\end{array}$ . For $k = 0$ , $\\Delta _ { 0 } = 0$ since $f _ { \\mathbf { w } } ^ { ( 0 ) } ( \\mathbf { x } ) = \\mathbf { x }$ and does not change with w. Given that (18) holds for $k$ we have ", + "bbox": [ + 173, + 102, + 828, + 137 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9411eba59e2e6b7633b2264e55f50def1c823d2ed86896c0fc2f367943109c8b.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbf { \\Phi } } & { = | \\nabla _ { x } \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} } \\\\ & { - \\nabla _ { x } \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} } \\\\ & - \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 142, + 861, + 516 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore, combining (17) and (18) the lemma’s proof is complete ", + "bbox": [ + 173, + 516, + 611, + 532 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Before presenting the perturbation bound for FGM attacks, we first prove the following simple lemma. ", + "bbox": [ + 173, + 546, + 826, + 575 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma 4. Consider vectors $\\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 }$ and norm function $\\| \\cdot \\|$ $| . \\ I f \\operatorname* { m a x } \\{ \\| \\mathbf { z } _ { 1 } \\| , \\| \\mathbf { z } _ { 2 } \\| \\} \\geq \\kappa ,$ , then ", + "bbox": [ + 168, + 578, + 772, + 595 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/bb6d09f59ac9d4c8021178bba3a389155f18244f4246218e1a3a14c165ca0ccd.jpg", + "text": "$$\n\\Bigl \\| \\frac { \\epsilon } { \\| \\mathbf { z } _ { 1 } \\| } \\mathbf { z } _ { 1 } - \\frac { \\epsilon } { \\| \\mathbf { z } _ { 2 } \\| } \\mathbf { z } _ { 2 } \\Bigr \\| \\leq \\frac { 2 \\epsilon } { \\kappa } \\| \\mathbf { z } _ { 1 } - \\mathbf { z } _ { 2 } \\| .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 598, + 627, + 632 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. Without loss of generality suppose $\\| \\mathbf { z } _ { 2 } \\| \\leq \\| \\mathbf { z } _ { 1 } \\|$ and therefore $\\kappa \\leq \\| \\mathbf { z } _ { 1 } \\|$ . Then, ", + "bbox": [ + 173, + 643, + 745, + 661 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/0a2a79269e79ddcada050effefbf9532cfe3eff5f79318d56794164f972219e0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\| \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } { \\bf z } _ { 1 } - \\frac { \\epsilon } { \\| { \\bf z } _ { 2 } \\| } { \\bf z } _ { 2 } \\| = \\epsilon \\big \\| \\frac { 1 } { \\| { \\bf z } _ { 1 } \\| } \\big ( { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\big ) - \\frac { \\| { \\bf z } _ { 1 } \\| - \\| { \\bf z } _ { 2 } \\| } { \\| { \\bf z } _ { 1 } \\| } \\frac { 1 } { \\| { \\bf z } _ { 2 } \\| } \\big \\| } \\\\ & { \\displaystyle \\quad \\quad \\quad \\leq \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| + \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\big | \\| { \\bf z } _ { 1 } \\| - \\| { \\bf z } _ { 2 } \\| \\big | } \\\\ & { \\displaystyle \\quad \\quad \\leq \\frac { 2 \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| } \\\\ & { \\displaystyle \\quad \\quad \\quad \\leq \\frac { 2 \\epsilon } { \\kappa } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 666, + 727, + 796 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma 5. Consider a $d$ -layer neural network function $f _ { \\mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\\sigma$ where $\\sigma ( 0 ) = 0$ . Consider FGM attacks with noise power \u000f according to Euclidean norm $| | \\cdot | | _ { 2 }$ . Suppose $\\kappa \\leq \\| \\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y ) \\| _ { 2 }$ holds over the $\\epsilon$ -ball around the support set $\\mathcal { X }$ . Then, for any norm-bounded perturbation vector u such that $\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } . \\forall i } \\end{array}$ , we have ", + "bbox": [ + 173, + 819, + 826, + 878 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/2c6501a3768cf0105073e9abf5fab638b04910c551d9d61ab88870d4f29f2cdf.jpg", + "text": "$$\n\\big \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) \\big \\| _ { 2 } \\leq \\frac { 2 e ^ { 2 } \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\| \\mathbf { x } \\| _ { 2 } \\big ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 883, + 812, + 928 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. The FGM attack according to Euclidean norm is simply the DNN loss’s gradient normalized to have $\\epsilon$ -Euclidean norm. The lemma is hence a direct result of combining Lemmas 3 and 4. ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To prove Theorem 2, we apply Lemma 1 together with the result in Lemma 5. Similar to the proof for Theorem 1, given weights $\\widetilde { \\mathbf { w } }$ we consider a zero-mean multivariate Gaussian perturbation vector u with diagonal covariance matrix where each element in the ith layer $\\mathbf { u } _ { i }$ varies with the scaled standard deviation kWfik2βw ξ with ξ properly chosen later in the proof. Consider weights w for which ", + "bbox": [ + 173, + 146, + 825, + 212 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/c5aa4ad25b6dc75fe11d604f2492413fa31778e1adb0aa25a30ce979050786b2.jpg", + "text": "$$\n\\forall i : \\ | \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\leq \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 215, + 624, + 247 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since $\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )$ , (Tropp, 2012) shows the following bound holds ", + "bbox": [ + 173, + 251, + 642, + 267 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/c8b4839acc43c95ac2b08a327946f30913ea6f35e2525c3fe925895d661125f2.jpg", + "text": "$$\n\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 271, + 635, + 309 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Then we apply a union bound over all layers for a maximum union probability of $1 / 2$ implying the normalized kUik2 for each layer is upper-bounded by ξp2h log(4hd). Now, if the assumptions of Lemma 5 hold for perturbation vector $\\mathbf { u }$ given the choice of $\\xi$ , for the FGM attack with noise power $\\epsilon$ according to Euclidean norm $\\| \\cdot \\| _ { 2 }$ we have ", + "bbox": [ + 173, + 313, + 825, + 378 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/1a4a44fe8973396015868da6f52f13006aea8e2fce4d2c351d1cae7f6e7587c6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| f _ { w + \\mathbf { n } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { w + \\mathbf { n } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + \\| f _ { w } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { w } ( \\mathbf x + \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w + \\mathbf { n } } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + ( \\underbrace { d } _ { \\mathbf { w - \\mathbf { n } } } ( \\mathbf x ) \\| _ { 2 } ) \\| \\delta _ { \\mathbf { w + \\mathbf { n } } } ^ { \\mathrm { t e m } } ( \\mathbf x ) - \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) \\| _ { 2 } } \\\\ & { \\leq e ( B + e ) \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } + 2 e ^ { 2 } \\frac { d } { \\kappa } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } + B ( \\frac { 1 } { \\sqrt { 1 + 1 } } W _ { i } ) \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { j } \\| } \\\\ & \\leq e ^ { 2 } ( B + e ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 168, + 383, + 816, + 608 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Hence we choose ", + "bbox": [ + 173, + 611, + 290, + 626 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b5ed57a060c53f12a5519d7a229e366729b908cfa2475c2b6a9b4133f7b017eb.jpg", + "text": "$$\n\\begin{array} { r } { \\dot { \\mathbf { \\xi } } = \\frac { \\gamma } { 8 e ^ { 5 } d ( B + \\epsilon ) \\sqrt { 2 h \\log ( 4 h d ) } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big ( 1 + \\frac { \\epsilon } { \\kappa } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ( 1 / B + \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } ) \\big ) } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 627, + 826, + 662 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "for which the assumptions of Lemmas 1 and 5 hold. Assuming $B \\geq 1$ , similar to Theorem 1’s proof we can show for any w such that $\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ we have ", + "bbox": [ + 173, + 672, + 826, + 708 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/71717b918d9f1b7e70060d175f91e8d1d446a6f3be849e9aa70815c660ff5e80.jpg", + "text": "$$\n\\begin{array} { l l } { \\displaystyle K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } \\\\ { \\displaystyle = \\mathcal { O } \\bigg ( d ^ { 2 } ( B + \\epsilon ) ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } \\{ 1 + \\frac { \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } \\big ) \\} ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ { \\leq \\mathcal { O } \\bigg ( d ^ { 2 } ( B + \\epsilon ) ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\{ 1 + \\frac { \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\} ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 715, + 839, + 844 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Then, applying Lemma 1 reveals that given any $\\eta > 0$ with probability at least $1 - \\eta$ for any w such that $\\begin{array} { r } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ we have ", + "bbox": [ + 171, + 847, + 828, + 881 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/5a34527ff5c90fb279655f299c65ef71c26ce9abaac1ce0c356c4d3ad0e6934d.jpg", + "text": "$$\nL _ { 0 } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\biggl ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\log \\frac { n } { \\eta } } { \\gamma ^ { 2 } n } } \\biggr )\n$$", + "text_format": "latex", + "bbox": [ + 246, + 886, + 750, + 929 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f o m } } ( f _ { \\mathbf { w } } ) : = \\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + \\frac { \\epsilon } { \\kappa } \\{ \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } \\right\\} . } \\end{array}$ Note that similar to our proof for Theorem 1 we can find a cover of size $O ( ( d \\log M ) ^ { d } d n ^ { 1 / 2 d } )$ for the spectral norms of the weights feasible set, where for any $\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }$ we have $a _ { i }$ such that $\\left| \\| \\mathbf { W } _ { i } \\| _ { 2 } - a _ { i } \\right| \\leq a _ { i } / d$ . Applying this covering number bound to (24) completes the proof. ", + "bbox": [ + 173, + 101, + 826, + 170 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.3 PROOF OF THEOREM 3 ", + "text_level": 1, + "bbox": [ + 174, + 184, + 375, + 199 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We use the following two lemmas to extend the proof of Theorem 2 for FGM attacks to show Theorem 3 for PGM attacks. ", + "bbox": [ + 174, + 210, + 825, + 239 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma 6. Consider a $d$ -layer neural network function $f _ { \\mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\\sigma$ where $\\sigma ( 0 ) = 0$ . We consider PGM attacks with noise power \u000f according to Euclidean norm $| | \\cdot | | _ { 2 } , r$ iterations and stepsize $\\alpha$ . Suppose $\\boldsymbol { \\kappa } \\leq \\| \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\| _ { 2 }$ holds over the $\\epsilon$ -ball around the support set $\\mathcal { X }$ . Then for any perturbation vector u such that $\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ for every $i$ we have ", + "bbox": [ + 176, + 242, + 825, + 304 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/482c33e88ba17f936b2535d14e0824270ae3711b3d6bfb3a21e9882eaf543d95.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) \\| _ { 2 } \\leq e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\times \\big ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\biggl [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + ( \\| \\mathbf { x } \\| _ { 2 } + \\epsilon ) \\big ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\biggr ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 309, + 736, + 391 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Here $\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )$ denotes the actual Lipschitz constant of $\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )$ . ", + "bbox": [ + 173, + 401, + 668, + 417 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. We use induction to show this lemma for different $r$ values. The result for case $r = 1$ is a direct consequence of Lemma 5. Suppose that the result is true for $r = k$ . Then, Lemmas 3 and 4 imply ", + "bbox": [ + 174, + 431, + 825, + 474 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/254fb01e44573d1f0cd6ad01c629bab08618283f883fcce0a78692a7328d78f0.jpg", + "text": "$$\n\\begin{array} { r l } & { | | E _ { \\frac { d } { d } } ^ { \\mathrm { L C } } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) | \\leq x _ { 0 } ^ { 3 } , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ } \\\\ & { \\quad + \\ \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) } \\\\ & \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 467, + 753, + 878 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where the last line follows from the equality $\\textstyle \\sum _ { i = 0 } ^ { k } s ^ { i } = { \\frac { 1 - s ^ { k + 1 } } { 1 - s } }$ . Therefore, by induction the lemma holds for every value $r \\geq 1$ . ", + "bbox": [ + 174, + 891, + 823, + 925 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma 7. Consider a $d$ -layer neural network function $f _ { \\mathbf { w } }$ with 1-Lipschitz, 1-smooth activation $\\sigma$ where $\\sigma ( 0 ) = 0$ . Also, assume that training loss $\\ell$ is 1-Lipschitz and 1-smooth. Then, ", + "bbox": [ + 169, + 103, + 825, + 133 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/ccfff99ac86d690054dbcb7e5eee8d466598fa27a5434265ae38f1184fb1f12c.jpg", + "text": "$$\n\\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\bigr ) \\bigr ) \\leq \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : = \\bigl ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\bigr ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 140, + 735, + 185 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. First of all note that according to the chain rule ", + "bbox": [ + 173, + 200, + 532, + 217 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/60522e9f0757d2290f2bebae880622c05525fd5e27a1431342a1fc567fa8614e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) = \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) ( \\nabla \\ell ) \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\operatorname* { l i p } ( f _ { \\mathbf { w } } ) ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 223, + 692, + 325 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Considering the above result, we complete the proof by inductively proving $\\begin{array} { r } { \\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\bigr ) \\ \\leq } \\end{array}$ $( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i - 1 } \\| \\mathbf { W } _ { j } \\| _ { 2 }$ . For $d = 1$ , $\\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } )$ is constant and hence the result holds. Assume the statement holds for $d = k$ . Due to the chain rule, ", + "bbox": [ + 174, + 332, + 825, + 381 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/472e9d0648034584ac6c5d561ea3806ea48c2cd4ce01585e3ea7860533a24a71.jpg", + "text": "$$\n\\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k + 1 ) } ( \\mathbf { x } ) = \\nabla _ { \\mathbf { x } } \\mathbf { W } _ { k + 1 } \\sigma \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) = \\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 259, + 386, + 736, + 406 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and therefore for any $\\mathbf { x }$ and $\\mathbf { v }$ ", + "bbox": [ + 173, + 412, + 369, + 428 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/2fd590cd10ae9b63167fde1bc62c7d54406897c3db9c6ef8b50c7a1ff08fb33f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\| _ { 2 } } \\\\ & { \\leq \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\nabla _ { x + 1 } ^ { F } - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T } \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\big \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\big \\| _ { 2 } } \\\\ & \\quad + \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ \\mathbf w \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 166, + 434, + 816, + 660 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "which shows the statement holds for $d = k + 1$ and therefore completes the proof via induction. ", + "bbox": [ + 169, + 665, + 797, + 681 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In order to prove Theorem 3, we note that for any norm-bounded $\\| \\mathbf { x } \\| _ { 2 } \\leq B$ and perturbation vector $\\mathbf { u }$ such that $\\forall i$ , $\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ we have ", + "bbox": [ + 174, + 699, + 826, + 729 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/c7e04340b7ce9aac15757ad69ecbf01d745063c55f27d965482570750302c538.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\times \\mathbf { n } } , \\mathbf { r } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\quad + \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\frac { \\| \\mathbf { u } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\qquad \\times \\displaystyle ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ^ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\Big [ \\| \\mathbf { u } _ { i } \\| _ { 2 } } \\\\ & \\qquad \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\mathbf { Z } _ { i } \\| _ { 2 } + ( B + \\epsilon ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 736, + 758, + 930 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/61c3273dca94419f51c819a971f35a8ddb1318009ecf768649c89039085fc627.jpg", + "text": "$$\n\\times \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + ( B + \\epsilon ) ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 264, + 99, + 715, + 145 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "The last inequality holds since as shown in Lemma $7 \\ : \\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y ) \\bigr ) \\leq \\ : \\varlimsup _ { \\mathbf { \\theta } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : =$ $\\begin{array} { r } { \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\end{array}$ . Here the upper-bound $\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\widetilde { \\mathbf { w } } } )$ for $\\widetilde { \\mathbf { w } }$ changes by a factor at most $e ^ { 2 / r }$ for w such that $| | \\mathbf { W } _ { i } | | _ { 2 } - | | \\widetilde { \\mathbf { W } } _ { i } | | _ { 2 } \\Big | \\leq \\frac { 1 } { r _ { i } ^ { d } } | | \\widetilde { \\mathbf { W } } _ { i } | | _ { 2 }$ . Therefore, given $\\widetilde { \\mathbf { w } }$ if similar to the proof for Theorem 2 we choose a zero-mean multivariate Gaussian distribution $Q$ efor $\\mathbf { u }$ with the ith layer $\\mathbf { u } _ { i }$ ’s standard deviation to be $\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}$ kWfik2 ξ where ", + "bbox": [ + 173, + 151, + 828, + 241 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/e4a8e622ee5947f40fdbc3fd0aaf1055f9cbad3fef7b63c347425ebff3fe7906.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { 8 d ( B + \\epsilon ) \\sqrt { 2 h \\log ( 4 h d ) } } } \\epsilon ^ { 4 } ( \\alpha / \\kappa ) \\frac { \\gamma } { 1 - e ^ { 2 \\left( 2 \\alpha / \\kappa \\right) \\sqrt { \\operatorname* { l i p } } ( \\nabla \\ell \\circ f _ { \\widetilde { \\infty } } ) ^ { r } } } ( \\prod _ { i = 1 } ^ { d } \\Vert \\widetilde { \\mathbf { W } } _ { i } \\Vert _ { 2 } ) \\big ( 1 + \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\Vert \\widetilde { \\mathbf { W } } _ { j } \\Vert _ { 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 246, + 861, + 286 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Then for any w satisfying $\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right| \\le \\frac { 1 } { r d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ , applying union bound shows that the assumptiholds for of Lemma 1 , and further $\\begin{array} { r } { \\operatorname* { P r } _ { \\mathbf { u } } \\bigl ( \\operatorname* { m a x } _ { \\mathbf { x } \\in \\mathcal { X } } \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) ) \\| _ { \\infty } \\leq \\frac { \\gamma } { 4 } \\bigr ) \\geq \\frac { 1 } { 2 } } \\end{array}$ $Q$ ", + "bbox": [ + 173, + 292, + 831, + 340 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/450a22f6c7ef092b92e89b2b103bc57b60aa60effb9d8c5c54fbe983401de269.jpg", + "text": "$$\n\\begin{array} { r l } & { K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { E } } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } \\\\ & { = \\mathcal { O } \\Big ( d ^ { 2 } ( B + c ) ^ { 2 } \\widehat { l } \\mathrm { l o g } ( i a ) \\Big ) \\times } \\\\ & { \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 \\frac { 1 - e ^ { 2 } ( 2 a \\rho \\gamma ) ^ { \\top } \\widehat { \\mathbb { H } } ( \\nabla \\xi \\circ f _ { \\varphi _ { i } } ) } { 1 - e ^ { 2 } \\gamma ( 2 a \\rho \\gamma ) \\widehat { l } \\mathrm { l o g } ( \\widehat { \\mathbf { W } } \\xi \\circ f _ { \\varphi _ { i } } ) } } ^ { 2 } \\{ 1 + \\frac { \\alpha } { \\kappa } \\Big ( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\Big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } \\Big \\} ^ { 2 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { E } } ^ { 2 } } { | \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } \\Big ) } \\\\ & { \\leq \\mathcal { O } \\Big ( d ^ { 2 } ( B + c ) ^ { 2 } \\widehat { l } \\mathrm { l o g } ( k a ) \\Big ) \\times } \\\\ & \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 \\frac { 1 - ( 2 \\sigma \\gamma ) \\widehat { l } \\mathrm { l o g } ( \\gamma \\circ f _ { \\varphi _ { i } } ) r } { 1 - ( 2 \\sigma \\gamma ) \\widehat { l } \\mathrm { l o g } ( \\widehat { \\mathbf { W } } \\xi \\circ f _ { \\varphi _ { i } } ) } ^ { 2 } \\Big \\{ 1 + \\frac { \\alpha } { \\kappa } \\Big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\Big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ^ { 2 } } \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { F } } ^ { 2 } } | \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 344, + 830, + 558 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Applying the above bound to Lemma 1 shows that for any $\\eta > 0$ the following holds with probability $1 - \\eta$ for any w where $\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right| \\le \\frac { 1 } { r d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ : ", + "bbox": [ + 173, + 561, + 826, + 595 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/32da9c48dda3f526194595c1bf1e637f687af91c91617b05957376c0944ea36f.jpg", + "text": "$$\nL _ { 0 } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + \\log \\frac { n } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 601, + 764, + 643 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where we consider $\\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )$ as the following expression ", + "bbox": [ + 173, + 648, + 570, + 665 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/97b7440ad3757a52be1c69c6019ec34bfd897c76ffb78076800818445fb923ff.jpg", + "text": "$$\n\\Big [ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + ( \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\Big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 671, + 828, + 717 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Using a similar argument to our proof of Theorem 2, we can properly cover the spectral norms for each $\\mathbf { W } _ { i }$ with $2 r d \\log M$ points, such that for any feasible $\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }$ value, satisfying the assumptions, we have value $a _ { i }$ in our cover where $\\begin{array} { r } { | \\| \\mathbf { W } _ { i } \\| _ { 2 } - a _ { i } | \\le \\frac { 1 } { r d } \\overset { \\cdot \\cdot } { a } _ { i } } \\end{array}$ . Therefore, we can cover all feasible combinations of spectral norms with $( 2 r d \\log M ) ^ { d } d n ^ { 1 / 2 d }$ , which combined with the above discussion completes the proof. ", + "bbox": [ + 173, + 722, + 826, + 796 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C.4 PROOF OF THEOREM 4 ", + "text_level": 1, + "bbox": [ + 174, + 813, + 375, + 828 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We first show the following lemma providing a perturbation bound for WRM attacks. ", + "bbox": [ + 173, + 838, + 733, + 854 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma 8. Consider a $d$ -layer neural net $f _ { \\mathbf { w } }$ satisfying the assumptions of Lemma 3. Then, for any weight perturbation u such that $\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ we have ", + "bbox": [ + 171, + 857, + 823, + 887 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/40db64043a69a03d0f1ef32da4efb78c3dabfcec9b2edaa316a8888247d1ad8e.jpg", + "text": "$$\n\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\| _ { 2 } \\leq \\frac { e ^ { 2 } } { \\lambda - \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) }\n$$", + "text_format": "latex", + "bbox": [ + 214, + 893, + 517, + 928 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/aeae3990564f0ec40a28759b9474da665e234136b48aa1264ffe84007a35d2b6.jpg", + "text": "$$\n\\times \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\bigg [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\big ( \\| \\mathbf { x } \\| _ { 2 } + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { j } \\| _ { 2 } } { \\lambda } \\big ) \\big ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\bigg ] .\n$$", + "text_format": "latex", + "bbox": [ + 218, + 99, + 784, + 145 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In the above inequality, $\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )$ denotes the Lipschitz constant of $\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )$ ", + "bbox": [ + 174, + 148, + 741, + 166 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. First of all note that for any $\\mathbf { x }$ we have $\\begin{array} { r } { \\| \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\| _ { 2 } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}$ , because we assume $\\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) < \\lambda$ implying WRM’s optimization is a convex optimization problem with the global solution $\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } )$ satisfying $\\begin{array} { r } { \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) = \\frac { 1 } { \\lambda } \\nabla \\ell \\circ f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) } \\end{array}$ which is norm-bounded by $\\begin{array} { r } { \\frac { \\operatorname* { l i p } ( \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}$ . Moreover, applying Lemma 3 we have ", + "bbox": [ + 173, + 181, + 825, + 244 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/5c09856ed51bcb3b14383e5b027fa549582d027dc9f57aaef98cb9f58dc51b8f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { = \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\frac { \\epsilon ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\Big [ \\big \\| \\mathbf { U } _ { i } \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\frac { \\operatorname* { l i p } ( \\nabla \\ell \\cup f _ { \\mathbf { w } } ) } { \\lambda } \\big \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) \\big \\| _ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 251, + 790, + 439 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "which shows the following inequality and hence completes the proof: ", + "bbox": [ + 173, + 443, + 629, + 458 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/1266de337953773da1104741952e4a8b54367e0f8b7d128949cf5b96105eef5a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\bigl ( 1 - \\frac { \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\bigr ) \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { \\leq \\frac { e ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\bigl ( \\| \\mathbf { x } \\| _ { 2 } + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { j } \\| _ { 2 } } { \\lambda } \\bigr ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 462, + 790, + 540 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Combining the above lemma with Lemma 2, for any norm-bounded $\\| \\mathbf { x } \\| _ { 2 } \\leq B$ and perturbation vector $\\mathbf { u }$ where $\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}$ , ", + "bbox": [ + 174, + 573, + 826, + 604 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/6e58575410104a826a198410cb638a01898b0e92cbdcde0ad935ddac65011fac.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left\\| \\int _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - \\int _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s e m } } ( \\mathbf { x } ) ) \\right\\| _ { 2 } } \\\\ & { \\leq \\Big \\| \\int _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s e m } } ( \\mathbf { x } ) ) \\Big \\| _ { 2 } + \\Big \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { m m } } ( \\mathbf { x } ) ) \\Big \\| _ { 2 } } \\\\ & { \\leq e ( B + \\frac { \\prod _ { j = 1 } ^ { d } | \\mathbf { w } _ { j } | } { \\lambda } ) ( \\underset { \\mathrm { L } ^ { - 1 } = 1 } { \\overset { d } { \\prod } } | \\mathbb { W } _ { 1 } | _ { 2 } ) \\underset { i = 1 } { \\overset { d } { \\sum } } \\Big \\| \\mathbb { W } _ { 1 } | _ { 2 } + \\Big ( \\underset { i = 1 } { \\overset { d } { \\prod } } | \\mathbb { W } _ { 1 } | _ { 2 } ) } \\\\ & { \\qquad \\times \\frac { e ^ { 2 } } { \\lambda - \\operatorname { l i p } ( \\nabla \\xi \\ell \\int _ { \\mathbf { w } } ) } \\underset { i = 1 } { \\overset { d } { \\sum } } \\bigg [ \\frac { \\| \\mathbf { U } _ { 1 } | _ { 2 } } { \\| \\mathbf { W } _ { 1 } \\| _ { 2 } } + \\big ( B + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { 2 } \\| _ { 2 } } { \\lambda } \\big ) ( \\underset { j = 1 } { \\overset { i } { \\prod } } | \\mathbf { W } _ { j } | | _ { 2 } ) \\underset { j = 1 } { \\overset { i } { \\sum } } \\frac { \\| \\mathbf { U } _ { j } | _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\bigg ] } \\\\ & \\leq e \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 608, + 816, + 832 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Similar to the proofs of Theorems 2,3, given $\\widetilde { \\mathbf { w } }$ we choose a zero-mean multivariate Gaussian distribution $Q$ e with diagonal covariance matrix for random perturbation u, with the $i$ th layer $\\mathbf { u } _ { i }$ ’s standard deviation parameter $\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}$ kWfik2β ξ where ", + "bbox": [ + 173, + 834, + 826, + 886 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/48f8600f3f149a9f539b7311237933b5fed9af5508b52d2ba30968065dc6334a.jpg", + "text": "$$\n\\begin{array} { r } { = \\frac { \\gamma } { 8 e ^ { 5 } d \\sqrt { 2 h \\log ( 4 h d ) } ( B + \\prod _ { j = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } / \\lambda ) \\left( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right) \\left( 1 + \\frac { 1 } { \\lambda - \\operatorname { l i p } ( \\nabla \\ell \\circ f _ { \\infty } ) } \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 890, + 839, + 928 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Using a union bound suggests the assumption of Lemma $\\begin{array} { r } { \\operatorname* { P r } _ { \\mathbf { u } } \\left( \\operatorname* { m a x } _ { \\mathbf { x } \\in \\mathcal { X } } \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) - \\right. } \\end{array}$ $\\begin{array} { r } { f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) \\| _ { \\infty } \\leq \\frac { \\gamma } { 4 } \\big ) \\geq \\frac { 1 } { 2 } } \\end{array}$ holds for $Q$ . Then, for any w satisfying $\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\leq } \\end{array}$ $\\frac { 1 } { 4 d / \\tau } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 }$ we have li $\\begin{array} { r } { \\overline { { \\mathsf { p } } } ( \\ell \\circ f _ { \\mathbf { w } } ) \\leq ( e ^ { \\tau / 4 } ) ^ { 2 } \\overline { { \\operatorname { l i p } } } ( \\ell \\circ f _ { \\widetilde { \\mathbf { w } } } ) \\leq ( e ^ { \\tau / 4 } ) ^ { 2 } \\lambda ( 1 - \\tau ) \\leq \\frac { 1 - \\tau } { 1 - \\tau / 2 } \\lambda \\leq ( 1 - \\frac { \\tau } { 2 } ) \\lambda } \\end{array}$ which implies the guard-band $\\tau$ for $\\widetilde { \\mathbf { w } }$ eapplies to w after being modified by a factor 2. Hence, ", + "bbox": [ + 174, + 127, + 826, + 198 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/d00365d665416434660ed0ce26ab274c5e10341be1c52706f1d7839d8ab6022b.jpg", + "text": "$$\n\\begin{array} { r l } & { K L ( P _ { \\Psi + \\Psi } | | Q ) \\le \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { F } } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } \\\\ & { \\le \\mathcal { O } \\Big ( d ^ { 2 } ( B + \\displaystyle \\prod _ { j = 1 } ^ { d } | | \\widehat { \\mathbf { W } } _ { i j } | | / \\lambda ) ^ { 2 } h \\log ( \\bar { h } d ) } \\\\ & { \\times \\displaystyle \\frac { ( | \\mathbf { I } _ { i - 1 } ^ { d } | | \\widehat { \\mathbf { W } } _ { i j } | | ^ { 2 } ) \\big ( 1 + \\frac { 1 } { \\lambda - \\log ( \\nabla \\xi _ { \\mathcal { F } _ { \\Psi } } ) } \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } | \\widehat { \\mathbf { W } } _ { i j } | \\big ) ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { | \\mathbf { W } _ { i } | | _ { \\mathcal { F } } ^ { 2 } } { | \\widehat { \\mathbf { W } } _ { i j } | | _ { 2 } ^ { 2 } } \\Big ) } \\\\ & { \\le \\mathcal { O } \\Big ( d ^ { 2 } ( B + \\displaystyle \\prod _ { j = 1 } ^ { d } | | \\mathbf { W } _ { i j } | | _ { \\mathcal { F } } | ) ^ { 2 } ) ^ { 2 } h \\log ( \\bar { h } d ) } \\\\ & { \\times \\displaystyle \\frac { ( | \\mathbf { I } _ { i - 1 } ^ { d } | | \\mathbf { W } _ { i } | | _ { 2 } ^ { 2 } ) \\big ( 1 + \\frac { 1 } { \\lambda - \\log ( \\nabla \\xi _ { \\Psi } ) } \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { d } | \\mathbf { W } _ { j } | | _ { 2 } \\big ) ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { | \\mathbf { W } _ { i } | | _ { \\mathcal { F } } ^ { 2 } } { | \\widehat { \\mathbf { W } } _ { i j } | | _ { 2 } ^ { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 204, + 753, + 436 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Using this bound in Lemma 1 implies that for any $\\eta > 0$ the following bound will hold with probability $1 - \\eta$ for any w where $\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { 4 d / \\tau } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}$ : ", + "bbox": [ + 173, + 458, + 823, + 493 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/ff4543e100a02c48592ad2bf71ed4b4e07d75fc663ead65fcc5f800e58887f75.jpg", + "text": "$$\nL _ { 0 } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { \\big ( B + \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } / \\lambda \\big ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\lambda } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { n } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 501, + 813, + 545 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where we define $\\Phi _ { \\lambda } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } )$ to be ", + "bbox": [ + 173, + 550, + 393, + 566 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/bf86ec09cd182e57c6ebca3332801aaa6ece085eba897c5c94a2bdc3d218e69c.jpg", + "text": "$$\n\\displaystyle \\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ( 1 + \\frac { 1 } { \\lambda - \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 571, + 769, + 618 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Using a similar argument to our proofs of Theorems 2 and 3, we can cover the possible spectral norms for each $\\mathbf { W } _ { i }$ with $O ( ( 8 d / \\tau ) \\log M )$ points, such that for any feasible $\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }$ value satisfying the theorem’s assumptions, we have value $a _ { i }$ in our cover where $\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\right\\| _ { 2 } - a _ { i } \\Big | \\le \\frac { 1 } { 4 d / \\tau } a _ { i } } \\end{array}$ . Therefore, we can cover all feasible combinations of spectral norms with $O ( ( ( 8 d / \\tau ) \\log M ) ^ { d } d n ^ { 1 / 2 d } )$ , which combined with the above discussion finishes the proof. ", + "bbox": [ + 173, + 623, + 826, + 702 + ], + "page_idx": 26 + } +] \ No newline at end of file diff --git a/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_middle.json b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..41b4c76862dd8aab574a6e8e88041c3bc75091f0 --- /dev/null +++ b/parse/train/Hyx4knR9Ym/Hyx4knR9Ym_middle.json @@ -0,0 +1,64341 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 506, + 97 + ], + "score": 1.0, + "content": "GENERALIZABLE ADVERSARIAL TRAINING VIA SPEC-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 281, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 281, + 118 + ], + "score": 1.0, + "content": "TRAL NORMALIZATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 134, + 317, + 179 + ], + "lines": [ + { + "bbox": [ + 112, + 134, + 316, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 134, + 316, + 147 + ], + "score": 1.0, + "content": "Farzan Farnia∗, Jesse M. 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A significant portion", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 313 + ], + "score": 1.0, + "content": "of this gap can be attributed to the decrease in generalization performance due to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 312, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 142, + 312, + 470, + 322 + ], + "score": 1.0, + "content": "adversarial training. 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We evaluate the power of spectral normalization extensively on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 387, + 471, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 387, + 471, + 400 + ], + "score": 1.0, + "content": "combinations of datasets, network architectures, and adversarial training schemes.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 109, + 422, + 206, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 208, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 208, + 437 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "Despite their impressive performance on many supervised learning tasks, deep neural networks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "(DNNs) are often highly susceptible to adversarial perturbations imperceptible to the human eye", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "(Szegedy et al., 2013; Goodfellow et al., 2014b). These “adversarial attacks\" have received enormous", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "attention in the machine learning literature over recent years (Goodfellow et al., 2014b; Moosavi Dez-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 492, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 503 + ], + "score": 1.0, + "content": "fooli et al., 2016; Carlini & Wagner, 2016; Kurakin et al., 2016; Papernot et al., 2016; Carlini &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "Wagner, 2017; Papernot et al., 2017; Madry et al., 2018; Tramèr et al., 2018). Adversarial attack stud-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 514, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 504, + 525 + ], + "score": 1.0, + "content": "ies have mainly focused on developing effective attack and defense schemes. While attack schemes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "attempt to mislead a trained classifier via additive perturbations to the input, defense mechanisms", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "aim to train classifiers robust to these perturbations. Although existing defense methods result in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "considerably better performance compared to standard training methods, the improved performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 507, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 507, + 570 + ], + "score": 1.0, + "content": "can still be far below the performance in non-adversarial settings (Athalye et al., 2018; Schmidt et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 136, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 136, + 581 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "A standard adversarial training scheme involves fitting a classifier using adversarially-perturbed", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "samples (Szegedy et al., 2013; Goodfellow et al., 2014b) with the intention of producing a trained", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "classifier with better robustness to attacks on future (i.e. test) samples. 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This discrepancy suggests that the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "overall adversarial test performance can be improved by applying effective regularization schemes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 316, + 217, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 217, + 332 + ], + "score": 1.0, + "content": "during adversarial training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 507, + 347 + ], + "score": 1.0, + "content": "In this work, we propose using spectral normalization (SN) (Miyato et al., 2018) as a computationally-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "efficient and statistically-powerful regularization scheme for adversarial training of DNNs. SN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "score": 1.0, + "content": "has been successfully implemented and applied for DNNs in the context of generative adversarial", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "networks (GANs) (Goodfellow et al., 2014a), resulting in state-of-the-art deep generative models", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "for several benchmark tasks (Miyato et al., 2018). Moreover, SN (Tsuzuku et al., 2018) and other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "similar Lipschitz regularization techniques (Cisse et al., 2017) have been successfully applied in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "non-adversarial training settings to improve the robustness of ERM-trained networks to adversarial", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "attacks. The theoretical results in (Bartlett et al., 2017; Neyshabur et al., 2017a) and empirical results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "in (Yoshida & Miyato, 2017) also suggest that SN can close the generalization gap for DNNs in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 226, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 226, + 447 + ], + "score": 1.0, + "content": "non-adversarial ERM setting.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "On the theoretical front, we extend the standard notion of margin loss to adversarial settings. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "leverage the PAC-Bayes generalization framework (McAllester, 1999) to prove generalization bounds", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "for spectrally-normalized DNNs in terms of our defined adversarial margin loss. 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For example, Figure 1 shows the training and validation performance for AlexNet", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "fit on the CIFAR10 dataset using FGM, PGM, and WRM, resulting in adversarial test accuracy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "improvements of 9, 11, and 4 percent, respectively. 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SN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "score": 1.0, + "content": "has been successfully implemented and applied for DNNs in the context of generative adversarial", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "networks (GANs) (Goodfellow et al., 2014a), resulting in state-of-the-art deep generative models", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "for several benchmark tasks (Miyato et al., 2018). Moreover, SN (Tsuzuku et al., 2018) and other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "similar Lipschitz regularization techniques (Cisse et al., 2017) have been successfully applied in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "non-adversarial training settings to improve the robustness of ERM-trained networks to adversarial", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "attacks. The theoretical results in (Bartlett et al., 2017; Neyshabur et al., 2017a) and empirical results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "in (Yoshida & Miyato, 2017) also suggest that SN can close the generalization gap for DNNs in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 226, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 226, + 447 + ], + "score": 1.0, + "content": "non-adversarial ERM setting.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 334, + 507, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "On the theoretical front, we extend the standard notion of margin loss to adversarial settings. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "leverage the PAC-Bayes generalization framework (McAllester, 1999) to prove generalization bounds", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "for spectrally-normalized DNNs in terms of our defined adversarial margin loss. We obtain adversarial", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "generalization error bounds for three well-known gradient-based attack schemes: fast gradient method", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "(FGM) (Goodfellow et al., 2014b), projected gradient method (PGM) (Kurakin et al., 2016), and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "Wasserstein risk minimization (WRM) (Sinha et al., 2018). Our theoretical analysis shows that the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 515, + 396, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 396, + 528 + ], + "score": 1.0, + "content": "adversarial generalization error will vanish by applying SN to all layers.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 450, + 506, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "On the empirical front, we show that SN can significantly improve the test performance of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "adversarially-trained DNNs. We perform numerical experiments over various standard datasets", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and DNN architectures. In almost all of our experiments, we obtain a better test performance after", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "applying SN. 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(2017a), we evaluate a DNN’s generalization", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 438, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 400, + 342 + ], + "score": 1.0, + "content": "performance using its expected margin loss defined for margin parameter", + "type": "text" + }, + { + "bbox": [ + 401, + 329, + 425, + 340 + ], + "score": 0.91, + "content": "\\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 329, + 438, + 342 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 282, + 506, + 342 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 343, + 408, + 371 + ], + "lines": [ + { + "bbox": [ + 201, + 343, + 408, + 371 + ], + "spans": [ + { + "bbox": [ + 201, + 343, + 408, + 371 + ], + "score": 0.93, + "content": "L _ { \\gamma } ( f _ { \\mathbf { w } } ) : = P \\bigg ( f _ { \\mathbf { w } } ( \\mathbf { X } ) [ Y ] \\leq \\gamma + \\operatorname* { m a x } _ { j \\neq Y } f _ { \\mathbf { w } } ( \\mathbf { X } ) [ j ] \\bigg ) ,", + "type": "interline_equation", + "image_path": "a628ba20a03d1f96572de5b8f76fd2200a4e39d6eca919f74c20ccd2cfcbde74.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 343, + 408, + 357.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 201, + 357.0, + 408, + 371.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 373, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 134, + 388 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 374, + 174, + 386 + ], + "score": 0.93, + "content": "f _ { \\mathbf { w } } ( \\mathbf { X } ) [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 373, + 224, + 388 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 224, + 375, + 230, + 385 + ], + "score": 0.7, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 373, + 275, + 388 + ], + "score": 1.0, + "content": "th entry of", + "type": "text" + }, + { + "bbox": [ + 275, + 374, + 332, + 386 + ], + "score": 0.92, + "content": "f _ { \\mathbf { w } } ( \\mathbf { X } ) \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 373, + 430, + 388 + ], + "score": 1.0, + "content": ". 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Given norm function", + "type": "text" + }, + { + "bbox": [ + 328, + 597, + 346, + 609 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 597, + 462, + 610 + ], + "score": 1.0, + "content": "and adversarial noise power", + "type": "text" + }, + { + "bbox": [ + 463, + 598, + 486, + 608 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 597, + 505, + 610 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 608, + 414, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 255, + 621 + ], + "score": 1.0, + "content": "adversarial additive noise for sample", + "type": "text" + }, + { + "bbox": [ + 256, + 608, + 280, + 620 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 608, + 336, + 621 + ], + "score": 1.0, + "content": "and classifier", + "type": "text" + }, + { + "bbox": [ + 336, + 609, + 349, + 620 + ], + "score": 0.88, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 608, + 414, + 621 + ], + "score": 1.0, + "content": "is defined to be", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 574, + 505, + 621 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 623, + 385, + 646 + ], + "lines": [ + { + "bbox": [ + 226, + 623, + 385, + 646 + ], + "spans": [ + { + "bbox": [ + 226, + 623, + 385, + 646 + ], + "score": 0.94, + "content": "\\delta _ { \\mathbf w } ^ { \\mathrm { a d v } } ( \\mathbf x ) : = \\operatorname * { a r g m a x } _ { \\| \\delta \\| \\leq \\epsilon } \\ell \\big ( f _ { \\mathbf w } ( \\mathbf x + \\pmb \\delta ) , y \\big ) .", + "type": "interline_equation", + "image_path": "e436fb0d7d7b2445b4dd44a4739193aaec85d8b390822261bb760b0cb52cca15.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 226, + 623, + 385, + 646 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 504, + 660 + ], + "score": 1.0, + "content": "To provide adversarial robustness against the above attack scheme, a standard technique, which is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 658, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 674 + ], + "score": 1.0, + "content": "called adversarial training, follows ERM training over the adversarially-perturbed samples by solving", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 649, + 505, + 674 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 674, + 476, + 707 + ], + "lines": [ + { + "bbox": [ + 136, + 674, + 476, + 707 + ], + "spans": [ + { + "bbox": [ + 136, + 674, + 476, + 707 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathcal { W } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell \\left( f _ { \\mathbf { w } } \\left( \\mathbf { x } _ { i } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { x } _ { i } ) \\right) , y _ { i } \\right) : = \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathcal { W } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\| \\delta _ { i } \\| \\leq \\epsilon } \\ell \\left( f _ { \\mathbf { w } } ( \\mathbf { x } _ { i } + \\delta _ { i } ) , y _ { i } \\right) .", + "type": "interline_equation", + "image_path": "0ef5ff3deff2585f04b6f6a366234cb82ddf00a783b310abb55d68c66342465b.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 136, + 674, + 476, + 685.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 136, + 685.0, + 476, + 696.0 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 136, + 696.0, + 476, + 707.0 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "However, (3) and (4) are intractable optimization problems. Therefore, several schemes have been", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "proposed in the literature to approximate the optimal solution of (3). In this work, we analyze the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "generalization performance of the following three gradient-based methods for approximating the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "solution to (3). We note that several other attack schemes such as DeepFool (Moosavi Dezfooli et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "2016), CW attacks (Carlini & Wagner, 2017), target and least-likely attacks (Kurakin et al., 2016)", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "have been introduced and examined in the literature, which can lead to interesting future directions", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 163, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 163, + 139 + ], + "score": 1.0, + "content": "for this work.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 710, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "generalization performance of the following three gradient-based methods for approximating the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "solution to (3). We note that several other attack schemes such as DeepFool (Moosavi Dezfooli et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "2016), CW attacks (Carlini & Wagner, 2017), target and least-likely attacks (Kurakin et al., 2016)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "have been introduced and examined in the literature, which can lead to interesting future directions", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 163, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 163, + 139 + ], + "score": 1.0, + "content": "for this work.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 121, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 121, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "1. Fast Gradient Method (FGM) (Goodfellow et al., 2014b): FGM approximates the solution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 480, + 172 + ], + "score": 1.0, + "content": "to (3) by considering a linearized DNN loss around a given data point. Hence, FGM perturbs", + "type": "text" + }, + { + "bbox": [ + 481, + 159, + 505, + 171 + ], + "score": 0.91, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 259, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 259, + 183 + ], + "score": 1.0, + "content": "by adding the following noise vector:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 186, + 389, + 210 + ], + "lines": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "spans": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "score": 0.94, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) : = \\underset { \\| \\delta \\| \\leq \\epsilon } { \\mathrm { a r g m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) .", + "type": "interline_equation", + "image_path": "a8103fe9da87a8078dd9c58670c3b37f7687231941d0b675721b45084e38b8d8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 198, + 230 + ], + "score": 1.0, + "content": "For the special case of", + "type": "text" + }, + { + "bbox": [ + 198, + 217, + 211, + 228 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 216, + 237, + 230 + ], + "score": 1.0, + "content": "-norm", + "type": "text" + }, + { + "bbox": [ + 238, + 217, + 264, + 229 + ], + "score": 0.89, + "content": "\\| \\cdot \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 216, + 505, + 230 + ], + "score": 1.0, + "content": ", the above representation of FGM recovers the fast gradient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 287, + 241 + ], + "score": 1.0, + "content": "sign method (FGSM) where each data point", + "type": "text" + }, + { + "bbox": [ + 288, + 228, + 312, + 240 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 228, + 391, + 241 + ], + "score": 1.0, + "content": "is perturbed by the", + "type": "text" + }, + { + "bbox": [ + 391, + 230, + 397, + 238 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "-normalized sign vector of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 202, + 252 + ], + "score": 1.0, + "content": "the loss’s gradient. For", + "type": "text" + }, + { + "bbox": [ + 203, + 239, + 212, + 250 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 238, + 239, + 252 + ], + "score": 1.0, + "content": "-norm", + "type": "text" + }, + { + "bbox": [ + 239, + 239, + 262, + 251 + ], + "score": 0.91, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 238, + 498, + 252 + ], + "score": 1.0, + "content": ", we similarly normalize the loss’s gradient vector to have", + "type": "text" + }, + { + "bbox": [ + 499, + 241, + 504, + 249 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 250, + 174, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 174, + 262 + ], + "score": 1.0, + "content": "Euclidean norm.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 120, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 120, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "2. Projected Gradient Method (PGM) (Kurakin et al., 2016): PGM is the iterative version of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "FGM and applies projected gradient descent to solve (3). PGM follows the following update rules for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 213, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 137, + 300 + ], + "score": 1.0, + "content": "a given", + "type": "text" + }, + { + "bbox": [ + 138, + 290, + 144, + 297 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 288, + 213, + 300 + ], + "score": 1.0, + "content": "number of steps:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 303, + 452, + 360 + ], + "lines": [ + { + "bbox": [ + 157, + 303, + 452, + 360 + ], + "spans": [ + { + "bbox": [ + 157, + 303, + 452, + 360 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\forall 1 \\le i \\le r : } & { \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i + 1 } ( \\mathbf { x } ) : = \\displaystyle \\prod _ { \\boldsymbol { \\epsilon } _ { \\epsilon , \\parallel } \\cdot \\parallel ^ { ( 0 ) } } \\bigl \\{ \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) + \\alpha \\nu _ { \\mathbf { w } } ^ { ( i ) } \\bigr \\} , } \\\\ & { \\nu _ { \\mathbf { w } } ^ { ( i ) } : = \\underset { \\parallel \\delta \\parallel \\le 1 } { \\arg \\operatorname* { m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) ) , y \\bigr ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6b7e6cd6e53ed6b5fe1283c39d11ed9b1865b5c19487a5861e666b531ebf7034.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 157, + 303, + 452, + 322.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 157, + 322.0, + 452, + 341.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 157, + 341.0, + 452, + 360.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 366, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 232, + 382 + ], + "score": 1.0, + "content": "Here, we first find the direction", + "type": "text" + }, + { + "bbox": [ + 232, + 366, + 248, + 380 + ], + "score": 0.92, + "content": "\\nu _ { \\mathbf { w } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 365, + 357, + 382 + ], + "score": 1.0, + "content": "along which the loss at the", + "type": "text" + }, + { + "bbox": [ + 357, + 370, + 362, + 378 + ], + "score": 0.31, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 365, + 506, + 382 + ], + "score": 1.0, + "content": "th perturbed point changes the most,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 387, + 392 + ], + "score": 1.0, + "content": "and then we move the perturbed point along this direction by stepsize", + "type": "text" + }, + { + "bbox": [ + 388, + 381, + 396, + 389 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "followed by projecting the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 395, + 403 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 244, + 403 + ], + "score": 1.0, + "content": "resulting perturbation onto the set", + "type": "text" + }, + { + "bbox": [ + 244, + 390, + 303, + 403 + ], + "score": 0.93, + "content": "\\left\\{ \\delta : \\left\\| \\delta \\right\\| \\leq \\epsilon \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 390, + 325, + 403 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 325, + 392, + 330, + 400 + ], + "score": 0.77, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 390, + 395, + 403 + ], + "score": 1.0, + "content": "-bounded norm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 119, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 119, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "3. Wasserstein Risk Minimization (WRM) (Sinha et al., 2018): WRM solves the following", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 221, + 429 + ], + "score": 1.0, + "content": "variant of (3) for data-point", + "type": "text" + }, + { + "bbox": [ + 221, + 417, + 245, + 429 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "where the norm constraint in (3) is replaced by a norm-squared", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 428, + 210, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 210, + 441 + ], + "score": 1.0, + "content": "Lagrangian penalty term:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 445, + 405, + 471 + ], + "lines": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "spans": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "score": 0.91, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) : = \\operatorname * { a r g m a x } _ { \\delta } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\pmb { \\delta } ) , y \\big ) - \\frac { \\lambda } { 2 } \\| \\pmb { \\delta } \\| ^ { 2 } .", + "type": "interline_equation", + "image_path": "f669b3f84fa5632204c604cc43c989f5b73c08aa68e89307525ded9698c0c27f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "As discussed earlier, the optimization problem (3) is generally intractable. However, in the case of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 171, + 501 + ], + "score": 1.0, + "content": "Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 171, + 488, + 192, + 500 + ], + "score": 0.91, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 487, + 250, + 501 + ], + "score": 1.0, + "content": ", if we assume", + "type": "text" + }, + { + "bbox": [ + 250, + 488, + 311, + 500 + ], + "score": 0.93, + "content": "\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 487, + 475, + 501 + ], + "score": 1.0, + "content": "’s Lipschitz constant is upper-bounded by", + "type": "text" + }, + { + "bbox": [ + 475, + 488, + 482, + 498 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 487, + 506, + 501 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "WRM optimization (7) results in solving a convex optimization problem and can be efficiently solved", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 204, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 204, + 522 + ], + "score": 1.0, + "content": "using gradient methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 506, + 565 + ], + "lines": [ + { + "bbox": [ + 102, + 527, + 509, + 549 + ], + "spans": [ + { + "bbox": [ + 102, + 527, + 364, + 549 + ], + "score": 1.0, + "content": "To obtain efficient adversarial defense schemes, we can substitute", + "type": "text" + }, + { + "bbox": [ + 365, + 531, + 384, + 543 + ], + "score": 0.76, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 527, + 387, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 388, + 531, + 408, + 543 + ], + "score": 0.61, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 527, + 423, + 549 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 423, + 531, + 445, + 543 + ], + "score": 0.89, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 527, + 460, + 549 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 460, + 531, + 479, + 543 + ], + "score": 0.9, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 527, + 509, + 549 + ], + "score": 1.0, + "content": "in (4).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Instead of fitting the classifier over true adversarial examples, which are NP-hard to obtain, we can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 440, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 440, + 565 + ], + "score": 1.0, + "content": "instead train the DNN over FGM, PGM, or WRM-adversarially perturbed samples.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 579, + 303, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 305, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 305, + 591 + ], + "score": 1.0, + "content": "2.3 ADVERSARIAL GENERALIZATION ERROR", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "The goal of adversarial training is to improve the robustness against adversarial attacks on not only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "the training samples but also on test samples; however, the adversarial training problem (4) focuses", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "only on the training samples. To evaluate the adversarial generalization performance, we extend the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "notion of margin loss defined earlier in (1) to adversarial training settings by defining the adversarial", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 167, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 167, + 657 + ], + "score": 1.0, + "content": "margin loss as", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 660, + 462, + 688 + ], + "lines": [ + { + "bbox": [ + 148, + 660, + 462, + 688 + ], + "spans": [ + { + "bbox": [ + 148, + 660, + 462, + 688 + ], + "score": 0.92, + "content": "L _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } ) = P \\bigg ( f _ { \\mathbf { w } } ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) ) [ Y ] \\leq \\gamma + \\operatorname* { m a x } _ { j \\neq Y } f _ { \\mathbf { w } } \\big ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) \\big ) [ j ] \\bigg ) .", + "type": "interline_equation", + "image_path": "fba65fc355814dd336ebccebd4538ef1d48d9e8e914b9f6b75fded54737a2eba.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 148, + 660, + 462, + 669.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 148, + 669.3333333333334, + 462, + 678.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 148, + 678.6666666666667, + 462, + 688.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 695, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 693, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 452, + 710 + ], + "score": 1.0, + "content": "Here, we measure the margin loss over adversarially-perturbed samples, and we use", + "type": "text" + }, + { + "bbox": [ + 452, + 694, + 493, + 709 + ], + "score": 0.92, + "content": "\\widehat { L } _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 693, + 506, + 710 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 341, + 723 + ], + "score": 1.0, + "content": "denote the empirical adversarial margin loss. We also use", + "type": "text" + }, + { + "bbox": [ + 341, + 708, + 429, + 722 + ], + "score": 0.79, + "content": "L _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) , L _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 707, + 450, + 723 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 450, + 710, + 493, + 723 + ], + "score": 0.9, + "content": "L _ { \\gamma } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 499, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 499, + 734 + ], + "score": 1.0, + "content": "denote the adversarial margin losses with FGM (5), PGM (6), and WRM (7) attacks, respectively.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 121, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 121, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "1. Fast Gradient Method (FGM) (Goodfellow et al., 2014b): FGM approximates the solution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 480, + 172 + ], + "score": 1.0, + "content": "to (3) by considering a linearized DNN loss around a given data point. Hence, FGM perturbs", + "type": "text" + }, + { + "bbox": [ + 481, + 159, + 505, + 171 + ], + "score": 0.91, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 259, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 259, + 183 + ], + "score": 1.0, + "content": "by adding the following noise vector:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 147, + 505, + 183 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 186, + 389, + 210 + ], + "lines": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "spans": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "score": 0.94, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) : = \\underset { \\| \\delta \\| \\leq \\epsilon } { \\mathrm { a r g m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) .", + "type": "interline_equation", + "image_path": "a8103fe9da87a8078dd9c58670c3b37f7687231941d0b675721b45084e38b8d8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 222, + 186, + 389, + 210 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 198, + 230 + ], + "score": 1.0, + "content": "For the special case of", + "type": "text" + }, + { + "bbox": [ + 198, + 217, + 211, + 228 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 216, + 237, + 230 + ], + "score": 1.0, + "content": "-norm", + "type": "text" + }, + { + "bbox": [ + 238, + 217, + 264, + 229 + ], + "score": 0.89, + "content": "\\| \\cdot \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 216, + 505, + 230 + ], + "score": 1.0, + "content": ", the above representation of FGM recovers the fast gradient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 287, + 241 + ], + "score": 1.0, + "content": "sign method (FGSM) where each data point", + "type": "text" + }, + { + "bbox": [ + 288, + 228, + 312, + 240 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 228, + 391, + 241 + ], + "score": 1.0, + "content": "is perturbed by the", + "type": "text" + }, + { + "bbox": [ + 391, + 230, + 397, + 238 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "-normalized sign vector of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 202, + 252 + ], + "score": 1.0, + "content": "the loss’s gradient. 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Projected Gradient Method (PGM) (Kurakin et al., 2016): PGM is the iterative version of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "FGM and applies projected gradient descent to solve (3). PGM follows the following update rules for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 213, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 137, + 300 + ], + "score": 1.0, + "content": "a given", + "type": "text" + }, + { + "bbox": [ + 138, + 290, + 144, + 297 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 288, + 213, + 300 + ], + "score": 1.0, + "content": "number of steps:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 265, + 506, + 300 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 303, + 452, + 360 + ], + "lines": [ + { + "bbox": [ + 157, + 303, + 452, + 360 + ], + "spans": [ + { + "bbox": [ + 157, + 303, + 452, + 360 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\forall 1 \\le i \\le r : } & { \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i + 1 } ( \\mathbf { x } ) : = \\displaystyle \\prod _ { \\boldsymbol { \\epsilon } _ { \\epsilon , \\parallel } \\cdot \\parallel ^ { ( 0 ) } } \\bigl \\{ \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) + \\alpha \\nu _ { \\mathbf { w } } ^ { ( i ) } \\bigr \\} , } \\\\ & { \\nu _ { \\mathbf { w } } ^ { ( i ) } : = \\underset { \\parallel \\delta \\parallel \\le 1 } { \\arg \\operatorname* { m a x } } \\delta ^ { T } \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , i } ( \\mathbf { x } ) ) , y \\bigr ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6b7e6cd6e53ed6b5fe1283c39d11ed9b1865b5c19487a5861e666b531ebf7034.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 157, + 303, + 452, + 322.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 157, + 322.0, + 452, + 341.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 157, + 341.0, + 452, + 360.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 366, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 232, + 382 + ], + "score": 1.0, + "content": "Here, we first find the direction", + "type": "text" + }, + { + "bbox": [ + 232, + 366, + 248, + 380 + ], + "score": 0.92, + "content": "\\nu _ { \\mathbf { w } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 365, + 357, + 382 + ], + "score": 1.0, + "content": "along which the loss at the", + "type": "text" + }, + { + "bbox": [ + 357, + 370, + 362, + 378 + ], + "score": 0.31, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 365, + 506, + 382 + ], + "score": 1.0, + "content": "th perturbed point changes the most,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 387, + 392 + ], + "score": 1.0, + "content": "and then we move the perturbed point along this direction by stepsize", + "type": "text" + }, + { + "bbox": [ + 388, + 381, + 396, + 389 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "followed by projecting the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 395, + 403 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 244, + 403 + ], + "score": 1.0, + "content": "resulting perturbation onto the set", + "type": "text" + }, + { + "bbox": [ + 244, + 390, + 303, + 403 + ], + "score": 0.93, + "content": "\\left\\{ \\delta : \\left\\| \\delta \\right\\| \\leq \\epsilon \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 390, + 325, + 403 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 325, + 392, + 330, + 400 + ], + "score": 0.77, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 390, + 395, + 403 + ], + "score": 1.0, + "content": "-bounded norm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 365, + 506, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 119, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 119, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "3. Wasserstein Risk Minimization (WRM) (Sinha et al., 2018): WRM solves the following", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 221, + 429 + ], + "score": 1.0, + "content": "variant of (3) for data-point", + "type": "text" + }, + { + "bbox": [ + 221, + 417, + 245, + 429 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "where the norm constraint in (3) is replaced by a norm-squared", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 428, + 210, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 210, + 441 + ], + "score": 1.0, + "content": "Lagrangian penalty term:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 404, + 505, + 441 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 445, + 405, + 471 + ], + "lines": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "spans": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "score": 0.91, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) : = \\operatorname * { a r g m a x } _ { \\delta } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\pmb { \\delta } ) , y \\big ) - \\frac { \\lambda } { 2 } \\| \\pmb { \\delta } \\| ^ { 2 } .", + "type": "interline_equation", + "image_path": "f669b3f84fa5632204c604cc43c989f5b73c08aa68e89307525ded9698c0c27f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 205, + 445, + 405, + 471 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "As discussed earlier, the optimization problem (3) is generally intractable. However, in the case of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 171, + 501 + ], + "score": 1.0, + "content": "Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 171, + 488, + 192, + 500 + ], + "score": 0.91, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 487, + 250, + 501 + ], + "score": 1.0, + "content": ", if we assume", + "type": "text" + }, + { + "bbox": [ + 250, + 488, + 311, + 500 + ], + "score": 0.93, + "content": "\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 487, + 475, + 501 + ], + "score": 1.0, + "content": "’s Lipschitz constant is upper-bounded by", + "type": "text" + }, + { + "bbox": [ + 475, + 488, + 482, + 498 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 487, + 506, + 501 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "WRM optimization (7) results in solving a convex optimization problem and can be efficiently solved", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 204, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 204, + 522 + ], + "score": 1.0, + "content": "using gradient methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 476, + 506, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 506, + 565 + ], + "lines": [ + { + "bbox": [ + 102, + 527, + 509, + 549 + ], + "spans": [ + { + "bbox": [ + 102, + 527, + 364, + 549 + ], + "score": 1.0, + "content": "To obtain efficient adversarial defense schemes, we can substitute", + "type": "text" + }, + { + "bbox": [ + 365, + 531, + 384, + 543 + ], + "score": 0.76, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 527, + 387, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 388, + 531, + 408, + 543 + ], + "score": 0.61, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 527, + 423, + 549 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 423, + 531, + 445, + 543 + ], + "score": 0.89, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 527, + 460, + 549 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 460, + 531, + 479, + 543 + ], + "score": 0.9, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 527, + 509, + 549 + ], + "score": 1.0, + "content": "in (4).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Instead of fitting the classifier over true adversarial examples, which are NP-hard to obtain, we can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 440, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 440, + 565 + ], + "score": 1.0, + "content": "instead train the DNN over FGM, PGM, or WRM-adversarially perturbed samples.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 102, + 527, + 509, + 565 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 579, + 303, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 305, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 305, + 591 + ], + "score": 1.0, + "content": "2.3 ADVERSARIAL GENERALIZATION ERROR", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "The goal of adversarial training is to improve the robustness against adversarial attacks on not only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "the training samples but also on test samples; however, the adversarial training problem (4) focuses", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "only on the training samples. To evaluate the adversarial generalization performance, we extend the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "notion of margin loss defined earlier in (1) to adversarial training settings by defining the adversarial", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 167, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 167, + 657 + ], + "score": 1.0, + "content": "margin loss as", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 599, + 505, + 657 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 660, + 462, + 688 + ], + "lines": [ + { + "bbox": [ + 148, + 660, + 462, + 688 + ], + "spans": [ + { + "bbox": [ + 148, + 660, + 462, + 688 + ], + "score": 0.92, + "content": "L _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } ) = P \\bigg ( f _ { \\mathbf { w } } ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) ) [ Y ] \\leq \\gamma + \\operatorname* { m a x } _ { j \\neq Y } f _ { \\mathbf { w } } \\big ( \\mathbf { X } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { a d v } } ( \\mathbf { X } ) \\big ) [ j ] \\bigg ) .", + "type": "interline_equation", + "image_path": "fba65fc355814dd336ebccebd4538ef1d48d9e8e914b9f6b75fded54737a2eba.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 148, + 660, + 462, + 669.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 148, + 669.3333333333334, + 462, + 678.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 148, + 678.6666666666667, + 462, + 688.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 695, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 693, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 452, + 710 + ], + "score": 1.0, + "content": "Here, we measure the margin loss over adversarially-perturbed samples, and we use", + "type": "text" + }, + { + "bbox": [ + 452, + 694, + 493, + 709 + ], + "score": 0.92, + "content": "\\widehat { L } _ { \\gamma } ^ { \\mathrm { a d v } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 693, + 506, + 710 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 341, + 723 + ], + "score": 1.0, + "content": "denote the empirical adversarial margin loss. We also use", + "type": "text" + }, + { + "bbox": [ + 341, + 708, + 429, + 722 + ], + "score": 0.79, + "content": "L _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) , L _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 707, + 450, + 723 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 450, + 710, + 493, + 723 + ], + "score": 0.9, + "content": "L _ { \\gamma } ^ { \\mathrm { w r m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 499, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 499, + 734 + ], + "score": 1.0, + "content": "denote the adversarial margin losses with FGM (5), PGM (6), and WRM (7) attacks, respectively.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 693, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 427, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 429, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 429, + 96 + ], + "score": 1.0, + "content": "3 MARGIN-BASED ADVERSARIAL GENERALIZATION BOUNDS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "As previously discussed, generalization performance can be different between adversarial and non-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "score": 1.0, + "content": "adversarial settings. In this section, we provide generalization bounds for DNN classifiers under", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "adversarial attacks in terms of the spectral norms of the trained DNN’s weight matrices. The bounds", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "motivate regularizing these spectral norms in order to limit the DNN’s capacity and improve its", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 322, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 322, + 162 + ], + "score": 1.0, + "content": "generalization performance under adversarial attacks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "We use the PAC-Bayes framework (McAllester, 1999; 2003) to prove our main results. To derive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 179, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 434, + 190 + ], + "score": 1.0, + "content": "adversarial generalization error bounds for DNNs with smooth activation functions", + "type": "text" + }, + { + "bbox": [ + 434, + 181, + 441, + 188 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 179, + 505, + 190 + ], + "score": 1.0, + "content": ", we first extend", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "a recent result on the margin-based generalization bound for the ReLU activation function (Neyshabur", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 334, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 334, + 213 + ], + "score": 1.0, + "content": "et al., 2017a) to general 1-Lipschitz activation functions.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 504, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 200, + 228 + ], + "score": 1.0, + "content": "Theorem 1. Consider", + "type": "text" + }, + { + "bbox": [ + 200, + 215, + 295, + 227 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { n n } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathbf { W } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 214, + 496, + 228 + ], + "score": 1.0, + "content": "the class of d hidden-layer neural networks with", + "type": "text" + }, + { + "bbox": [ + 497, + 216, + 504, + 225 + ], + "score": 0.69, + "content": "h", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 300, + 238 + ], + "score": 1.0, + "content": "units per hidden-layer with 1-Lipschitz activation", + "type": "text" + }, + { + "bbox": [ + 300, + 229, + 307, + 236 + ], + "score": 0.59, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 226, + 348, + 238 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 348, + 226, + 386, + 238 + ], + "score": 0.92, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 226, + 442, + 238 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 442, + 227, + 451, + 236 + ], + "score": 0.78, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 226, + 455, + 238 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 455, + 227, + 465, + 236 + ], + "score": 0.41, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "’s support", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 236, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 209, + 251 + ], + "score": 1.0, + "content": "set, is norm-bounded as", + "type": "text" + }, + { + "bbox": [ + 209, + 237, + 254, + 249 + ], + "score": 0.87, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 236, + 258, + 251 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 237, + 295, + 248 + ], + "score": 0.77, + "content": "\\forall \\mathbf { x } \\in { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 236, + 408, + 251 + ], + "score": 1.0, + "content": ". Also assume for constant", + "type": "text" + }, + { + "bbox": [ + 409, + 237, + 442, + 248 + ], + "score": 0.91, + "content": "M \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 236, + 461, + 251 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 461, + 237, + 504, + 249 + ], + "score": 0.91, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 248, + 142, + 261 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 142, + 261 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 258, + 415, + 293 + ], + "lines": [ + { + "bbox": [ + 195, + 258, + 415, + 293 + ], + "spans": [ + { + "bbox": [ + 195, + 258, + 415, + 293 + ], + "score": 0.93, + "content": "\\forall i : ~ \\frac { 1 } { M } \\leq \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } } { \\beta _ { \\mathbf { w } } } \\leq M , \\quad \\beta _ { \\mathbf { w } } : = \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) ^ { 1 / d } .", + "type": "interline_equation", + "image_path": "65149fb37096cc18401f4f83d66787664661b763120128178de050f90d13c0c9.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 258, + 415, + 275.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 195, + 275.5, + 415, + 293.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 127, + 309 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 296, + 141, + 307 + ], + "score": 0.89, + "content": "\\beta _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 294, + 263, + 309 + ], + "score": 1.0, + "content": "denotes the geometric mean of", + "type": "text" + }, + { + "bbox": [ + 264, + 296, + 276, + 307 + ], + "score": 0.87, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 294, + 468, + 309 + ], + "score": 1.0, + "content": "’s spectral norms across all layers. Then, for any", + "type": "text" + }, + { + "bbox": [ + 469, + 296, + 503, + 307 + ], + "score": 0.9, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 294, + 506, + 309 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 340, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 204, + 320 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 307, + 228, + 318 + ], + "score": 0.82, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 305, + 260, + 320 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 260, + 307, + 300, + 318 + ], + "score": 0.91, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 305, + 340, + 320 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 323, + 453, + 357 + ], + "lines": [ + { + "bbox": [ + 157, + 323, + 453, + 357 + ], + "spans": [ + { + "bbox": [ + 157, + 323, + 453, + 357 + ], + "score": 0.93, + "content": "L _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { B ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) ,", + "type": "interline_equation", + "image_path": "2c3f8efecae8bdf8059492f399508d55a04cd934671cf4808d438540b9a69797.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 157, + 323, + 453, + 334.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 157, + 334.3333333333333, + 453, + 345.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 157, + 345.66666666666663, + 453, + 356.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 422, + 381 + ], + "lines": [ + { + "bbox": [ + 242, + 363, + 421, + 382 + ], + "spans": [ + { + "bbox": [ + 242, + 363, + 421, + 382 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) : = \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 282, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 282, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 282, + 407 + ], + "score": 1.0, + "content": "Proof. We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 419, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "We now generalize this result to adversarial settings where the DNN’s performance is evaluated under", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "adversarial attacks. We prove three separate adversarial generalization error bounds for FGM, PGM,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 441, + 183, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 183, + 453 + ], + "score": 1.0, + "content": "and WRM attacks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 259, + 471 + ], + "score": 1.0, + "content": "For the following results, we consider", + "type": "text" + }, + { + "bbox": [ + 259, + 459, + 275, + 469 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { \\mathrm { n n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 457, + 506, + 471 + ], + "score": 1.0, + "content": ", the class of neural nets defined in Theorem 1. Moreover,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 234, + 482 + ], + "score": 1.0, + "content": "we assume that the training loss", + "type": "text" + }, + { + "bbox": [ + 234, + 469, + 262, + 481 + ], + "score": 0.93, + "content": "\\ell ( \\hat { y } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "and its first-order derivative are 1-Lipschitz. Similar to Sinha", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 261, + 492 + ], + "score": 1.0, + "content": "et al. (2018), we assume the activation", + "type": "text" + }, + { + "bbox": [ + 262, + 483, + 269, + 490 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 480, + 380, + 492 + ], + "score": 1.0, + "content": "is smooth and its derivative", + "type": "text" + }, + { + "bbox": [ + 381, + 480, + 391, + 490 + ], + "score": 0.88, + "content": "\\sigma ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "is 1-Lipschitz. This class of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "activations include ELU (Clevert et al., 2015) and tanh functions but not the ReLU function. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "our numerical results in Table 1 from the Appendix suggest similar generalization performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 513, + 255, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 255, + 525 + ], + "score": 1.0, + "content": "between ELU and ReLU activations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 199, + 541 + ], + "score": 1.0, + "content": "Theorem 2. Consider", + "type": "text" + }, + { + "bbox": [ + 199, + 528, + 214, + 540 + ], + "score": 0.86, + "content": "\\mathcal { F } _ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 527, + 219, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 219, + 529, + 229, + 538 + ], + "score": 0.75, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 527, + 390, + 541 + ], + "score": 1.0, + "content": "in Theorem 1 and training loss function", + "type": "text" + }, + { + "bbox": [ + 391, + 529, + 397, + 538 + ], + "score": 0.42, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "satisfying the assumptions", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 351, + 552 + ], + "score": 1.0, + "content": "stated above. 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Then, for any", + "type": "text" + }, + { + "bbox": [ + 329, + 564, + 365, + 575 + ], + "score": 0.88, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 562, + 429, + 577 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 429, + 564, + 451, + 575 + ], + "score": 0.84, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "the following", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 573, + 327, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 287, + 588 + ], + "score": 1.0, + "content": "bound holds for the FGM margin loss of any", + "type": "text" + }, + { + "bbox": [ + 287, + 575, + 327, + 586 + ], + "score": 0.92, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 591, + 476, + 625 + ], + "lines": [ + { + "bbox": [ + 134, + 591, + 476, + 625 + ], + "spans": [ + { + "bbox": [ + 134, + 591, + 476, + 625 + ], + "score": 0.88, + "content": "\\begin{array} { r } { L _ { 0 } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) , } \\end{array}", + "type": "interline_equation", + "image_path": "4dfa8e7000269871e712b2d591a10c764c3edb7ad81fad4be3ed77e7f6821266.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 134, + 591, + 476, + 602.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 134, + 602.3333333333334, + 476, + 613.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 134, + 613.6666666666667, + 476, + 625.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 630, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "score": 0.69, + "content": "\\begin{array} { r } { \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) : = \\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + ( \\epsilon / \\kappa ) ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\right\\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "d13a266f750f4645802622beec38ccec9b17630427cfc888afb53d947d477897.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 282, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 282, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 282, + 676 + ], + "score": 1.0, + "content": "Proof. We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "Note that the above theorem assumes that the change rate for the loss function around test samples is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 136, + 711 + ], + "score": 1.0, + "content": "at least", + "type": "text" + }, + { + "bbox": [ + 137, + 701, + 143, + 709 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 699, + 361, + 711 + ], + "score": 1.0, + "content": ", which gives a baseline for measuring the attack power", + "type": "text" + }, + { + "bbox": [ + 362, + 701, + 367, + 709 + ], + "score": 0.34, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". In our numerical experiments, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "validate this assumption over standard image recognition tasks. Next, we generalize this result to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 418, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 418, + 732 + ], + "score": 1.0, + "content": "adversarial settings with PGM attack, i.e. the iterative version of FGM attack.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 394, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 496, + 397, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 496, + 397, + 504, + 405 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 663, + 504, + 673 + ], + "lines": [ + { + "bbox": [ + 496, + 664, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 496, + 664, + 504, + 673 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 427, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 429, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 429, + 96 + ], + "score": 1.0, + "content": "3 MARGIN-BASED ADVERSARIAL GENERALIZATION BOUNDS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "As previously discussed, generalization performance can be different between adversarial and non-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "score": 1.0, + "content": "adversarial settings. In this section, we provide generalization bounds for DNN classifiers under", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "adversarial attacks in terms of the spectral norms of the trained DNN’s weight matrices. The bounds", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "motivate regularizing these spectral norms in order to limit the DNN’s capacity and improve its", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 322, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 322, + 162 + ], + "score": 1.0, + "content": "generalization performance under adversarial attacks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 106, + 506, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "We use the PAC-Bayes framework (McAllester, 1999; 2003) to prove our main results. To derive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 179, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 434, + 190 + ], + "score": 1.0, + "content": "adversarial generalization error bounds for DNNs with smooth activation functions", + "type": "text" + }, + { + "bbox": [ + 434, + 181, + 441, + 188 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 179, + 505, + 190 + ], + "score": 1.0, + "content": ", we first extend", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "a recent result on the margin-based generalization bound for the ReLU activation function (Neyshabur", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 334, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 334, + 213 + ], + "score": 1.0, + "content": "et al., 2017a) to general 1-Lipschitz activation functions.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 167, + 505, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 504, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 200, + 228 + ], + "score": 1.0, + "content": "Theorem 1. Consider", + "type": "text" + }, + { + "bbox": [ + 200, + 215, + 295, + 227 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { n n } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathbf { W } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 214, + 496, + 228 + ], + "score": 1.0, + "content": "the class of d hidden-layer neural networks with", + "type": "text" + }, + { + "bbox": [ + 497, + 216, + 504, + 225 + ], + "score": 0.69, + "content": "h", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 300, + 238 + ], + "score": 1.0, + "content": "units per hidden-layer with 1-Lipschitz activation", + "type": "text" + }, + { + "bbox": [ + 300, + 229, + 307, + 236 + ], + "score": 0.59, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 226, + 348, + 238 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 348, + 226, + 386, + 238 + ], + "score": 0.92, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 226, + 442, + 238 + ], + "score": 1.0, + "content": ". 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Then, for any", + "type": "text" + }, + { + "bbox": [ + 469, + 296, + 503, + 307 + ], + "score": 0.9, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 294, + 506, + 309 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 340, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 204, + 320 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 307, + 228, + 318 + ], + "score": 0.82, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 305, + 260, + 320 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 260, + 307, + 300, + 318 + ], + "score": 0.91, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 305, + 340, + 320 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 294, + 506, + 320 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 323, + 453, + 357 + ], + "lines": [ + { + "bbox": [ + 157, + 323, + 453, + 357 + ], + "spans": [ + { + "bbox": [ + 157, + 323, + 453, + 357 + ], + "score": 0.93, + "content": "L _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { B ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) ,", + "type": "interline_equation", + "image_path": "2c3f8efecae8bdf8059492f399508d55a04cd934671cf4808d438540b9a69797.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 157, + 323, + 453, + 334.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 157, + 334.3333333333333, + 453, + 345.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 157, + 345.66666666666663, + 453, + 356.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 422, + 381 + ], + "lines": [ + { + "bbox": [ + 242, + 363, + 421, + 382 + ], + "spans": [ + { + "bbox": [ + 242, + 363, + 421, + 382 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\Phi ^ { \\mathrm { e r m } } ( f _ { \\mathbf { w } } ) : = \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 242, + 363, + 421, + 382 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 282, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 282, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 282, + 407 + ], + "score": 1.0, + "content": "Proof. We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 393, + 282, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 419, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "We now generalize this result to adversarial settings where the DNN’s performance is evaluated under", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "adversarial attacks. We prove three separate adversarial generalization error bounds for FGM, PGM,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 441, + 183, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 183, + 453 + ], + "score": 1.0, + "content": "and WRM attacks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 418, + 506, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 259, + 471 + ], + "score": 1.0, + "content": "For the following results, we consider", + "type": "text" + }, + { + "bbox": [ + 259, + 459, + 275, + 469 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { \\mathrm { n n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 457, + 506, + 471 + ], + "score": 1.0, + "content": ", the class of neural nets defined in Theorem 1. Moreover,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 234, + 482 + ], + "score": 1.0, + "content": "we assume that the training loss", + "type": "text" + }, + { + "bbox": [ + 234, + 469, + 262, + 481 + ], + "score": 0.93, + "content": "\\ell ( \\hat { y } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "and its first-order derivative are 1-Lipschitz. Similar to Sinha", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 261, + 492 + ], + "score": 1.0, + "content": "et al. (2018), we assume the activation", + "type": "text" + }, + { + "bbox": [ + 262, + 483, + 269, + 490 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 480, + 380, + 492 + ], + "score": 1.0, + "content": "is smooth and its derivative", + "type": "text" + }, + { + "bbox": [ + 381, + 480, + 391, + 490 + ], + "score": 0.88, + "content": "\\sigma ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "is 1-Lipschitz. This class of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "activations include ELU (Clevert et al., 2015) and tanh functions but not the ReLU function. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "our numerical results in Table 1 from the Appendix suggest similar generalization performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 513, + 255, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 255, + 525 + ], + "score": 1.0, + "content": "between ELU and ReLU activations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 457, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 199, + 541 + ], + "score": 1.0, + "content": "Theorem 2. Consider", + "type": "text" + }, + { + "bbox": [ + 199, + 528, + 214, + 540 + ], + "score": 0.86, + "content": "\\mathcal { F } _ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 527, + 219, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 219, + 529, + 229, + 538 + ], + "score": 0.75, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 527, + 390, + 541 + ], + "score": 1.0, + "content": "in Theorem 1 and training loss function", + "type": "text" + }, + { + "bbox": [ + 391, + 529, + 397, + 538 + ], + "score": 0.42, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "satisfying the assumptions", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 351, + 552 + ], + "score": 1.0, + "content": "stated above. 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Then, for any", + "type": "text" + }, + { + "bbox": [ + 329, + 564, + 365, + 575 + ], + "score": 0.88, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 562, + 429, + 577 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 429, + 564, + 451, + 575 + ], + "score": 0.84, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "the following", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 573, + 327, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 287, + 588 + ], + "score": 1.0, + "content": "bound holds for the FGM margin loss of any", + "type": "text" + }, + { + "bbox": [ + 287, + 575, + 327, + 586 + ], + "score": 0.92, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 527, + 506, + 588 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 591, + 476, + 625 + ], + "lines": [ + { + "bbox": [ + 134, + 591, + 476, + 625 + ], + "spans": [ + { + "bbox": [ + 134, + 591, + 476, + 625 + ], + "score": 0.88, + "content": "\\begin{array} { r } { L _ { 0 } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) , } \\end{array}", + "type": "interline_equation", + "image_path": "4dfa8e7000269871e712b2d591a10c764c3edb7ad81fad4be3ed77e7f6821266.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 134, + 591, + 476, + 602.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 134, + 602.3333333333334, + 476, + 613.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 134, + 613.6666666666667, + 476, + 625.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 630, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "score": 0.69, + "content": "\\begin{array} { r } { \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) : = \\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + ( \\epsilon / \\kappa ) ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\right\\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "d13a266f750f4645802622beec38ccec9b17630427cfc888afb53d947d477897.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 131, + 630, + 505, + 651 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 282, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 282, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 282, + 676 + ], + "score": 1.0, + "content": "Proof. We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 660, + 282, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "Note that the above theorem assumes that the change rate for the loss function around test samples is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 136, + 711 + ], + "score": 1.0, + "content": "at least", + "type": "text" + }, + { + "bbox": [ + 137, + 701, + 143, + 709 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 699, + 361, + 711 + ], + "score": 1.0, + "content": ", which gives a baseline for measuring the attack power", + "type": "text" + }, + { + "bbox": [ + 362, + 701, + 367, + 709 + ], + "score": 0.34, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". In our numerical experiments, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "validate this assumption over standard image recognition tasks. Next, we generalize this result to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 418, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 418, + 732 + ], + "score": 1.0, + "content": "adversarial settings with PGM attack, i.e. the iterative version of FGM attack.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 688, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 199, + 96 + ], + "score": 1.0, + "content": "Theorem 3. Consider", + "type": "text" + }, + { + "bbox": [ + 199, + 83, + 228, + 94 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { n n } , \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "and training loss function ` for which the assumptions in Theorem 2", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 415, + 106 + ], + "score": 1.0, + "content": "hold. We consider a PGM attack with noise power \u000f given Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 415, + 94, + 437, + 106 + ], + "score": 0.8, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 93, + 441, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 442, + 96, + 448, + 104 + ], + "score": 0.31, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "iterations for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 186, + 118 + ], + "score": 1.0, + "content": "attack, and stepsize", + "type": "text" + }, + { + "bbox": [ + 187, + 107, + 194, + 114 + ], + "score": 0.46, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 104, + 254, + 118 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 255, + 105, + 289, + 116 + ], + "score": 0.9, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 104, + 356, + 118 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 357, + 105, + 380, + 116 + ], + "score": 0.85, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "the following bound applies to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 262, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 221, + 129 + ], + "score": 1.0, + "content": "the PGM margin loss of any", + "type": "text" + }, + { + "bbox": [ + 222, + 116, + 262, + 127 + ], + "score": 0.91, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 132, + 486, + 166 + ], + "lines": [ + { + "bbox": [ + 124, + 132, + 486, + 166 + ], + "spans": [ + { + "bbox": [ + 124, + 132, + 486, + 166 + ], + "score": 0.94, + "content": "L _ { 0 } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { r d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) .", + "type": "interline_equation", + "image_path": "d306a56826d0e7ad984fa7002dccb30221035d090d09bcf4a30f9a96f4a1c5bf.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 124, + 132, + 486, + 143.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 124, + 143.33333333333334, + 486, + 154.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 124, + 154.66666666666669, + 486, + 166.00000000000003 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 332, + 184 + ], + "lines": [ + { + "bbox": [ + 104, + 167, + 335, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 167, + 168, + 189 + ], + "score": 1.0, + "content": "Here we define", + "type": "text" + }, + { + "bbox": [ + 168, + 171, + 220, + 184 + ], + "score": 0.93, + "content": "\\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 167, + 335, + 189 + ], + "score": 1.0, + "content": "as the following expression", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 104, + 190, + 500, + 225 + ], + "lines": [ + { + "bbox": [ + 104, + 190, + 500, + 225 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 500, + 225 + ], + "score": 0.86, + "content": "\\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\left( 1 + ( \\alpha / \\kappa ) { \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\right) \\right\\} ^ { 2 } \\sum _ { i = 1 } ^ { d } { \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } }", + "type": "interline_equation", + "image_path": "0ff186e4f785a9950d0f414c3d2f8a72571c1056f6477a93831335f5c9ee6a6c.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 104, + 190, + 500, + 201.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 104, + 201.66666666666666, + 500, + 213.33333333333331 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 104, + 213.33333333333331, + 500, + 224.99999999999997 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 229, + 508, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 133, + 248 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 230, + 349, + 247 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\varlimsup ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : = \\bigl ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\bigr ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 229, + 508, + 248 + ], + "score": 1.0, + "content": "provides an upper-bound on the Lips-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 242, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 174, + 261 + ], + "score": 1.0, + "content": "chitz constant of", + "type": "text" + }, + { + "bbox": [ + 175, + 247, + 237, + 260 + ], + "score": 0.93, + "content": "\\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 245, + 242, + 261 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 282, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 282, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 282, + 284 + ], + "score": 1.0, + "content": "Proof. We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 235, + 309 + ], + "score": 1.0, + "content": "In the above result, notice that if", + "type": "text" + }, + { + "bbox": [ + 235, + 295, + 341, + 308 + ], + "score": 0.92, + "content": "\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) / \\kappa < 1 / ( 2 \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "then for any number of gradient steps the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "PGM margin-based generalization bound will grow the FGM generalization error bound in Theorem", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 152, + 333 + ], + "score": 1.0, + "content": "2 by factor", + "type": "text" + }, + { + "bbox": [ + 153, + 318, + 271, + 332 + ], + "score": 0.9, + "content": "1 / \\big ( 1 - ( \\overline { { 2 \\alpha / \\kappa } } ) \\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 318, + 505, + 333 + ], + "score": 1.0, + "content": ". We next extend our adversarial generalization analysis to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 329, + 167, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 167, + 342 + ], + "score": 1.0, + "content": "WRM attacks.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 237, + 358 + ], + "score": 1.0, + "content": "Theorem 4. For neural net class", + "type": "text" + }, + { + "bbox": [ + 237, + 346, + 253, + 356 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 344, + 320, + 358 + ], + "score": 1.0, + "content": "and training loss", + "type": "text" + }, + { + "bbox": [ + 321, + 346, + 326, + 355 + ], + "score": 0.62, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "satisfying Theorem 2’s assumptions, consider", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 274, + 370 + ], + "score": 1.0, + "content": "a WRM attack with Lagrangian coefficient", + "type": "text" + }, + { + "bbox": [ + 275, + 357, + 282, + 366 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 356, + 365, + 370 + ], + "score": 1.0, + "content": "and Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 365, + 356, + 386, + 368 + ], + "score": 0.89, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 356, + 460, + 370 + ], + "score": 1.0, + "content": ". 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We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 303, + 546 + ], + "score": 1.0, + "content": "As discussed by Sinha et al. (2018), the condition", + "type": "text" + }, + { + "bbox": [ + 303, + 533, + 376, + 545 + ], + "score": 0.9, + "content": "\\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) < \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "for the actual Lipschitz constant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 117, + 557 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 545, + 152, + 556 + ], + "score": 0.92, + "content": "\\nabla \\ell \\circ f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "is in fact required to guarantee WRM’s convergence to the global solution. Notice that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 433, + 567 + ], + "score": 1.0, + "content": "the WRM generalization error bound in Theorem 4 is bounded by the product of", + "type": "text" + }, + { + "bbox": [ + 434, + 555, + 486, + 571 + ], + "score": 0.93, + "content": "\\frac { 1 } { \\lambda - \\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 290, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 290, + 581 + ], + "score": 1.0, + "content": "the FGM generalization bound in Theorem 2.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 597, + 422, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 423, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 423, + 613 + ], + "score": 1.0, + "content": "4 SPECTRAL NORMALIZATION OF CONVOLUTIONAL LAYERS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "To control the Lipschitz constant of our trained network, we need to ensure that the spectral norm", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 474, + 646 + ], + "score": 1.0, + "content": "associated with each linear operation in the network does not exceed some pre-specified", + "type": "text" + }, + { + "bbox": [ + 475, + 633, + 482, + 644 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 633, + 505, + 646 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "fully-connected layers (i.e. regular matrix multiplication), please see Appendix B. For a general class", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "of linear operations including convolution, Tsuzuku et al. (2018) propose to compute the operation’s", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "spectral norm through computing the gradient of the Euclidean norm of the operation’s output. Here,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "we leverage the deconvolution operation to further simplify and accelerate computing the spectral", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "norm of the convolution operation. Additionally, Sedghi et al. (2018) develop a method for computing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "all the singular values including the largest one, i.e. the spectral norm. While elegant, the method only", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "applies to convolution filters with stride 1 and zero-padding. However, in practice the normalization", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "factor depends on the stride size and padding scheme governing the convolution operation. Here", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 271, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 496, + 273, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 496, + 273, + 504, + 281 + ], + "score": 0.995, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 509, + 504, + 520 + ], + "lines": [ + { + "bbox": [ + 496, + 511, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 496, + 511, + 504, + 519 + ], + "score": 0.996, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 199, + 96 + ], + "score": 1.0, + "content": "Theorem 3. Consider", + "type": "text" + }, + { + "bbox": [ + 199, + 83, + 228, + 94 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { n n } , \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "and training loss function ` for which the assumptions in Theorem 2", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 415, + 106 + ], + "score": 1.0, + "content": "hold. We consider a PGM attack with noise power \u000f given Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 415, + 94, + 437, + 106 + ], + "score": 0.8, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 93, + 441, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 442, + 96, + 448, + 104 + ], + "score": 0.31, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "iterations for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 186, + 118 + ], + "score": 1.0, + "content": "attack, and stepsize", + "type": "text" + }, + { + "bbox": [ + 187, + 107, + 194, + 114 + ], + "score": 0.46, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 104, + 254, + 118 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 255, + 105, + 289, + 116 + ], + "score": 0.9, + "content": "\\eta , \\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 104, + 356, + 118 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 357, + 105, + 380, + 116 + ], + "score": 0.85, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "the following bound applies to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 262, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 221, + 129 + ], + "score": 1.0, + "content": "the PGM margin loss of any", + "type": "text" + }, + { + "bbox": [ + 222, + 116, + 262, + 127 + ], + "score": 0.91, + "content": "f _ { \\mathbf { w } } \\in \\mathcal { F } _ { n n }", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 506, + 129 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 132, + 486, + 166 + ], + "lines": [ + { + "bbox": [ + 124, + 132, + 486, + 166 + ], + "spans": [ + { + "bbox": [ + 124, + 132, + 486, + 166 + ], + "score": 0.94, + "content": "L _ { 0 } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\bigg ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } ) + d \\log \\frac { r d n \\log M } { \\eta } } { \\gamma ^ { 2 } n } } \\bigg ) .", + "type": "interline_equation", + "image_path": "d306a56826d0e7ad984fa7002dccb30221035d090d09bcf4a30f9a96f4a1c5bf.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 124, + 132, + 486, + 143.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 124, + 143.33333333333334, + 486, + 154.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 124, + 154.66666666666669, + 486, + 166.00000000000003 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 332, + 184 + ], + "lines": [ + { + "bbox": [ + 104, + 167, + 335, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 167, + 168, + 189 + ], + "score": 1.0, + "content": "Here we define", + "type": "text" + }, + { + "bbox": [ + 168, + 171, + 220, + 184 + ], + "score": 0.93, + "content": "\\Phi _ { \\epsilon , \\kappa , r , \\alpha } ^ { \\mathrm { p g m } } ( f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 167, + 335, + 189 + ], + "score": 1.0, + "content": "as the following expression", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 167, + 335, + 189 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 104, + 190, + 500, + 225 + ], + "lines": [ + { + "bbox": [ + 104, + 190, + 500, + 225 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 500, + 225 + ], + "score": 0.86, + "content": "\\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\left( 1 + ( \\alpha / \\kappa ) { \\frac { 1 - 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We defer the proof to the Appendix.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 270, + 282, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 235, + 309 + ], + "score": 1.0, + "content": "In the above result, notice that if", + "type": "text" + }, + { + "bbox": [ + 235, + 295, + 341, + 308 + ], + "score": 0.92, + "content": "\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) / \\kappa < 1 / ( 2 \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "then for any number of gradient steps the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "PGM margin-based generalization bound will grow the FGM generalization error bound in Theorem", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 152, + 333 + ], + "score": 1.0, + "content": "2 by factor", + "type": "text" + }, + { + "bbox": [ + 153, + 318, + 271, + 332 + ], + "score": 0.9, + "content": "1 / \\big ( 1 - ( \\overline { { 2 \\alpha / \\kappa } } ) \\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 318, + 505, + 333 + ], + "score": 1.0, + "content": ". 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This allows us to efficiently performs matrix multiplication with", + "type": "text" + }, + { + "bbox": [ + 412, + 320, + 430, + 331 + ], + "score": 0.89, + "content": "W ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "without explicitly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 159, + 344 + ], + "score": 1.0, + "content": "constructing", + "type": "text" + }, + { + "bbox": [ + 159, + 331, + 171, + 342 + ], + "score": 0.57, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 331, + 302, + 344 + ], + "score": 1.0, + "content": ". Therefore we can approximate", + "type": "text" + }, + { + "bbox": [ + 302, + 331, + 327, + 343 + ], + "score": 0.92, + "content": "\\sigma ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "using a modified version of power iteration", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "(Algorithm 1), wrapping the appropriate stride size and padding arguments into the convolution and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 302, + 366 + ], + "score": 1.0, + "content": "convolution transpose operations. After obtaining", + "type": "text" + }, + { + "bbox": [ + 302, + 353, + 327, + 365 + ], + "score": 0.92, + "content": "\\sigma ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 353, + 381, + 366 + ], + "score": 1.0, + "content": ", we compute", + "type": "text" + }, + { + "bbox": [ + 381, + 353, + 402, + 364 + ], + "score": 0.9, + "content": "W _ { \\mathrm { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "in the same manner as for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "the fully-connected layers. 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Algorithm1 Convolutional power iteration
Initialize ü with a random vector matching the shape of the convolution input
for t = 0,...,T-1do
ν ← conv(W,u)/llconv(W,u)ll2
ü ← conv_transpose(W,v)/llconv_transpose(W,v)ll2
end for
σ ←v· conv(W,u)
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We show that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "spectral normalization improves both test accuracy and generalization for a variety of adversarial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 323, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 323, + 639 + ], + "score": 1.0, + "content": "training schemes, datasets, and network architectures.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "All experiments are implemented in TensorFlow (Abadi et al., 2016). For each experiment, we cross", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 205, + 667 + ], + "score": 1.0, + "content": "validate 4 to 6 values of", + "type": "text" + }, + { + "bbox": [ + 205, + 655, + 212, + 666 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "(see (9)) using a fixed validation set of 500 samples. 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Algorithm1 Convolutional power iteration
Initialize ü with a random vector matching the shape of the convolution input
for t = 0,...,T-1do
ν ← conv(W,u)/llconv(W,u)ll2
ü ← conv_transpose(W,v)/llconv_transpose(W,v)ll2
end for
σ ←v· conv(W,u)
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The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 253, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 253, + 733 + ], + "score": 1.0, + "content": "code will be made readily available.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 644, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "We first demonstrate the effect of the proposed spectral normalization approach on the final DNN", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 208, + 128 + ], + "score": 1.0, + "content": "weights by comparing the", + "type": "text" + }, + { + "bbox": [ + 209, + 116, + 219, + 126 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 115, + 289, + 128 + ], + "score": 1.0, + "content": "norm of the input", + "type": "text" + }, + { + "bbox": [ + 289, + 117, + 297, + 125 + ], + "score": 0.37, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 115, + 376, + 128 + ], + "score": 1.0, + "content": "to that of the output", + "type": "text" + }, + { + "bbox": [ + 376, + 115, + 403, + 127 + ], + "score": 0.92, + "content": "f _ { \\mathbf { w } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 115, + 507, + 128 + ], + "score": 1.0, + "content": ". 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At", + "type": "text" + }, + { + "bbox": [ + 426, + 160, + 453, + 171 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 159, + 506, + 172 + ], + "score": 1.0, + "content": ", the gain of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "the network cannot be greater than 1, which is consistent with what we observe. Additionally, we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "score": 1.0, + "content": "provide a comparison of our method to that of Miyato et al. 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Figure 5 shows that even without adversarial training, AlexNet with SN becomes", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "more robust to FGM, PGM, and WRM attacks. Adversarial training improves adversarial robustness", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 113, + 84, + 455, + 93 + ], + "lines": [ + { + "bbox": [ + 111, + 83, + 457, + 95 + ], + "spans": [ + { + "bbox": [ + 111, + 83, + 457, + 95 + ], + "score": 1.0, + "content": ".1 VALIDATION OF SPECTRAL NORMALIZATION IMPLEMENTATION AND BOUNDS", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "We first demonstrate the effect of the proposed spectral normalization approach on the final DNN", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 208, + 128 + ], + "score": 1.0, + "content": "weights by comparing the", + "type": "text" + }, + { + "bbox": [ + 209, + 116, + 219, + 126 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 115, + 289, + 128 + ], + "score": 1.0, + "content": "norm of the input", + "type": "text" + }, + { + "bbox": [ + 289, + 117, + 297, + 125 + ], + "score": 0.37, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 115, + 376, + 128 + ], + "score": 1.0, + "content": "to that of the output", + "type": "text" + }, + { + "bbox": [ + 376, + 115, + 403, + 127 + ], + "score": 0.92, + "content": "f _ { \\mathbf { w } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 115, + 507, + 128 + ], + "score": 1.0, + "content": ". As shown in Figure 2(a),", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 237, + 140 + ], + "score": 1.0, + "content": "without spectral normalization (", + "type": "text" + }, + { + "bbox": [ + 237, + 127, + 268, + 138 + ], + "score": 0.9, + "content": "\\beta = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "in (9)), the norm gain can be large. 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As we decrease", + "type": "text" + }, + { + "bbox": [ + 439, + 149, + 447, + 159 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 148, + 506, + 161 + ], + "score": 1.0, + "content": ", however, we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 426, + 172 + ], + "score": 1.0, + "content": "produce more constrained networks, resulting in a decrease in norm gain. At", + "type": "text" + }, + { + "bbox": [ + 426, + 160, + 453, + 171 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 159, + 506, + 172 + ], + "score": 1.0, + "content": ", the gain of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "the network cannot be greater than 1, which is consistent with what we observe. Additionally, we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "score": 1.0, + "content": "provide a comparison of our method to that of Miyato et al. (2018) in Appendix A.1, empirically", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "demonstrating that Miyato et al.’s method does not properly control the spectral norm of convolutional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 322, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 322, + 215 + ], + "score": 1.0, + "content": "layers, resulting in worse generalization performance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 104, + 507, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 220, + 504, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 214, + 233 + ], + "score": 1.0, + "content": "Figure 2(b) shows that the", + "type": "text" + }, + { + "bbox": [ + 214, + 220, + 224, + 231 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "norms of the gradients with respect to the training samples are nicely", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "distributed after spectral normalization. Additionally, this figure suggests that the minimum gradient", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 116, + 253 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 242, + 206, + 254 + ], + "score": 1.0, + "content": "-norm assumption (the", + "type": "text" + }, + { + "bbox": [ + 207, + 244, + 214, + 252 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "condition in Theorems 2 and 3) holds for spectrally-normalized networks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 219, + 505, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 273 + ], + "score": 1.0, + "content": "The first column of Figure 3 shows that, as observed by Bartlett et al. (2017), AlexNet trained using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "ERM generates similar margin distributions for both random and true labels on CIFAR10 unless we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "normalize the margins appropriately. We see that even without further correction, ERM training with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "SN allows AlexNet to have distinguishable performance between the two datasets. This observation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "suggests that SN as a regularization scheme enforces the generalization error bounds shown for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "spectrally-normalized DNNs by Bartlett et al. (2017) and Neyshabur et al. (2017a). 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As demonstrated by the other columns in Figure 3, a smaller normalization factor", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "results in larger normalized margin values and much tighter margin-based generalization bounds (a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 357, + 439, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 143, + 370 + ], + "score": 1.0, + "content": "factor of", + "type": "text" + }, + { + "bbox": [ + 143, + 357, + 159, + 368 + ], + "score": 0.88, + "content": "1 0 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 357, + 258, + 370 + ], + "score": 1.0, + "content": "for ERM and a factor of", + "type": "text" + }, + { + "bbox": [ + 258, + 357, + 274, + 368 + ], + "score": 0.88, + "content": "1 0 ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 357, + 439, + 370 + ], + "score": 1.0, + "content": "for FGM and PGM) (see Theorems 1-4).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 257, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 126, + 510, + 286, + 530 + ], + "lines": [ + { + "bbox": [ + 125, + 509, + 288, + 521 + ], + "spans": [ + { + "bbox": [ + 125, + 509, + 137, + 521 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 137, + 510, + 147, + 519 + ], + "score": 0.84, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 509, + 253, + 521 + ], + "score": 1.0, + "content": "norm gain due to the network", + "type": "text" + }, + { + "bbox": [ + 253, + 510, + 259, + 520 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 509, + 288, + 521 + ], + "score": 1.0, + "content": "trained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 125, + 519, + 173, + 531 + ], + "spans": [ + { + "bbox": [ + 125, + 519, + 173, + 531 + ], + "score": 1.0, + "content": "using ERM.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 125, + 509, + 288, + 531 + ] + }, + { + "type": "image", + "bbox": [ + 327, + 384, + 478, + 495 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 327, + 384, + 478, + 495 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 327, + 384, + 478, + 495 + ], + "spans": [ + { + "bbox": [ + 327, + 384, + 478, + 495 + ], + "score": 0.952, + "type": "image", + "image_path": "2d10a59f5082d7d00e28922000204f0b02852c3f5bbd768c93322e043f96d3e2.jpg" + } + ] + } + ], + "index": 25.0, + "virtual_lines": [ + { + "bbox": [ + 327, + 384, + 478, + 439.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 327, + 439.5, + 478, + 495.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 324, + 502, + 486, + 533 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 324, + 502, + 486, + 513 + ], + "spans": [ + { + "bbox": [ + 324, + 502, + 406, + 513 + ], + "score": 1.0, + "content": "(b) Distributions of the", + "type": "text" + }, + { + "bbox": [ + 407, + 502, + 416, + 512 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 502, + 486, + 513 + ], + "score": 1.0, + "content": "norms of the gradi-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 323, + 512, + 486, + 523 + ], + "spans": [ + { + "bbox": [ + 323, + 512, + 486, + 523 + ], + "score": 1.0, + "content": "ents with respect to training samples. 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(2016) can be observed even", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "score": 1.0, + "content": "for adversarial training methods. Figure 4 shows how the FGM, PGM, or WRM adversarial training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "schemes only slightly delay the rate at which AlexNet fits random labels on CIFAR10, and therefore", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "the generalization gap can be quite large without proper regularization. After introducing spectral", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "normalization, however, we see that the network has a much harder time fitting both the random and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "true labels. With the proper amount of SN (chosen via cross validation), we can obtain networks that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "struggle to fit random labels while still obtaining the same or better test performance on true labels.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 616, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We also observe that training schemes regularized with SN result in networks more robust to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "adversarial attacks. 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While our work does not provide sample complexity lower-bounds, we study the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "broader function class of DNNs where we provide upper-bounds on adversarial generalization error", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 617, + 434, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 434, + 628 + ], + "score": 1.0, + "content": "and suggest an explicit regularization scheme for adversarial training over DNNs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "Generalization in deep learning has been a topic of great interest in machine learning (Zhang et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "2016). 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While all of these works seek to understand", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "the robustness properties of different classification function classes, unlike our work they do not focus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 422, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 422, + 523 + ], + "score": 1.0, + "content": "on the generalization aspects of learning over DNNs under adversarial attacks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 412, + 505, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Concerning the generalization aspect of adversarial training, Sinha et al. (2018) provides optimization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "and generalization guarantees for WRM under the assumptions discussed after Theorem 4. However,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "their generalization guarantee only applies to the Wasserstein cost function, which is different from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "the 0-1 or margin loss and does not explicitly suggest a regularization scheme. In a recent related", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "work, Schmidt et al. (2018) numerically shows the wide generalization gap in PGM adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "training and theoretically establishes lower-bounds on the sample complexity of linear classifiers in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Gaussian settings. While our work does not provide sample complexity lower-bounds, we study the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "broader function class of DNNs where we provide upper-bounds on adversarial generalization error", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 617, + 434, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 434, + 628 + ], + "score": 1.0, + "content": "and suggest an explicit regularization scheme for adversarial training over DNNs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 528, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "Generalization in deep learning has been a topic of great interest in machine learning (Zhang et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "2016). In addition to margin-based bounds (Bartlett et al., 2017; Neyshabur et al., 2017a), various", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "other tools including VC dimension (Anthony & Bartlett, 2009), norm-based capacity scores (Bartlett", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "& Mendelson, 2002; Neyshabur et al., 2015), and flatness of local minima (Keskar et al., 2016;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "Neyshabur et al., 2017b) have been used to analyze generalization properties of DNNs. Recently,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "Arora et al. (2018) introduced a compression approach to further improve the margin-based bounds", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "presented by Bartlett et al. (2017); Neyshabur et al. (2017a). The PAC-Bayes bound has also been", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "considered and computed by Dziugaite & Roy (2017), resulting in non-vacuous bounds for the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 172, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 172, + 732 + ], + "score": 1.0, + "content": "MNIST dataset.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 633, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 218, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 219, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 219, + 96 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "We are grateful for support under the National Science Foundation grant under CCF-1563098, and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "the Center for Science of Information (CSoI), an NSF Science and Technology Center under grant", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 212, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 212, + 139 + ], + "score": 1.0, + "content": "agreement CCF-0939370.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 156, + 176, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 176, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 176, + 170 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 161, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 114, + 182, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 114, + 182, + 506, + 199 + ], + "score": 1.0, + "content": "Andy Davis, Jeffrey Dean, Matthieu Devin, et al. 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The amount of spectral normalization was selected from 4-6", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 145, + 181 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 169, + 154, + 180 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "via cross validation on 500 samples. 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DatasetArchitectureTrainingTrain accTest accTrain acc (SN)Test acc (SN)
CIFAR10AlexNetERM1.000.791.000.79
CIFAR10AlexNetFGM l20.980.540.930.63
CIFAR10AlexNetFGM lo1.000.510.670.56
CIFAR10AlexNetPGM l20.990.500.920.62
CIFAR10AlexNetPGMlo0.990.440.860.54
CIFAR10AlexNetWRM1.000.610.760.65
CIFAR10ELU-AlexNetERM1.000.791.000.79
CIFAR10ELU-AlexNetFGM l20.970.520.680.60
CIFAR10ELU-AlexNetPGMl20.980.530.880.61
CIFAR10ELU-AlexNetWRM1.000.601.000.60
CIFAR10InceptionERM1.000.851.000.86
CIFAR10InceptionPGM l20.990.531.000.58
CIFAR10InceptionPGM lo0.980.480.620.56
CIFAR10InceptionWRM1.000.661.000.67
CIFAR101-layer MLPERM0.980.490.680.53
CIFAR101-layer MLPFGM l20.600.360.600.46
CIFAR101-layer MLPPGM l20.570.360.550.46
CIFAR101-layer MLPWRM0.600.410.620.50
CIFAR102-layer MLPERM0.990.510.790.56
CIFAR102-layer MLPFGM l20.570.360.660.49
CIFAR102-layer MLPPGM l20.930.350.660.48
CIFAR102-layer MLPWRM0.870.350.730.52
CIFAR10ResNetERM1.000.801.000.83
CIFAR10ResNetPGM l20.990.491.000.55
CIFAR10ResNetPGM lo0.980.440.720.53
CIFAR10ResNetWRM1.000.631.000.66
MNISTELU-NetERM1.000.991.000.99*
MNISTELU-NetFGM l20.980.971.000.97
MNISTELU-NetPGM l20.990.971.000.97
MNISTELU-NetWRM0.950.920.950.93
MNIST1-layer MLPERM1.000.981.000.98*
MNIST1-layer MLPFGM l20.880.881.000.96
MNIST1-layer MLPPGM l21.000.961.000.96
MNIST1-layer MLPWRM0.920.880.920.88
MNIST2-layer MLPERM1.000.981.000.98
MNIST2-layer MLPFGM l20.970.911.000.96
MNIST2-layer MLPPGM l21.000.961.000.97
MNIST2-layer MLPWRM0.970.880.980.90
SVHNAlexNetERM1.000.931.000.93*
SVHNAlexNetFGM l20.970.760.950.83
SVHNAlexNetPGM l21.000.780.850.81
SVHNAlexNetWRM1.000.830.870.84
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DatasetArchitectureTrainingTrain accTest accTrain acc (SN)Test acc (SN)
CIFAR10AlexNetERM1.000.791.000.79
CIFAR10AlexNetFGM l20.980.540.930.63
CIFAR10AlexNetFGM lo1.000.510.670.56
CIFAR10AlexNetPGM l20.990.500.920.62
CIFAR10AlexNetPGMlo0.990.440.860.54
CIFAR10AlexNetWRM1.000.610.760.65
CIFAR10ELU-AlexNetERM1.000.791.000.79
CIFAR10ELU-AlexNetFGM l20.970.520.680.60
CIFAR10ELU-AlexNetPGMl20.980.530.880.61
CIFAR10ELU-AlexNetWRM1.000.601.000.60
CIFAR10InceptionERM1.000.851.000.86
CIFAR10InceptionPGM l20.990.531.000.58
CIFAR10InceptionPGM lo0.980.480.620.56
CIFAR10InceptionWRM1.000.661.000.67
CIFAR101-layer MLPERM0.980.490.680.53
CIFAR101-layer MLPFGM l20.600.360.600.46
CIFAR101-layer MLPPGM l20.570.360.550.46
CIFAR101-layer MLPWRM0.600.410.620.50
CIFAR102-layer MLPERM0.990.510.790.56
CIFAR102-layer MLPFGM l20.570.360.660.49
CIFAR102-layer MLPPGM l20.930.350.660.48
CIFAR102-layer MLPWRM0.870.350.730.52
CIFAR10ResNetERM1.000.801.000.83
CIFAR10ResNetPGM l20.990.491.000.55
CIFAR10ResNetPGM lo0.980.440.720.53
CIFAR10ResNetWRM1.000.631.000.66
MNISTELU-NetERM1.000.991.000.99*
MNISTELU-NetFGM l20.980.971.000.97
MNISTELU-NetPGM l20.990.971.000.97
MNISTELU-NetWRM0.950.920.950.93
MNIST1-layer MLPERM1.000.981.000.98*
MNIST1-layer MLPFGM l20.880.881.000.96
MNIST1-layer MLPPGM l21.000.961.000.96
MNIST1-layer MLPWRM0.920.880.920.88
MNIST2-layer MLPERM1.000.981.000.98
MNIST2-layer MLPFGM l20.970.911.000.96
MNIST2-layer MLPPGM l21.000.961.000.97
MNIST2-layer MLPWRM0.970.880.980.90
SVHNAlexNetERM1.000.931.000.93*
SVHNAlexNetFGM l20.970.760.950.83
SVHNAlexNetPGM l21.000.780.850.81
SVHNAlexNetWRM1.000.830.870.84
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DatasetArchitectureTrainingno SN runtime SN runtimeratio
CIFAR10AlexNetERM229 s283 s1.24
CIFAR10AlexNetFGM l2407 s463 s1.14
CIFAR10AlexNetFGM loo408 s465 s1.14
CIFAR10AlexNetPGM l22917 s3077 s1.05
CIFAR10AlexNetPGM lo2896 s3048 s1.05
CIFAR10AlexNetWRM3076 s3151 s1.02
CIFAR10ELU-AlexNetERM231 s283 s1.23
CIFAR10ELU-AlexNetFGM l2410 s466 s1.14
CIFAR10ELU-AlexNetPGM l22939 s3093 s1.05
CIFAR10ELU-AlexNetWRM3094 s3150 s1.02
CIFAR10InceptionERM632 s734 s1.16
CIFAR10InceptionPGM l29994 s6082 s0.61
CIFAR10InceptionPGM lo9948 s6063 s0.61
CIFAR10InceptionWRM10247 s6356 s0.62
CIFAR101-layer MLPERM22 s31 s1.42
CIFAR101-layer MLPFGM l225 s35s1.43
CIFAR101-layer MLPPGM l279 s93 s1.18
CIFAR101-layer MLPWRM73 s86 s1.18
CIFAR102-layer MLPERM23 s37 s1.59
CIFAR102-layer MLPFGM l227 s41 s1.51
CIFAR102-layer MLPPGMl291 s108 s1.19
CIFAR102-layer MLPWRM85 s103 s1.21
CIFAR10ResNetERM315 s547 s1.73
CIFAR10ResNetPGM l22994 s3300 s1.10
CIFAR10ResNetPGM lo2980 s3300 s1.11
CIFAR10ResNetWRM3187 s3457 s1.08
MNISTELU-NetERM55 s97s1.76
MNISTELU-NetFGM l291s136 s1.49
MNISTELU-NetPGM l2614 s676 s1.10
MNISTELU-NetWRM635 s670 s1.06
MNIST1-layer MLPERM15 s24 s1.60
MNIST1-layer MLPFGM l217 s27s1.57
MNIST1-layer MLPPGM l257s71 s1.24
MNIST1-layer MLPWRM51 s63 s1.24
MNIST2-layer MLPERM17 s31 s1.84
MNIST2-layer MLPFGM l220 s35 s1.77
MNIST2-layer MLPPGM l267 s89 s1.32
MNIST2-layer MLPWRM62 s81 s1.30
SVHNAlexNetERM334 s412 s1.23
SVHNAlexNetFGM l2596 s676 s1.13
SVHNAlexNetPGM l24270 s4495 s1.05
SVHNAlexNetWRM4501 s4572 s1.02
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DatasetArchitectureTrainingno SN runtime SN runtimeratio
CIFAR10AlexNetERM229 s283 s1.24
CIFAR10AlexNetFGM l2407 s463 s1.14
CIFAR10AlexNetFGM loo408 s465 s1.14
CIFAR10AlexNetPGM l22917 s3077 s1.05
CIFAR10AlexNetPGM lo2896 s3048 s1.05
CIFAR10AlexNetWRM3076 s3151 s1.02
CIFAR10ELU-AlexNetERM231 s283 s1.23
CIFAR10ELU-AlexNetFGM l2410 s466 s1.14
CIFAR10ELU-AlexNetPGM l22939 s3093 s1.05
CIFAR10ELU-AlexNetWRM3094 s3150 s1.02
CIFAR10InceptionERM632 s734 s1.16
CIFAR10InceptionPGM l29994 s6082 s0.61
CIFAR10InceptionPGM lo9948 s6063 s0.61
CIFAR10InceptionWRM10247 s6356 s0.62
CIFAR101-layer MLPERM22 s31 s1.42
CIFAR101-layer MLPFGM l225 s35s1.43
CIFAR101-layer MLPPGM l279 s93 s1.18
CIFAR101-layer MLPWRM73 s86 s1.18
CIFAR102-layer MLPERM23 s37 s1.59
CIFAR102-layer MLPFGM l227 s41 s1.51
CIFAR102-layer MLPPGMl291 s108 s1.19
CIFAR102-layer MLPWRM85 s103 s1.21
CIFAR10ResNetERM315 s547 s1.73
CIFAR10ResNetPGM l22994 s3300 s1.10
CIFAR10ResNetPGM lo2980 s3300 s1.11
CIFAR10ResNetWRM3187 s3457 s1.08
MNISTELU-NetERM55 s97s1.76
MNISTELU-NetFGM l291s136 s1.49
MNISTELU-NetPGM l2614 s676 s1.10
MNISTELU-NetWRM635 s670 s1.06
MNIST1-layer MLPERM15 s24 s1.60
MNIST1-layer MLPFGM l217 s27s1.57
MNIST1-layer MLPPGM l257s71 s1.24
MNIST1-layer MLPWRM51 s63 s1.24
MNIST2-layer MLPERM17 s31 s1.84
MNIST2-layer MLPFGM l220 s35 s1.77
MNIST2-layer MLPPGM l267 s89 s1.32
MNIST2-layer MLPWRM62 s81 s1.30
SVHNAlexNetERM334 s412 s1.23
SVHNAlexNetFGM l2596 s676 s1.13
SVHNAlexNetPGM l24270 s4495 s1.05
SVHNAlexNetWRM4501 s4572 s1.02
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DatasetArchitectureTrainingproposed SN runtime Miyato SN runtime
CIFAR10AlexNetERM1.11
CIFAR10AlexNetFGM l21.06
CIFAR10AlexNetFGMloo1.11
CIFAR10AlexNetPGM l21.01
CIFAR10AlexNetPGM lo1.11
CIFAR10AlexNetWRM1.02
CIFAR10InceptionERM0.98
CIFAR10InceptionPGM l21.04
CIFAR10InceptionPGM loo1.06
CIFAR10InceptionWRM1.03
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DatasetArchitectureTrainingproposed SN runtime Miyato SN runtime
CIFAR10AlexNetERM1.11
CIFAR10AlexNetFGM l21.06
CIFAR10AlexNetFGMloo1.11
CIFAR10AlexNetPGM l21.01
CIFAR10AlexNetPGM lo1.11
CIFAR10AlexNetWRM1.02
CIFAR10InceptionERM0.98
CIFAR10InceptionPGM l21.04
CIFAR10InceptionPGM loo1.06
CIFAR10InceptionWRM1.03
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(2018):", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 474, + 506, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 524, + 367, + 538 + ], + "lines": [ + { + "bbox": [ + 242, + 524, + 367, + 538 + ], + "spans": [ + { + "bbox": [ + 242, + 524, + 367, + 538 + ], + "score": 0.92, + "content": "W _ { \\mathrm { S N } } = W / \\operatorname* { m a x } ( 1 , \\sigma ( W ) / \\beta ) ,", + "type": "interline_equation", + "image_path": "fa45b9adad6d632c2f957c14da0851003d412fba397a6bd2ce7a27d051bb6e26.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 242, + 524, + 367, + 538 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 446, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 446, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 446, + 563 + ], + "score": 1.0, + "content": "which we observe to result in faster training in practice for supervised learning tasks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 548, + 446, + 563 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 577, + 169, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 171, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 171, + 594 + ], + "score": 1.0, + "content": "C PROOFS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 601, + 229, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 231, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 231, + 615 + ], + "score": 1.0, + "content": "C.1 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 409, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 409, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 409, + 636 + ], + "score": 1.0, + "content": "First let us quote the following two lemmas from (Neyshabur et al., 2017a).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 622, + 409, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 104, + 636, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 304, + 652 + ], + "score": 1.0, + "content": "Lemma 1 (Neyshabur et al. (2017a)). Consider", + "type": "text" + }, + { + "bbox": [ + 304, + 638, + 396, + 650 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { n n } = \\{ f _ { \\mathbf { w } } : \\mathbf { w } \\in \\mathcal { W } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 636, + 506, + 652 + ], + "score": 1.0, + "content": "as the class of neural nets", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 236, + 662 + ], + "score": 1.0, + "content": "parameterized by w where each", + "type": "text" + }, + { + "bbox": [ + 236, + 649, + 249, + 660 + ], + "score": 0.88, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 648, + 296, + 662 + ], + "score": 1.0, + "content": "maps input", + "type": "text" + }, + { + "bbox": [ + 296, + 650, + 325, + 659 + ], + "score": 0.88, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 648, + 336, + 662 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 336, + 650, + 351, + 659 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 648, + 371, + 662 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 371, + 650, + 380, + 660 + ], + "score": 0.77, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 648, + 507, + 662 + ], + "score": 1.0, + "content": "be a distribution on parameter", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 409, + 672 + ], + "score": 1.0, + "content": "vector chosen independently from the n training samples. Then, for each", + "type": "text" + }, + { + "bbox": [ + 409, + 660, + 436, + 671 + ], + "score": 0.85, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 140, + 684 + ], + "score": 1.0, + "content": "at least", + "type": "text" + }, + { + "bbox": [ + 140, + 671, + 166, + 682 + ], + "score": 0.83, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 670, + 392, + 684 + ], + "score": 1.0, + "content": "for any w and any random perturbation u satisfying", + "type": "text" + }, + { + "bbox": [ + 392, + 671, + 505, + 684 + ], + "score": 0.88, + "content": "\\operatorname* { P r } _ { \\mathbf { u } } \\left( \\operatorname* { m a x } _ { \\mathbf { x } \\in \\mathcal { X } } \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) - \\right.", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 683, + 228, + 697 + ], + "spans": [ + { + "bbox": [ + 107, + 683, + 192, + 697 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f _ { \\mathbf { w } } ( \\mathbf { x } ) \\lVert _ { \\infty } \\leq \\frac { \\gamma } { 4 } ) \\geq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 684, + 228, + 697 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 636, + 507, + 697 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 702, + 410, + 736 + ], + "lines": [ + { + "bbox": [ + 200, + 702, + 410, + 736 + ], + "spans": [ + { + "bbox": [ + 200, + 702, + 410, + 736 + ], + "score": 0.94, + "content": "L _ { 0 } ( f _ { \\mathbf { w } } ) \\leq \\widehat { L } _ { \\gamma } ( f _ { \\mathbf { w } } ) + 4 \\sqrt { \\frac { K L ( P _ { \\mathbf { w } + \\mathbf { u } } \\| Q ) + \\log \\frac { 6 n } { \\eta } } { n - 1 } } .", + "type": "interline_equation", + "image_path": "2032f981072cad7dea010ab4eb3a7554675ba428a604ffc9b987dc40106ff5a7.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 702, + 410, + 719.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 200, + 719.0, + 410, + 736.0 + ], + "spans": [], + "index": 31 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 308, + 96 + ], + "score": 1.0, + "content": "Lemma 2 (Neyshabur et al. (2017a)). Consider a", + "type": "text" + }, + { + "bbox": [ + 308, + 83, + 314, + 92 + ], + "score": 0.58, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 81, + 382, + 96 + ], + "score": 1.0, + "content": "-layer neural net", + "type": "text" + }, + { + "bbox": [ + 383, + 83, + 395, + 94 + ], + "score": 0.86, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "with 1-Lipschitz activation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 142, + 107 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 96, + 150, + 104 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 93, + 177, + 107 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 178, + 93, + 216, + 105 + ], + "score": 0.9, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 93, + 360, + 107 + ], + "score": 1.0, + "content": ". Then for any norm-bounded input", + "type": "text" + }, + { + "bbox": [ + 360, + 94, + 403, + 106 + ], + "score": 0.91, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "and weight perturbation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 377, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 104, + 199, + 117 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbf { \\dot { u } } : \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 104, + 377, + 118 + ], + "score": 1.0, + "content": ", we have the following perturbation bound:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 120, + 420, + 154 + ], + "lines": [ + { + "bbox": [ + 190, + 120, + 420, + 154 + ], + "spans": [ + { + "bbox": [ + 190, + 120, + 420, + 154 + ], + "score": 0.93, + "content": "\\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) - f _ { \\mathbf { w } } ( \\mathbf { x } ) \\| _ { 2 } \\leq e B \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } .", + "type": "interline_equation", + "image_path": "87b4d53b6814707767de659c98397b67d7e33d71416fb1722cd8b32231162e97.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 120, + 420, + 137.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 190, + 137.0, + 420, + 154.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 161, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 226, + 178 + ], + "score": 1.0, + "content": "To prove Theorem 1, consider", + "type": "text" + }, + { + "bbox": [ + 226, + 164, + 239, + 176 + ], + "score": 0.89, + "content": "f _ { \\widetilde { \\mathbf { w } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 161, + 293, + 178 + ], + "score": 1.0, + "content": "with weights", + "type": "text" + }, + { + "bbox": [ + 294, + 164, + 303, + 174 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 161, + 332, + 178 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 332, + 163, + 387, + 177 + ], + "score": 0.93, + "content": "( 1 + \\textstyle { \\frac { 1 } { d } } ) ^ { d } \\leq e", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 161, + 405, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 405, + 163, + 471, + 177 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { e } \\leq ( 1 - \\frac { 1 } { d } ) ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 161, + 505, + 178 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 424, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 212, + 192 + ], + "score": 1.0, + "content": "weight vector w such that", + "type": "text" + }, + { + "bbox": [ + 213, + 176, + 340, + 192 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 177, + 380, + 192 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 381, + 180, + 385, + 189 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 177, + 424, + 192 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 194, + 411, + 228 + ], + "lines": [ + { + "bbox": [ + 199, + 194, + 411, + 228 + ], + "spans": [ + { + "bbox": [ + 199, + 194, + 411, + 228 + ], + "score": 0.93, + "content": "( 1 / e ) ^ { \\frac { d } { d - 1 } } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq e \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "fad54310d0374c9f4833c0c3598de2616f71377b0620a9acc8b76c2f36ce6bb1.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 194, + 411, + 211.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 199, + 211.0, + 411, + 228.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 227, + 244 + ], + "score": 1.0, + "content": "We apply Lemma 1, choosing", + "type": "text" + }, + { + "bbox": [ + 227, + 231, + 236, + 242 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "to be a zero-mean multivariate Gaussian distribution with diagonal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 283, + 258 + ], + "score": 1.0, + "content": "covariance matrix, where each entry of the", + "type": "text" + }, + { + "bbox": [ + 283, + 247, + 288, + 255 + ], + "score": 0.44, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 245, + 320, + 258 + ], + "score": 1.0, + "content": "th layer", + "type": "text" + }, + { + "bbox": [ + 320, + 245, + 333, + 257 + ], + "score": 0.88, + "content": "\\mathbf { U } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 245, + 429, + 258 + ], + "score": 1.0, + "content": "has standard deviation", + "type": "text" + }, + { + "bbox": [ + 429, + 242, + 483, + 260 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 241, + 506, + 258 + ], + "score": 1.0, + "content": "kWfik2 ξ with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 113, + 270 + ], + "score": 0.8, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 258, + 257, + 271 + ], + "score": 1.0, + "content": "chosen later in the proof. Note that", + "type": "text" + }, + { + "bbox": [ + 257, + 258, + 271, + 270 + ], + "score": 0.89, + "content": "\\beta _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "edefined earlier in the theorem is the geometric average of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 437, + 282 + ], + "score": 1.0, + "content": "spectral norms across all layers. Then for the ith layer’s random perturbation vector", + "type": "text" + }, + { + "bbox": [ + 437, + 269, + 502, + 281 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 315, + 294 + ], + "score": 1.0, + "content": "we get the following bound from (Tropp, 2012) with", + "type": "text" + }, + { + "bbox": [ + 315, + 281, + 322, + 290 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 280, + 507, + 294 + ], + "score": 1.0, + "content": "representing the width of the ith hidden layer:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 294, + 389, + 324 + ], + "lines": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "spans": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "score": 0.95, + "content": "\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "506da5167bfdd899767c975aa46e9bfba65fc3a47e9f7d01b83bac8d4dac1384.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 414, + 339 + ], + "score": 1.0, + "content": "We now use a union bound over all layers for a maximum union probability of", + "type": "text" + }, + { + "bbox": [ + 414, + 326, + 429, + 339 + ], + "score": 0.76, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 325, + 505, + 339 + ], + "score": 1.0, + "content": ", which implies the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 336, + 507, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 153, + 359 + ], + "score": 1.0, + "content": "normalized", + "type": "text" + }, + { + "bbox": [ + 163, + 336, + 507, + 356 + ], + "score": 1.0, + "content": "w kUik2 for each layer is upper-bounded by ξp2h log(4hd). Then for any w satisfying", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 357, + 277, + 372 + ], + "spans": [ + { + "bbox": [ + 107, + 357, + 235, + 371 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 357, + 263, + 372 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 263, + 360, + 268, + 368 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 357, + 277, + 372 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 374, + 438, + 494 + ], + "lines": [ + { + "bbox": [ + 172, + 374, + 438, + 494 + ], + "spans": [ + { + "bbox": [ + 172, + 374, + 438, + 494 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\displaystyle \\operatorname* { m a x } _ { \\| \\mathbf x \\| _ { 2 } \\leq B } \\| f _ { \\mathbf w + \\mathbf u } ( \\mathbf x ) - f _ { \\mathbf w } ( \\mathbf x ) \\| _ { 2 } \\leq e B \\left( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } } & { } \\\\ { \\displaystyle } & { \\overset { ( a ) } \\leq e ^ { 2 } B \\left( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } } \\\\ & { = e ^ { 2 } B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } } \\\\ & { \\leq e ^ { 2 } d B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\xi \\sqrt { 2 h \\log ( 4 h d ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "be569788850b7e55151611a430c41a1d3e886b24248d085fab4b1ade54c03fa9.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 172, + 374, + 438, + 414.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 172, + 414.0, + 438, + 454.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 172, + 454.0, + 438, + 494.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 506, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 508, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 190, + 513 + ], + "score": 1.0, + "content": "Here (a) holds, since", + "type": "text" + }, + { + "bbox": [ + 191, + 498, + 362, + 517 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq \\frac { e } { \\| \\widetilde { \\mathbf { W } } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 495, + 424, + 517 + ], + "score": 1.0, + "content": "is true for each", + "type": "text" + }, + { + "bbox": [ + 424, + 502, + 430, + 512 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 495, + 508, + 517 + ], + "score": 1.0, + "content": ". Hence we choose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 515, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 209, + 534 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\xi = \\frac { \\gamma } { 3 0 d B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\sqrt { h \\log ( 4 h d ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 515, + 507, + 539 + ], + "score": 1.0, + "content": "for which the perturbation vector satisfies the assumptions of Lemma 2.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 532, + 333, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 333, + 544 + ], + "score": 1.0, + "content": "eThen, we bound the KL-divergence term in Lemma 1 as", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 546, + 445, + 686 + ], + "lines": [ + { + "bbox": [ + 165, + 546, + 445, + 686 + ], + "spans": [ + { + "bbox": [ + 165, + 546, + 445, + 686 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } } \\\\ & { } & { = \\frac { 3 0 ^ { 2 } d ^ { 2 } B ^ { 2 } \\beta _ { \\mathbf { w } } ^ { 2 } \\| \\mathbf { b } \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\widehat { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\overset { ( b ) } { \\leq } \\frac { 3 0 ^ { 2 } e ^ { 2 } d ^ { 2 } B ^ { 2 } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } h \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { = \\mathcal { O } \\Big ( d ^ { 2 } B ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } \\Big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "410d4e9bc8a295499f4d3f96be2a810cec09e76be2ebd041b970549d6a3a3982.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 165, + 546, + 445, + 592.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 592.6666666666666, + 445, + 639.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 165, + 639.3333333333333, + 445, + 685.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 689, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 305, + 705 + ], + "score": 1.0, + "content": "Note that (b) holds, because we assume", + "type": "text" + }, + { + "bbox": [ + 306, + 689, + 460, + 704 + ], + "score": 0.88, + "content": "\\begin{array} { r l r } { | \\| { \\bf W } _ { i } \\| _ { 2 } ~ - ~ \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } \\Big | } & { { } \\le ~ } & { \\frac { 1 } { d } \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 688, + 506, + 705 + ], + "score": 1.0, + "content": "implying", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 695, + 511, + 725 + ], + "spans": [ + { + "bbox": [ + 102, + 695, + 281, + 724 + ], + "score": 1.0, + "content": "1kW k Qdi=1 kWfik2 ≤ (1 − 1d )−(d−1) 1kWjk", + "type": "text" + }, + { + "bbox": [ + 106, + 703, + 431, + 721 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 1 } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq ( 1 - \\frac { 1 } { d } ) ^ { - ( d - 1 ) } \\frac { 1 } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq \\frac { e } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 698, + 467, + 725 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 467, + 706, + 473, + 717 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 698, + 511, + 725 + ], + "score": 1.0, + "content": ". There-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 263, + 734 + ], + "score": 1.0, + "content": "fore, Lemma 1 implies with probability", + "type": "text" + }, + { + "bbox": [ + 263, + 721, + 286, + 732 + ], + "score": 0.88, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "we have the following bound hold for any w satisfying", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 308, + 96 + ], + "score": 1.0, + "content": "Lemma 2 (Neyshabur et al. (2017a)). Consider a", + "type": "text" + }, + { + "bbox": [ + 308, + 83, + 314, + 92 + ], + "score": 0.58, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 81, + 382, + 96 + ], + "score": 1.0, + "content": "-layer neural net", + "type": "text" + }, + { + "bbox": [ + 383, + 83, + 395, + 94 + ], + "score": 0.86, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "with 1-Lipschitz activation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 142, + 107 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 96, + 150, + 104 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 93, + 177, + 107 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 178, + 93, + 216, + 105 + ], + "score": 0.9, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 93, + 360, + 107 + ], + "score": 1.0, + "content": ". Then for any norm-bounded input", + "type": "text" + }, + { + "bbox": [ + 360, + 94, + 403, + 106 + ], + "score": 0.91, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "and weight perturbation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 377, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 104, + 199, + 117 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbf { \\dot { u } } : \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 104, + 377, + 118 + ], + "score": 1.0, + "content": ", we have the following perturbation bound:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 505, + 118 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 120, + 420, + 154 + ], + "lines": [ + { + "bbox": [ + 190, + 120, + 420, + 154 + ], + "spans": [ + { + "bbox": [ + 190, + 120, + 420, + 154 + ], + "score": 0.93, + "content": "\\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) - f _ { \\mathbf { w } } ( \\mathbf { x } ) \\| _ { 2 } \\leq e B \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } .", + "type": "interline_equation", + "image_path": "87b4d53b6814707767de659c98397b67d7e33d71416fb1722cd8b32231162e97.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 120, + 420, + 137.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 190, + 137.0, + 420, + 154.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 161, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 226, + 178 + ], + "score": 1.0, + "content": "To prove Theorem 1, consider", + "type": "text" + }, + { + "bbox": [ + 226, + 164, + 239, + 176 + ], + "score": 0.89, + "content": "f _ { \\widetilde { \\mathbf { w } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 161, + 293, + 178 + ], + "score": 1.0, + "content": "with weights", + "type": "text" + }, + { + "bbox": [ + 294, + 164, + 303, + 174 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 161, + 332, + 178 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 332, + 163, + 387, + 177 + ], + "score": 0.93, + "content": "( 1 + \\textstyle { \\frac { 1 } { d } } ) ^ { d } \\leq e", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 161, + 405, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 405, + 163, + 471, + 177 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { e } \\leq ( 1 - \\frac { 1 } { d } ) ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 161, + 505, + 178 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 424, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 212, + 192 + ], + "score": 1.0, + "content": "weight vector w such that", + "type": "text" + }, + { + "bbox": [ + 213, + 176, + 340, + 192 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 177, + 380, + 192 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 381, + 180, + 385, + 189 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 177, + 424, + 192 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 161, + 505, + 192 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 194, + 411, + 228 + ], + "lines": [ + { + "bbox": [ + 199, + 194, + 411, + 228 + ], + "spans": [ + { + "bbox": [ + 199, + 194, + 411, + 228 + ], + "score": 0.93, + "content": "( 1 / e ) ^ { \\frac { d } { d - 1 } } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq e \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "fad54310d0374c9f4833c0c3598de2616f71377b0620a9acc8b76c2f36ce6bb1.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 194, + 411, + 211.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 199, + 211.0, + 411, + 228.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 227, + 244 + ], + "score": 1.0, + "content": "We apply Lemma 1, choosing", + "type": "text" + }, + { + "bbox": [ + 227, + 231, + 236, + 242 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "to be a zero-mean multivariate Gaussian distribution with diagonal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 283, + 258 + ], + "score": 1.0, + "content": "covariance matrix, where each entry of the", + "type": "text" + }, + { + "bbox": [ + 283, + 247, + 288, + 255 + ], + "score": 0.44, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 245, + 320, + 258 + ], + "score": 1.0, + "content": "th layer", + "type": "text" + }, + { + "bbox": [ + 320, + 245, + 333, + 257 + ], + "score": 0.88, + "content": "\\mathbf { U } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 245, + 429, + 258 + ], + "score": 1.0, + "content": "has standard deviation", + "type": "text" + }, + { + "bbox": [ + 429, + 242, + 483, + 260 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 241, + 506, + 258 + ], + "score": 1.0, + "content": "kWfik2 ξ with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 113, + 270 + ], + "score": 0.8, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 258, + 257, + 271 + ], + "score": 1.0, + "content": "chosen later in the proof. Note that", + "type": "text" + }, + { + "bbox": [ + 257, + 258, + 271, + 270 + ], + "score": 0.89, + "content": "\\beta _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "edefined earlier in the theorem is the geometric average of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 437, + 282 + ], + "score": 1.0, + "content": "spectral norms across all layers. Then for the ith layer’s random perturbation vector", + "type": "text" + }, + { + "bbox": [ + 437, + 269, + 502, + 281 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 315, + 294 + ], + "score": 1.0, + "content": "we get the following bound from (Tropp, 2012) with", + "type": "text" + }, + { + "bbox": [ + 315, + 281, + 322, + 290 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 280, + 507, + 294 + ], + "score": 1.0, + "content": "representing the width of the ith hidden layer:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 230, + 507, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 294, + 389, + 324 + ], + "lines": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "spans": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "score": 0.95, + "content": "\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "506da5167bfdd899767c975aa46e9bfba65fc3a47e9f7d01b83bac8d4dac1384.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 223, + 294, + 389, + 324 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 414, + 339 + ], + "score": 1.0, + "content": "We now use a union bound over all layers for a maximum union probability of", + "type": "text" + }, + { + "bbox": [ + 414, + 326, + 429, + 339 + ], + "score": 0.76, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 325, + 505, + 339 + ], + "score": 1.0, + "content": ", which implies the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 336, + 507, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 153, + 359 + ], + "score": 1.0, + "content": "normalized", + "type": "text" + }, + { + "bbox": [ + 163, + 336, + 507, + 356 + ], + "score": 1.0, + "content": "w kUik2 for each layer is upper-bounded by ξp2h log(4hd). 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Hence we choose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 515, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 209, + 534 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\xi = \\frac { \\gamma } { 3 0 d B \\beta _ { \\widetilde { \\mathbf { w } } } ^ { d - 1 } \\sqrt { h \\log ( 4 h d ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 515, + 507, + 539 + ], + "score": 1.0, + "content": "for which the perturbation vector satisfies the assumptions of Lemma 2.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 532, + 333, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 333, + 544 + ], + "score": 1.0, + "content": "eThen, we bound the KL-divergence term in Lemma 1 as", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 495, + 508, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 546, + 445, + 686 + ], + "lines": [ + { + "bbox": [ + 165, + 546, + 445, + 686 + ], + "spans": [ + { + "bbox": [ + 165, + 546, + 445, + 686 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } } \\\\ & { } & { = \\frac { 3 0 ^ { 2 } d ^ { 2 } B ^ { 2 } \\beta _ { \\mathbf { w } } ^ { 2 } \\| \\mathbf { b } \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\widehat { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\overset { ( b ) } { \\leq } \\frac { 3 0 ^ { 2 } e ^ { 2 } d ^ { 2 } B ^ { 2 } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } h \\log ( 4 h d ) } { 2 \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { = \\mathcal { O } \\Big ( d ^ { 2 } B ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } \\Big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "410d4e9bc8a295499f4d3f96be2a810cec09e76be2ebd041b970549d6a3a3982.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 165, + 546, + 445, + 592.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 592.6666666666666, + 445, + 639.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 165, + 639.3333333333333, + 445, + 685.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 689, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 305, + 705 + ], + "score": 1.0, + "content": "Note that (b) holds, because we assume", + "type": "text" + }, + { + "bbox": [ + 306, + 689, + 460, + 704 + ], + "score": 0.88, + "content": "\\begin{array} { r l r } { | \\| { \\bf W } _ { i } \\| _ { 2 } ~ - ~ \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } \\Big | } & { { } \\le ~ } & { \\frac { 1 } { d } \\| \\widetilde { { \\bf W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 688, + 506, + 705 + ], + "score": 1.0, + "content": "implying", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 695, + 511, + 725 + ], + "spans": [ + { + "bbox": [ + 102, + 695, + 281, + 724 + ], + "score": 1.0, + "content": "1kW k Qdi=1 kWfik2 ≤ (1 − 1d )−(d−1) 1kWjk", + "type": "text" + }, + { + "bbox": [ + 106, + 703, + 431, + 721 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 1 } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\leq ( 1 - \\frac { 1 } { d } ) ^ { - ( d - 1 ) } \\frac { 1 } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\leq \\frac { e } { \\| \\mathbf { W } _ { j } \\| } \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 698, + 467, + 725 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 467, + 706, + 473, + 717 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 698, + 511, + 725 + ], + "score": 1.0, + "content": ". 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This together with (15) completes the proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 252, + 230, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 231, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 231, + 265 + ], + "score": 1.0, + "content": "C.2 PROOF OF THEOREM 2", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 104, + 272, + 466, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 467, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 467, + 286 + ], + "score": 1.0, + "content": "We start by proving the following lemmas providing perturbation bound for FGM attacks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 200, + 302 + ], + "score": 1.0, + "content": "Lemma 3. 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For this purpose we can use a cover of size", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 240, + 227 + ], + "score": 0.89, + "content": "2 \\log _ { 1 + 1 / d } M \\leq 2 ( d + 1 ) \\log M ,", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 212, + 339, + 227 + ], + "score": 1.0, + "content": "1 which combined for all", + "type": "text" + }, + { + "bbox": [ + 339, + 214, + 344, + 223 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 212, + 443, + 227 + ], + "score": 1.0, + "content": "’s gives a cover with size", + "type": "text" + }, + { + "bbox": [ + 444, + 213, + 505, + 226 + ], + "score": 0.92, + "content": "{ \\mathcal { O } } ( ( d \\log M ) ^ { d } )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 474, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 231, + 239 + ], + "score": 1.0, + "content": "whose logarithm is growing as", + "type": "text" + }, + { + "bbox": [ + 231, + 226, + 291, + 238 + ], + "score": 0.92, + "content": "d \\log ( d \\log M )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 225, + 474, + 239 + ], + "score": 1.0, + "content": ". This together with (15) completes the proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 101, + 141, + 510, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 252, + 230, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 231, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 231, + 265 + ], + "score": 1.0, + "content": "C.2 PROOF OF THEOREM 2", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 104, + 272, + 466, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 467, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 467, + 286 + ], + "score": 1.0, + "content": "We start by proving the following lemmas providing perturbation bound for FGM attacks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 272, + 467, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 200, + 302 + ], + "score": 1.0, + "content": "Lemma 3. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 290, + 207, + 299 + ], + "score": 0.65, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 288, + 274, + 302 + ], + "score": 1.0, + "content": "-layer neural net", + "type": "text" + }, + { + "bbox": [ + 275, + 290, + 287, + 301 + ], + "score": 0.88, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "with 1-Lipschitz and 1-smooth (1-Lipschitz derivative)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 149, + 313 + ], + "score": 1.0, + "content": "activation", + "type": "text" + }, + { + "bbox": [ + 150, + 302, + 157, + 310 + ], + "score": 0.65, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 298, + 185, + 313 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 185, + 300, + 223, + 312 + ], + "score": 0.91, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 298, + 297, + 313 + ], + "score": 1.0, + "content": ". Let training loss", + "type": "text" + }, + { + "bbox": [ + 297, + 300, + 367, + 312 + ], + "score": 0.92, + "content": "\\ell : ( \\mathbb { R } ^ { m } , \\mathcal { Y } ) \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 298, + 506, + 313 + ], + "score": 1.0, + "content": "also be 1-Lipschitz and 1-smooth", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 186, + 325 + ], + "score": 1.0, + "content": "for any fixed label", + "type": "text" + }, + { + "bbox": [ + 186, + 312, + 214, + 322 + ], + "score": 0.88, + "content": "y \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 311, + 305, + 325 + ], + "score": 1.0, + "content": ". 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Since for a fixed", + "type": "text" + }, + { + "bbox": [ + 206, + 409, + 219, + 420 + ], + "score": 0.73, + "content": "y \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 407, + 474, + 421 + ], + "score": 1.0, + "content": "satisfies the same Lipschitzness and smoothness properties as", + "type": "text" + }, + { + "bbox": [ + 474, + 411, + 481, + 419 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 407, + 505, + 421 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 420, + 327, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 420, + 180, + 432 + ], + "score": 0.92, + "content": "\\| \\nabla _ { \\mathbf { z } } \\ell ( \\mathbf { z } , y ) \\| _ { 2 } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 420, + 327, + 433 + ], + "score": 1.0, + "content": "and applying the chain rule implies:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 407, + 505, + 433 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 437, + 488, + 610 + ], + "lines": [ + { + "bbox": [ + 121, + 437, + 488, + 610 + ], + "spans": [ + { + "bbox": [ + 121, + 437, + 488, + 610 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\mathbf x } \\ell ( \\mathbf r _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - \\nabla _ { \\mathbf x } \\ell ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { = \\| ( \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| ( \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\quad + \\| ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) ) ( \\nabla \\ell ) ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| _ { 2 } + ( \\prod _ { i } \\| \\mathbf x _ { i } \\| _ { 2 } ) \\| ( \\| \\nabla \\ell ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & { \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| _ { 2 } + ( \\prod _ { i = 1 } ^ { d } \\| \\nabla _ { \\mathbf x } ) \\| ( \\nabla \\ell ( f _ { \\infty + \\mathbf n } ( \\mathbf x ) , y ) - ( \\nabla \\ell ) ( f _ { \\infty } ( \\mathbf x ) , y ) \\| _ { 2 } } \\\\ & \\le \\| \\nabla _ { \\mathbf x } f _ { \\infty + \\mathbf n } ( \\mathbf x ) - \\nabla _ { \\mathbf x } f _ { \\infty } ( \\mathbf x ) \\| \\end{array}", + "type": "interline_equation", + "image_path": "7f3ade952da620e1f0ddfd6b6cef205bc1187f12c0e1f0f6bcb9ca2671ec19ff.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 121, + 437, + 488, + 494.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 121, + 494.6666666666667, + 488, + 552.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 121, + 552.3333333333334, + 488, + 610.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 104, + 612, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 612, + 437, + 631 + ], + "score": 1.0, + "content": "The above result is a conclusion of Lemma 2 and the lemma’s assumptions implying", + "type": "text" + }, + { + "bbox": [ + 438, + 615, + 505, + 629 + ], + "score": 0.92, + "content": "\\left\\| \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\right\\| _ { 2 } \\leq", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 623, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 162, + 644 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 623, + 280, + 650 + ], + "score": 1.0, + "content": "for every x. Now, we define", + "type": "text" + }, + { + "bbox": [ + 280, + 629, + 429, + 645 + ], + "score": 0.92, + "content": "\\Delta _ { k } = \\left. \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } + \\mathbf { u } } ^ { ( k ) } ( \\mathbf { x } ) - \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\right. _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 623, + 459, + 650 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 459, + 630, + 505, + 644 + ], + "score": 0.9, + "content": "f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) : =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 234, + 656 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { k } \\sigma ( \\mathbf { W } _ { k - 1 } \\cdot \\cdot \\cdot \\sigma ( \\mathbf { W } _ { 1 } \\mathbf { x } ) ) \\cdot \\cdot \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 643, + 369, + 657 + ], + "score": 1.0, + "content": "denotes the DNN’s output at layer", + "type": "text" + }, + { + "bbox": [ + 369, + 645, + 375, + 654 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ". With (17) in mind, we complete", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 655, + 388, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 388, + 668 + ], + "score": 1.0, + "content": "this lemma’s proof by showing the following inequality via induction:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 612, + 505, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 673, + 468, + 708 + ], + "lines": [ + { + "bbox": [ + 143, + 673, + 468, + 708 + ], + "spans": [ + { + "bbox": [ + 143, + 673, + 468, + 708 + ], + "score": 0.93, + "content": "\\Delta _ { k } \\leq e ( 1 + \\frac { 1 } { d } ) ^ { k } \\big ( \\prod _ { i = 1 } ^ { k } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { k } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\| \\mathbf { x } \\| _ { 2 } \\big ( \\prod _ { j = 1 } ^ { i - 1 } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] .", + "type": "interline_equation", + "image_path": "c9200bcc6c904b97cfecd1c24a74e527c09706625c3d87ff8b92938e04023675.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 143, + 673, + 468, + 684.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 143, + 684.6666666666666, + 468, + 696.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 143, + 696.3333333333333, + 468, + 707.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 115, + 717, + 460, + 735 + ], + "lines": [ + { + "bbox": [ + 117, + 712, + 461, + 737 + ], + "spans": [ + { + "bbox": [ + 117, + 712, + 292, + 737 + ], + "score": 1.0, + "content": "1Note that log 1x ≥ 1 − x implying log 11− 1d+1", + "type": "text" + }, + { + "bbox": [ + 259, + 717, + 316, + 735 + ], + "score": 0.78, + "content": "\\textstyle { \\frac { 1 } { 1 - { \\frac { 1 } { d + 1 } } } } \\geq { \\frac { 1 } { d + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 717, + 357, + 732 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + }, + { + "bbox": [ + 357, + 718, + 456, + 730 + ], + "score": 0.89, + "content": "( \\log ( 1 + 1 / d ) ) ^ { - 1 } \\leq d + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 717, + 461, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 117, + 712, + 461, + 737 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 507, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 318, + 97 + ], + "score": 1.0, + "content": "The above equation will prove the lemma because for", + "type": "text" + }, + { + "bbox": [ + 319, + 83, + 345, + 93 + ], + "score": 0.93, + "content": "k \\leq d", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 80, + 380, + 97 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 381, + 81, + 485, + 96 + ], + "score": 0.94, + "content": "\\begin{array} { r } { ( 1 + \\frac { 1 } { d } ) ^ { k } \\le ( 1 + \\frac { 1 } { d } ) ^ { d } \\le e } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 80, + 506, + 97 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 131, + 108 + ], + "score": 0.72, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 94, + 135, + 111 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 135, + 97, + 168, + 109 + ], + "score": 0.76, + "content": "\\Delta _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 94, + 192, + 111 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 192, + 95, + 243, + 109 + ], + "score": 0.94, + "content": "f _ { \\mathbf { w } } ^ { ( 0 ) } ( \\mathbf { x } ) = \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 94, + 462, + 111 + ], + "score": 1.0, + "content": "and does not change with w. Given that (18) holds for", + "type": "text" + }, + { + "bbox": [ + 462, + 98, + 469, + 107 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 94, + 506, + 111 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 113, + 527, + 409 + ], + "lines": [ + { + "bbox": [ + 127, + 113, + 527, + 409 + ], + "spans": [ + { + "bbox": [ + 127, + 113, + 527, + 409 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\mathbf { \\Phi } } & { = | \\nabla _ { x } \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} } \\\\ & { - \\nabla _ { x } \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} } \\\\ & - \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\} \\{ \\mathbf { S } _ { 0 } ^ { \\varepsilon } \\ \\end{array}", + "type": "interline_equation", + "image_path": "9411eba59e2e6b7633b2264e55f50def1c823d2ed86896c0fc2f367943109c8b.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 127, + 113, + 527, + 211.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 127, + 211.66666666666669, + 527, + 310.33333333333337 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 127, + 310.33333333333337, + 527, + 409.00000000000006 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 374, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 374, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 374, + 424 + ], + "score": 1.0, + "content": "Therefore, combining (17) and (18) the lemma’s proof is complete", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 506, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "Before presenting the perturbation bound for FGM attacks, we first prove the following simple", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 444, + 139, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 139, + 457 + ], + "score": 1.0, + "content": "lemma.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 103, + 458, + 473, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 474, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 224, + 472 + ], + "score": 1.0, + "content": "Lemma 4. Consider vectors", + "type": "text" + }, + { + "bbox": [ + 224, + 461, + 249, + 470 + ], + "score": 0.89, + "content": "\\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 458, + 326, + 472 + ], + "score": 1.0, + "content": "and norm function", + "type": "text" + }, + { + "bbox": [ + 326, + 459, + 344, + 471 + ], + "score": 0.71, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 459, + 450, + 471 + ], + "score": 0.86, + "content": "| . \\ I f \\operatorname* { m a x } \\{ \\| \\mathbf { z } _ { 1 } \\| , \\| \\mathbf { z } _ { 2 } \\| \\} \\geq \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 458, + 474, + 472 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 474, + 384, + 501 + ], + "lines": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "spans": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "score": 0.93, + "content": "\\Bigl \\| \\frac { \\epsilon } { \\| \\mathbf { z } _ { 1 } \\| } \\mathbf { z } _ { 1 } - \\frac { \\epsilon } { \\| \\mathbf { z } _ { 2 } \\| } \\mathbf { z } _ { 2 } \\Bigr \\| \\leq \\frac { 2 \\epsilon } { \\kappa } \\| \\mathbf { z } _ { 1 } - \\mathbf { z } _ { 2 } \\| .", + "type": "interline_equation", + "image_path": "bb6d09f59ac9d4c8021178bba3a389155f18244f4246218e1a3a14c165ca0ccd.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 456, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 457, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 277, + 525 + ], + "score": 1.0, + "content": "Proof. Without loss of generality suppose", + "type": "text" + }, + { + "bbox": [ + 277, + 511, + 330, + 524 + ], + "score": 0.93, + "content": "\\| \\mathbf { z } _ { 2 } \\| \\leq \\| \\mathbf { z } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 510, + 387, + 525 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 387, + 511, + 426, + 524 + ], + "score": 0.93, + "content": "\\kappa \\leq \\| \\mathbf { z } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 510, + 457, + 525 + ], + "score": 1.0, + "content": ". 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Consider a", + "type": "text" + }, + { + "bbox": [ + 203, + 651, + 209, + 660 + ], + "score": 0.72, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 649, + 336, + 663 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 336, + 651, + 349, + 662 + ], + "score": 0.87, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 660, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 114, + 671 + ], + "score": 0.57, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 660, + 142, + 675 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 142, + 661, + 180, + 673 + ], + "score": 0.88, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 660, + 479, + 675 + ], + "score": 1.0, + "content": ". 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Consider vectors", + "type": "text" + }, + { + "bbox": [ + 224, + 461, + 249, + 470 + ], + "score": 0.89, + "content": "\\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 458, + 326, + 472 + ], + "score": 1.0, + "content": "and norm function", + "type": "text" + }, + { + "bbox": [ + 326, + 459, + 344, + 471 + ], + "score": 0.71, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 459, + 450, + 471 + ], + "score": 0.86, + "content": "| . \\ I f \\operatorname* { m a x } \\{ \\| \\mathbf { z } _ { 1 } \\| , \\| \\mathbf { z } _ { 2 } \\| \\} \\geq \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 458, + 474, + 472 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 458, + 474, + 472 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 474, + 384, + 501 + ], + "lines": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "spans": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "score": 0.93, + "content": "\\Bigl \\| \\frac { \\epsilon } { \\| \\mathbf { z } _ { 1 } \\| } \\mathbf { z } _ { 1 } - \\frac { \\epsilon } { \\| \\mathbf { z } _ { 2 } \\| } \\mathbf { z } _ { 2 } \\Bigr \\| \\leq \\frac { 2 \\epsilon } { \\kappa } \\| \\mathbf { z } _ { 1 } - \\mathbf { z } _ { 2 } \\| .", + "type": "interline_equation", + "image_path": "bb6d09f59ac9d4c8021178bba3a389155f18244f4246218e1a3a14c165ca0ccd.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 227, + 474, + 384, + 501 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 456, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 457, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 277, + 525 + ], + "score": 1.0, + "content": "Proof. 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Then,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 510, + 457, + 525 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 528, + 445, + 631 + ], + "lines": [ + { + "bbox": [ + 166, + 528, + 445, + 631 + ], + "spans": [ + { + "bbox": [ + 166, + 528, + 445, + 631 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\| \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } { \\bf z } _ { 1 } - \\frac { \\epsilon } { \\| { \\bf z } _ { 2 } \\| } { \\bf z } _ { 2 } \\| = \\epsilon \\big \\| \\frac { 1 } { \\| { \\bf z } _ { 1 } \\| } \\big ( { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\big ) - \\frac { \\| { \\bf z } _ { 1 } \\| - \\| { \\bf z } _ { 2 } \\| } { \\| { \\bf z } _ { 1 } \\| } \\frac { 1 } { \\| { \\bf z } _ { 2 } \\| } \\big \\| } \\\\ & { \\displaystyle \\quad \\quad \\quad \\leq \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| + \\frac { \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\big | \\| { \\bf z } _ { 1 } \\| - \\| { \\bf z } _ { 2 } \\| \\big | } \\\\ & { \\displaystyle \\quad \\quad \\leq \\frac { 2 \\epsilon } { \\| { \\bf z } _ { 1 } \\| } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| } \\\\ & { \\displaystyle \\quad \\quad \\quad \\leq \\frac { 2 \\epsilon } { \\kappa } \\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "0a2a79269e79ddcada050effefbf9532cfe3eff5f79318d56794164f972219e0.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 166, + 528, + 445, + 562.3333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 166, + 562.3333333333334, + 445, + 596.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 166, + 596.6666666666667, + 445, + 631.0000000000001 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 506, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 203, + 663 + ], + "score": 1.0, + "content": "Lemma 5. Consider a", + "type": "text" + }, + { + "bbox": [ + 203, + 651, + 209, + 660 + ], + "score": 0.72, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 649, + 336, + 663 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 336, + 651, + 349, + 662 + ], + "score": 0.87, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 660, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 114, + 671 + ], + "score": 0.57, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 660, + 142, + 675 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 142, + 661, + 180, + 673 + ], + "score": 0.88, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 660, + 479, + 675 + ], + "score": 1.0, + "content": ". Consider FGM attacks with noise power \u000f according to Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 479, + 663, + 502, + 673 + ], + "score": 0.9, + "content": "| | \\cdot | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 660, + 506, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 144, + 686 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 144, + 673, + 241, + 684 + ], + "score": 0.9, + "content": "\\kappa \\leq \\| \\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 671, + 307, + 686 + ], + "score": 1.0, + "content": "holds over the", + "type": "text" + }, + { + "bbox": [ + 307, + 677, + 311, + 682 + ], + "score": 0.57, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 671, + 430, + 686 + ], + "score": 1.0, + "content": "-ball around the support set", + "type": "text" + }, + { + "bbox": [ + 430, + 675, + 439, + 682 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 671, + 506, + 686 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 683, + 426, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 296, + 696 + ], + "score": 1.0, + "content": "norm-bounded perturbation vector u such that", + "type": "text" + }, + { + "bbox": [ + 296, + 684, + 386, + 696 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } . \\forall i } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 683, + 426, + 696 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 649, + 506, + 696 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 700, + 497, + 735 + ], + "lines": [ + { + "bbox": [ + 113, + 700, + 497, + 735 + ], + "spans": [ + { + "bbox": [ + 113, + 700, + 497, + 735 + ], + "score": 0.93, + "content": "\\big \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { f g m } } ( \\mathbf { x } ) \\big \\| _ { 2 } \\leq \\frac { 2 e ^ { 2 } \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\| \\mathbf { x } \\| _ { 2 } \\big ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] .", + "type": "interline_equation", + "image_path": "2c6501a3768cf0105073e9abf5fab638b04910c551d9d61ab88870d4f29f2cdf.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 113, + 700, + 497, + 711.6666666666666 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 113, + 711.6666666666666, + 497, + 723.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 113, + 723.3333333333333, + 497, + 734.9999999999999 + ], + "spans": [], + "index": 20 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Proof. The FGM attack according to Euclidean norm is simply the DNN loss’s gradient normalized", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 137, + 106 + ], + "score": 1.0, + "content": "to have", + "type": "text" + }, + { + "bbox": [ + 138, + 96, + 143, + 104 + ], + "score": 0.77, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "-Euclidean norm. The lemma is hence a direct result of combining Lemmas 3 and 4.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "To prove Theorem 2, we apply Lemma 1 together with the result in Lemma 5. Similar to the proof for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 213, + 141 + ], + "score": 1.0, + "content": "Theorem 1, given weights", + "type": "text" + }, + { + "bbox": [ + 213, + 128, + 223, + 138 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "we consider a zero-mean multivariate Gaussian perturbation vector u", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 371, + 151 + ], + "score": 1.0, + "content": "with diagonal covariance matrix where each element in the ith layer", + "type": "text" + }, + { + "bbox": [ + 372, + 140, + 383, + 150 + ], + "score": 0.86, + "content": "\\mathbf { u } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "varies with the scaled standard", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 493, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 146, + 166 + ], + "score": 1.0, + "content": "deviation", + "type": "text" + }, + { + "bbox": [ + 164, + 149, + 493, + 171 + ], + "score": 1.0, + "content": "kWfik2βw ξ with ξ properly chosen later in the proof. Consider weights w for which", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 171, + 382, + 196 + ], + "lines": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "spans": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "score": 0.94, + "content": "\\forall i : \\ | \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\leq \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "c5aa4ad25b6dc75fe11d604f2492413fa31778e1adb0aa25a30ce979050786b2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 393, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 393, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 131, + 214 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 199, + 196, + 212 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 198, + 393, + 214 + ], + "score": 1.0, + "content": ", (Tropp, 2012) shows the following bound holds", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 215, + 389, + 245 + ], + "lines": [ + { + "bbox": [ + 223, + 215, + 389, + 245 + ], + "spans": [ + { + "bbox": [ + 223, + 215, + 389, + 245 + ], + "score": 0.95, + "content": "\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "c8b4839acc43c95ac2b08a327946f30913ea6f35e2525c3fe925895d661125f2.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 215, + 389, + 230.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 223, + 230.0, + 389, + 245.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 434, + 261 + ], + "score": 1.0, + "content": "Then we apply a union bound over all layers for a maximum union probability of", + "type": "text" + }, + { + "bbox": [ + 434, + 249, + 450, + 261 + ], + "score": 0.73, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "implying the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 507, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 154, + 278 + ], + "score": 1.0, + "content": "normalized", + "type": "text" + }, + { + "bbox": [ + 165, + 258, + 507, + 278 + ], + "score": 1.0, + "content": "kUik2 for each layer is upper-bounded by ξp2h log(4hd). Now, if the assumptions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 276, + 289 + ], + "score": 1.0, + "content": "of Lemma 5 hold for perturbation vector", + "type": "text" + }, + { + "bbox": [ + 277, + 280, + 285, + 287 + ], + "score": 0.57, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 277, + 367, + 289 + ], + "score": 1.0, + "content": "given the choice of", + "type": "text" + }, + { + "bbox": [ + 367, + 278, + 373, + 289 + ], + "score": 0.82, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 277, + 505, + 289 + ], + "score": 1.0, + "content": ", for the FGM attack with noise", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 318, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 133, + 301 + ], + "score": 1.0, + "content": "power", + "type": "text" + }, + { + "bbox": [ + 134, + 290, + 139, + 298 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 288, + 258, + 301 + ], + "score": 1.0, + "content": "according to Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 258, + 289, + 280, + 300 + ], + "score": 0.9, + "content": "\\| \\cdot \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 288, + 318, + 301 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 103, + 304, + 500, + 482 + ], + "lines": [ + { + "bbox": [ + 103, + 304, + 500, + 482 + ], + "spans": [ + { + "bbox": [ + 103, + 304, + 500, + 482 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\| f _ { w + \\mathbf { n } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { w + \\mathbf { n } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + \\| f _ { w } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { w } ( \\mathbf x + \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w + \\mathbf { n } } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + ( \\underbrace { d } _ { \\mathbf { w - \\mathbf { n } } } ( \\mathbf x ) \\| _ { 2 } ) \\| \\delta _ { \\mathbf { w + \\mathbf { n } } } ^ { \\mathrm { t e m } } ( \\mathbf x ) - \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) \\| _ { 2 } } \\\\ & { \\leq e ( B + e ) \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } + 2 e ^ { 2 } \\frac { d } { \\kappa } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } + B ( \\frac { 1 } { \\sqrt { 1 + 1 } } W _ { i } ) \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { j } \\| } \\\\ & \\leq e ^ { 2 } ( B + e ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ \\end{array}", + "type": "interline_equation", + "image_path": "1a4a44fe8973396015868da6f52f13006aea8e2fce4d2c351d1cae7f6e7587c6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 103, + 304, + 500, + 363.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 103, + 363.3333333333333, + 500, + 422.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 103, + 422.66666666666663, + 500, + 481.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 178, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 179, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 179, + 497 + ], + "score": 1.0, + "content": "Hence we choose", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 497, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\dot { \\mathbf { \\xi } } = \\frac { \\gamma } { 8 e ^ { 5 } d ( B + \\epsilon ) \\sqrt { 2 h \\log ( 4 h d ) } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big ( 1 + \\frac { \\epsilon } { \\kappa } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ( 1 / B + \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } ) \\big ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "b5ed57a060c53f12a5519d7a229e366729b908cfa2475c2b6a9b4133f7b017eb.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 357, + 547 + ], + "score": 1.0, + "content": "for which the assumptions of Lemmas 1 and 5 hold. Assuming", + "type": "text" + }, + { + "bbox": [ + 358, + 534, + 385, + 545 + ], + "score": 0.9, + "content": "B \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 533, + 506, + 547 + ], + "score": 1.0, + "content": ", similar to Theorem 1’s proof", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 545, + 405, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 240, + 560 + ], + "score": 1.0, + "content": "we can show for any w such that", + "type": "text" + }, + { + "bbox": [ + 240, + 545, + 368, + 560 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 546, + 405, + 560 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 567, + 514, + 669 + ], + "lines": [ + { + "bbox": [ + 111, + 567, + 514, + 669 + ], + "spans": [ + { + "bbox": [ + 111, + 567, + 514, + 669 + ], + "score": 0.92, + "content": "\\begin{array} { l l } { \\displaystyle K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } \\\\ { \\displaystyle = \\mathcal { O } \\bigg ( d ^ { 2 } ( B + \\epsilon ) ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } \\{ 1 + \\frac { \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } \\big ) \\} ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } } \\\\ { \\leq \\mathcal { O } \\bigg ( d ^ { 2 } ( B + \\epsilon ) ^ { 2 } h \\log ( h d ) \\frac { \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\{ 1 + \\frac { \\epsilon } { \\kappa } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\} ^ { 2 } } { \\gamma ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "71717b918d9f1b7e70060d175f91e8d1d446a6f3be849e9aa70815c660ff5e80.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 111, + 567, + 514, + 601.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 111, + 601.0, + 514, + 635.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 111, + 635.0, + 514, + 669.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 671, + 507, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 296, + 685 + ], + "score": 1.0, + "content": "Then, applying Lemma 1 reveals that given any", + "type": "text" + }, + { + "bbox": [ + 297, + 673, + 321, + 684 + ], + "score": 0.94, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 671, + 418, + 685 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 418, + 673, + 442, + 684 + ], + "score": 0.88, + "content": "1 - \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 671, + 505, + 685 + ], + "score": 1.0, + "content": "for any w such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 683, + 290, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 125, + 699 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 683, + 252, + 698 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\le \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 684, + 290, + 699 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 702, + 459, + 736 + ], + "lines": [ + { + "bbox": [ + 151, + 702, + 459, + 736 + ], + "spans": [ + { + "bbox": [ + 151, + 702, + 459, + 736 + ], + "score": 0.93, + "content": "L _ { 0 } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) \\le \\widehat { L } _ { \\gamma } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\mathcal { O } \\biggl ( \\sqrt { \\frac { ( B + \\epsilon ) ^ { 2 } d ^ { 2 } h \\log ( d h ) \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f g m } } ( f _ { \\mathbf { w } } ) + \\log \\frac { n } { \\eta } } { \\gamma ^ { 2 } n } } \\biggr )", + "type": "interline_equation", + "image_path": "5a34527ff5c90fb279655f299c65ef71c26ce9abaac1ce0c356c4d3ad0e6934d.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 151, + 702, + 459, + 713.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 151, + 713.3333333333334, + 459, + 724.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 151, + 724.6666666666667, + 459, + 736.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Proof. The FGM attack according to Euclidean norm is simply the DNN loss’s gradient normalized", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 137, + 106 + ], + "score": 1.0, + "content": "to have", + "type": "text" + }, + { + "bbox": [ + 138, + 96, + 143, + 104 + ], + "score": 0.77, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "-Euclidean norm. The lemma is hence a direct result of combining Lemmas 3 and 4.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "To prove Theorem 2, we apply Lemma 1 together with the result in Lemma 5. Similar to the proof for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 213, + 141 + ], + "score": 1.0, + "content": "Theorem 1, given weights", + "type": "text" + }, + { + "bbox": [ + 213, + 128, + 223, + 138 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "we consider a zero-mean multivariate Gaussian perturbation vector u", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 371, + 151 + ], + "score": 1.0, + "content": "with diagonal covariance matrix where each element in the ith layer", + "type": "text" + }, + { + "bbox": [ + 372, + 140, + 383, + 150 + ], + "score": 0.86, + "content": "\\mathbf { u } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "varies with the scaled standard", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 493, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 146, + 166 + ], + "score": 1.0, + "content": "deviation", + "type": "text" + }, + { + "bbox": [ + 164, + 149, + 493, + 171 + ], + "score": 1.0, + "content": "kWfik2βw ξ with ξ properly chosen later in the proof. Consider weights w for which", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 117, + 506, + 171 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 171, + 382, + 196 + ], + "lines": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "spans": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "score": 0.94, + "content": "\\forall i : \\ | \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big | \\leq \\frac { 1 } { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "c5aa4ad25b6dc75fe11d604f2492413fa31778e1adb0aa25a30ce979050786b2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 171, + 382, + 196 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 393, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 393, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 131, + 214 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 199, + 196, + 212 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { i } \\sim \\mathcal { N } ( 0 , \\xi _ { i } ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 198, + 393, + 214 + ], + "score": 1.0, + "content": ", (Tropp, 2012) shows the following bound holds", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 198, + 393, + 214 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 215, + 389, + 245 + ], + "lines": [ + { + "bbox": [ + 223, + 215, + 389, + 245 + ], + "spans": [ + { + "bbox": [ + 223, + 215, + 389, + 245 + ], + "score": 0.95, + "content": "\\mathrm { P r } \\big ( \\beta _ { \\widetilde { \\mathbf { w } } } \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } > t \\big ) \\le 2 h \\exp ( - \\frac { t ^ { 2 } } { 2 h \\xi ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "c8b4839acc43c95ac2b08a327946f30913ea6f35e2525c3fe925895d661125f2.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 215, + 389, + 230.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 223, + 230.0, + 389, + 245.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 434, + 261 + ], + "score": 1.0, + "content": "Then we apply a union bound over all layers for a maximum union probability of", + "type": "text" + }, + { + "bbox": [ + 434, + 249, + 450, + 261 + ], + "score": 0.73, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "implying the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 507, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 154, + 278 + ], + "score": 1.0, + "content": "normalized", + "type": "text" + }, + { + "bbox": [ + 165, + 258, + 507, + 278 + ], + "score": 1.0, + "content": "kUik2 for each layer is upper-bounded by ξp2h log(4hd). 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f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { w + \\mathbf { n } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + \\| f _ { w } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { w } ( \\mathbf x + \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w + \\mathbf { n } } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) - f _ { \\mathbf { w } } ( \\mathbf x + \\delta _ { w + \\mathbf { n } } ^ { \\mathrm { t e m } } ( \\mathbf x ) ) \\| _ { 2 } + ( \\underbrace { d } _ { \\mathbf { w - \\mathbf { n } } } ( \\mathbf x ) \\| _ { 2 } ) \\| \\delta _ { \\mathbf { w + \\mathbf { n } } } ^ { \\mathrm { t e m } } ( \\mathbf x ) - \\delta _ { w } ^ { \\mathrm { t e m } } ( \\mathbf x ) \\| _ { 2 } } \\\\ & { \\leq e ( B + e ) \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } + 2 e ^ { 2 } \\frac { d } { \\kappa } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } + B ( \\frac { 1 } { \\sqrt { 1 + 1 } } W _ { i } ) \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\displaystyle \\frac { d } { i - 1 } \\| \\mathbf { W } _ { j } \\| } \\\\ & \\leq e ^ { 2 } ( B + e ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ \\end{array}", + "type": "interline_equation", + "image_path": "1a4a44fe8973396015868da6f52f13006aea8e2fce4d2c351d1cae7f6e7587c6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 103, + 304, + 500, + 363.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 103, + 363.3333333333333, + 500, + 422.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 103, + 422.66666666666663, + 500, + 481.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 178, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 179, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 179, + 497 + ], + "score": 1.0, + "content": "Hence we choose", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 483, + 179, + 497 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 497, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\dot { \\mathbf { \\xi } } = \\frac { \\gamma } { 8 e ^ { 5 } d ( B + \\epsilon ) \\sqrt { 2 h \\log ( 4 h d ) } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\big ( 1 + \\frac { \\epsilon } { \\kappa } \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ( 1 / B + \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } ) \\big ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "b5ed57a060c53f12a5519d7a229e366729b908cfa2475c2b6a9b4133f7b017eb.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 497, + 506, + 525 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 357, + 547 + ], + "score": 1.0, + "content": "for which the assumptions of Lemmas 1 and 5 hold. 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a _ { i } \\right| \\leq a _ { i } / d", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 121, + 470, + 136 + ], + "score": 1.0, + "content": ". Applying this covering number bound to (24) completes the proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 146, + 230, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 231, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 231, + 160 + ], + "score": 1.0, + "content": "C.3 PROOF OF THEOREM 3", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 181 + ], + "score": 1.0, + "content": "We use the following two lemmas to extend the proof of Theorem 2 for FGM attacks to show Theorem", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 185, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 185, + 190 + ], + "score": 1.0, + "content": "3 for PGM attacks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 108, + 192, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 504, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 200, + 205 + ], + "score": 1.0, + "content": "Lemma 6. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 194, + 207, + 203 + ], + "score": 0.56, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 192, + 331, + 205 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 331, + 194, + 344, + 205 + ], + "score": 0.88, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 192, + 496, + 205 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + }, + { + "bbox": [ + 497, + 196, + 504, + 203 + ], + "score": 0.69, + "content": "\\sigma", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 204, + 504, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 132, + 217 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 204, + 171, + 216 + ], + "score": 0.93, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 204, + 470, + 217 + ], + "score": 1.0, + "content": ". 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Suppose", + "type": "text" + }, + { + "bbox": [ + 245, + 215, + 342, + 229 + ], + "score": 0.93, + "content": "\\boldsymbol { \\kappa } \\leq \\| \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 216, + 402, + 229 + ], + "score": 1.0, + "content": "holds over the", + "type": "text" + }, + { + "bbox": [ + 402, + 218, + 407, + 226 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "-ball around the support", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 473, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 120, + 242 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 230, + 129, + 239 + ], + "score": 0.7, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 228, + 316, + 242 + ], + "score": 1.0, + "content": ". Then for any perturbation vector u such that", + "type": "text" + }, + { + "bbox": [ + 316, + 228, + 393, + 242 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 228, + 432, + 242 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 432, + 231, + 437, + 239 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 228, + 473, + 242 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 245, + 451, + 310 + ], + "lines": [ + { + "bbox": [ + 159, + 245, + 451, + 310 + ], + "spans": [ + { + "bbox": [ + 159, + 245, + 451, + 310 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) \\| _ { 2 } \\leq e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\times \\big ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\biggl [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + ( \\| \\mathbf { x } \\| _ { 2 } + \\epsilon ) \\big ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\biggr ] . } \\end{array}", + "type": "interline_equation", + "image_path": "482c33e88ba17f936b2535d14e0824270ae3711b3d6bfb3a21e9882eaf543d95.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 159, + 245, + 451, + 266.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 159, + 266.6666666666667, + 451, + 288.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 159, + 288.33333333333337, + 451, + 310.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 409, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 410, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 128, + 333 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 318, + 182, + 331 + ], + "score": 0.86, + "content": "\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 317, + 344, + 333 + ], + "score": 1.0, + "content": "denotes the actual Lipschitz constant of", + "type": "text" + }, + { + "bbox": [ + 344, + 318, + 405, + 331 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 317, + 410, + 333 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 344, + 354 + ], + "score": 1.0, + "content": "Proof. We use induction to show this lemma for different", + "type": "text" + }, + { + "bbox": [ + 344, + 345, + 350, + 353 + ], + "score": 0.74, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 342, + 462, + 354 + ], + "score": 1.0, + "content": "values. The result for case", + "type": "text" + }, + { + "bbox": [ + 462, + 344, + 487, + 353 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 379, + 366 + ], + "score": 1.0, + "content": "direct consequence of Lemma 5. Suppose that the result is true for", + "type": "text" + }, + { + "bbox": [ + 379, + 354, + 404, + 363 + ], + "score": 0.9, + "content": "r = k", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 353, + 505, + 366 + ], + "score": 1.0, + "content": ". Then, Lemmas 3 and 4", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 362, + 134, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 134, + 379 + ], + "score": 1.0, + "content": "imply", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 370, + 461, + 696 + ], + "lines": [ + { + "bbox": [ + 146, + 370, + 461, + 696 + ], + "spans": [ + { + "bbox": [ + 146, + 370, + 461, + 696 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { | | E _ { \\frac { d } { d } } ^ { \\mathrm { L C } } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) | \\leq x _ { 0 } ^ { 3 } , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ } \\\\ & { \\quad + \\ \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) } \\\\ & \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ \\end{array}", + "type": "interline_equation", + "image_path": "254fb01e44573d1f0cd6ad01c629bab08618283f883fcce0a78692a7328d78f0.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 146, + 370, + 461, + 478.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 146, + 478.6666666666667, + 461, + 587.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 146, + 587.3333333333334, + 461, + 696.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 703, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 284, + 725 + ], + "score": 1.0, + "content": "where the last line follows from the equality", + "type": "text" + }, + { + "bbox": [ + 284, + 705, + 361, + 722 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { i = 0 } ^ { k } s ^ { i } = { \\frac { 1 - s ^ { k + 1 } } { 1 - s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 706, + 506, + 721 + ], + "score": 1.0, + "content": ". Therefore, by induction the lemma", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 192, + 732 + ], + "score": 1.0, + "content": "holds for every value", + "type": "text" + }, + { + "bbox": [ + 193, + 721, + 217, + 731 + ], + "score": 0.89, + "content": "r \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 720, + 221, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 135 + ], + "lines": [ + { + "bbox": [ + 101, + 70, + 504, + 107 + ], + "spans": [ + { + "bbox": [ + 101, + 70, + 133, + 107 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 80, + 504, + 98 + ], + "score": 0.72, + "content": "\\begin{array} { r } { \\Phi _ { \\epsilon , \\kappa } ^ { \\mathrm { f o m } } ( f _ { \\mathbf { w } } ) : = \\left\\{ \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ( 1 + \\frac { \\epsilon } { \\kappa } \\{ \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } \\right\\} . } \\end{array}", + "type": "inline_equation", + "image_path": "2c4d4a6522abcd240952167daaa1a97e0931d8e9503ba0389893ad7fc86dd064.jpg" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 97, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 104, + 97, + 416, + 112 + ], + "score": 1.0, + "content": "Note that similar to our proof for Theorem 1 we can find a cover of size", + "type": "text" + }, + { + "bbox": [ + 416, + 99, + 505, + 111 + ], + "score": 0.9, + "content": "O ( ( d \\log M ) ^ { d } d n ^ { 1 / 2 d } )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 381, + 123 + ], + "score": 1.0, + "content": "for the spectral norms of the weights feasible set, where for any", + "type": "text" + }, + { + "bbox": [ + 382, + 110, + 412, + 123 + ], + "score": 0.93, + "content": "\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 110, + 452, + 123 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 452, + 113, + 462, + 122 + ], + "score": 0.85, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 121, + 470, + 136 + ], + "spans": [ + { + "bbox": [ + 107, + 122, + 197, + 136 + ], + "score": 0.92, + "content": "\\left| \\| \\mathbf { W } _ { i } \\| _ { 2 } - a _ { i } \\right| \\leq a _ { i } / d", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 121, + 470, + 136 + ], + "score": 1.0, + "content": ". Applying this covering number bound to (24) completes the proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 101, + 70, + 506, + 136 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 146, + 230, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 231, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 231, + 160 + ], + "score": 1.0, + "content": "C.3 PROOF OF THEOREM 3", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 181 + ], + "score": 1.0, + "content": "We use the following two lemmas to extend the proof of Theorem 2 for FGM attacks to show Theorem", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 185, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 185, + 190 + ], + "score": 1.0, + "content": "3 for PGM attacks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 166, + 505, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 192, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 504, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 200, + 205 + ], + "score": 1.0, + "content": "Lemma 6. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 194, + 207, + 203 + ], + "score": 0.56, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 192, + 331, + 205 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 331, + 194, + 344, + 205 + ], + "score": 0.88, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 192, + 496, + 205 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + }, + { + "bbox": [ + 497, + 196, + 504, + 203 + ], + "score": 0.69, + "content": "\\sigma", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 204, + 504, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 132, + 217 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 204, + 171, + 216 + ], + "score": 0.93, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 204, + 470, + 217 + ], + "score": 1.0, + "content": ". We consider PGM attacks with noise power \u000f according to Euclidean norm", + "type": "text" + }, + { + "bbox": [ + 471, + 204, + 504, + 216 + ], + "score": 0.8, + "content": "| | \\cdot | | _ { 2 } , r", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 215, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 197, + 229 + ], + "score": 1.0, + "content": "iterations and stepsize", + "type": "text" + }, + { + "bbox": [ + 198, + 218, + 205, + 226 + ], + "score": 0.51, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 216, + 245, + 229 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 245, + 215, + 342, + 229 + ], + "score": 0.93, + "content": "\\boldsymbol { \\kappa } \\leq \\| \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 216, + 402, + 229 + ], + "score": 1.0, + "content": "holds over the", + "type": "text" + }, + { + "bbox": [ + 402, + 218, + 407, + 226 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "-ball around the support", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 473, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 120, + 242 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 230, + 129, + 239 + ], + "score": 0.7, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 228, + 316, + 242 + ], + "score": 1.0, + "content": ". Then for any perturbation vector u such that", + "type": "text" + }, + { + "bbox": [ + 316, + 228, + 393, + 242 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 228, + 432, + 242 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 432, + 231, + 437, + 239 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 228, + 473, + 242 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 192, + 506, + 242 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 245, + 451, + 310 + ], + "lines": [ + { + "bbox": [ + 159, + 245, + 451, + 310 + ], + "spans": [ + { + "bbox": [ + 159, + 245, + 451, + 310 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) \\| _ { 2 } \\leq e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\times \\big ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\biggl [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + ( \\| \\mathbf { x } \\| _ { 2 } + \\epsilon ) \\big ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\displaystyle \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\biggr ] . } \\end{array}", + "type": "interline_equation", + "image_path": "482c33e88ba17f936b2535d14e0824270ae3711b3d6bfb3a21e9882eaf543d95.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 159, + 245, + 451, + 266.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 159, + 266.6666666666667, + 451, + 288.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 159, + 288.33333333333337, + 451, + 310.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 409, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 410, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 128, + 333 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 318, + 182, + 331 + ], + "score": 0.86, + "content": "\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 317, + 344, + 333 + ], + "score": 1.0, + "content": "denotes the actual Lipschitz constant of", + "type": "text" + }, + { + "bbox": [ + 344, + 318, + 405, + 331 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 317, + 410, + 333 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 317, + 410, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 344, + 354 + ], + "score": 1.0, + "content": "Proof. We use induction to show this lemma for different", + "type": "text" + }, + { + "bbox": [ + 344, + 345, + 350, + 353 + ], + "score": 0.74, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 342, + 462, + 354 + ], + "score": 1.0, + "content": "values. The result for case", + "type": "text" + }, + { + "bbox": [ + 462, + 344, + 487, + 353 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 379, + 366 + ], + "score": 1.0, + "content": "direct consequence of Lemma 5. Suppose that the result is true for", + "type": "text" + }, + { + "bbox": [ + 379, + 354, + 404, + 363 + ], + "score": 0.9, + "content": "r = k", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 353, + 505, + 366 + ], + "score": 1.0, + "content": ". Then, Lemmas 3 and 4", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 362, + 134, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 134, + 379 + ], + "score": 1.0, + "content": "imply", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 342, + 506, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 370, + 461, + 696 + ], + "lines": [ + { + "bbox": [ + 146, + 370, + 461, + 696 + ], + "spans": [ + { + "bbox": [ + 146, + 370, + 461, + 696 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { | | E _ { \\frac { d } { d } } ^ { \\mathrm { L C } } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) | \\leq x _ { 0 } ^ { 3 } , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) , } \\\\ & { \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ } \\\\ & { \\quad + \\ \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } \\ \\mathbf { S } ^ { \\varepsilon } \\ \\exp ( \\frac { 1 } { \\varepsilon } ) \\cdot \\mathbf { S } ^ { \\varepsilon } \\ ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) } \\\\ & \\leq \\frac { \\varepsilon ^ { 2 } } { \\varepsilon } \\log ( x _ { 0 } ^ { 4 } + \\varepsilon ( \\frac { 1 } { \\varepsilon } ) ^ { 2 } ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ { \\varepsilon } \\cdot \\mathbf { S } ^ { \\varepsilon } ) \\cdot ( \\mathbf { S } _ \\end{array}", + "type": "interline_equation", + "image_path": "254fb01e44573d1f0cd6ad01c629bab08618283f883fcce0a78692a7328d78f0.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 146, + 370, + 461, + 478.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 146, + 478.6666666666667, + 461, + 587.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 146, + 587.3333333333334, + 461, + 696.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 703, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 284, + 725 + ], + "score": 1.0, + "content": "where the last line follows from the equality", + "type": "text" + }, + { + "bbox": [ + 284, + 705, + 361, + 722 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { i = 0 } ^ { k } s ^ { i } = { \\frac { 1 - s ^ { k + 1 } } { 1 - s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 706, + 506, + 721 + ], + "score": 1.0, + "content": ". Therefore, by induction the lemma", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 192, + 732 + ], + "score": 1.0, + "content": "holds for every value", + "type": "text" + }, + { + "bbox": [ + 193, + 721, + 217, + 731 + ], + "score": 0.89, + "content": "r \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 720, + 221, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 703, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 200, + 95 + ], + "score": 1.0, + "content": "Lemma 7. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 83, + 207, + 92 + ], + "score": 0.59, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 81, + 331, + 95 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 331, + 83, + 344, + 94 + ], + "score": 0.87, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 81, + 497, + 95 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + }, + { + "bbox": [ + 497, + 87, + 504, + 92 + ], + "score": 0.71, + "content": "\\sigma", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 448, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 133, + 106 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 93, + 171, + 106 + ], + "score": 0.92, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 93, + 300, + 106 + ], + "score": 1.0, + "content": ". Also, assume that training loss", + "type": "text" + }, + { + "bbox": [ + 300, + 94, + 306, + 104 + ], + "score": 0.7, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 93, + 448, + 106 + ], + "score": 1.0, + "content": "is 1-Lipschitz and 1-smooth. Then,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 111, + 450, + 147 + ], + "lines": [ + { + "bbox": [ + 162, + 111, + 450, + 147 + ], + "spans": [ + { + "bbox": [ + 162, + 111, + 450, + 147 + ], + "score": 0.94, + "content": "\\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\bigr ) \\bigr ) \\leq \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : = \\bigl ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\bigr ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "ccfff99ac86d690054dbcb7e5eee8d466598fa27a5434265ae38f1184fb1f12c.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 111, + 450, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 162, + 123.0, + 450, + 135.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 162, + 135.0, + 450, + 147.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 326, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 326, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 326, + 173 + ], + "score": 1.0, + "content": "Proof. First of all note that according to the chain rule", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 177, + 424, + 258 + ], + "lines": [ + { + "bbox": [ + 187, + 177, + 424, + 258 + ], + "spans": [ + { + "bbox": [ + 187, + 177, + 424, + 258 + ], + "score": 0.96, + "content": "\\begin{array} { r l } & { \\displaystyle \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) = \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) ( \\nabla \\ell ) \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\operatorname* { l i p } ( f _ { \\mathbf { w } } ) ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "60522e9f0757d2290f2bebae880622c05525fd5e27a1431342a1fc567fa8614e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 187, + 177, + 424, + 193.2 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 187, + 193.2, + 424, + 209.39999999999998 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 187, + 209.39999999999998, + 424, + 225.59999999999997 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 187, + 225.59999999999997, + 424, + 241.79999999999995 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 187, + 241.79999999999995, + 424, + 257.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 429, + 277 + ], + "score": 1.0, + "content": "Considering the above result, we complete the proof by inductively proving", + "type": "text" + }, + { + "bbox": [ + 430, + 262, + 505, + 276 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\bigr ) \\ \\leq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 271, + 510, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 251, + 291 + ], + "score": 0.92, + "content": "( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i - 1 } \\| \\mathbf { W } _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 271, + 274, + 295 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 275, + 277, + 300, + 289 + ], + "score": 0.81, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 271, + 305, + 295 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 277, + 346, + 290 + ], + "score": 0.83, + "content": "\\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 271, + 510, + 295 + ], + "score": 1.0, + "content": "is constant and hence the result holds.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 352, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 234, + 302 + ], + "score": 1.0, + "content": "Assume the statement holds for", + "type": "text" + }, + { + "bbox": [ + 235, + 291, + 259, + 300 + ], + "score": 0.89, + "content": "d = k", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 289, + 352, + 302 + ], + "score": 1.0, + "content": ". Due to the chain rule,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 306, + 451, + 322 + ], + "lines": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "spans": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "score": 0.88, + "content": "\\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k + 1 ) } ( \\mathbf { x } ) = \\nabla _ { \\mathbf { x } } \\mathbf { W } _ { k + 1 } \\sigma \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) = \\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T }", + "type": "interline_equation", + "image_path": "472e9d0648034584ac6c5d561ea3806ea48c2cd4ce01585e3ea7860533a24a71.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 226, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 226, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 192, + 341 + ], + "score": 1.0, + "content": "and therefore for any", + "type": "text" + }, + { + "bbox": [ + 193, + 330, + 200, + 338 + ], + "score": 0.58, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 326, + 218, + 341 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 330, + 226, + 338 + ], + "score": 0.59, + "content": "\\mathbf { v }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 102, + 344, + 500, + 523 + ], + "lines": [ + { + "bbox": [ + 102, + 344, + 500, + 523 + ], + "spans": [ + { + "bbox": [ + 102, + 344, + 500, + 523 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\| _ { 2 } } \\\\ & { \\leq \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\nabla _ { x + 1 } ^ { F } - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T } \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\big \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\big \\| _ { 2 } } \\\\ & \\quad + \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ \\mathbf w \\end{array}", + "type": "interline_equation", + "image_path": "2fd590cd10ae9b63167fde1bc62c7d54406897c3db9c6ef8b50c7a1ff08fb33f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 102, + 344, + 500, + 403.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 102, + 403.6666666666667, + 500, + 463.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 102, + 463.33333333333337, + 500, + 523.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 527, + 488, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 487, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 252, + 541 + ], + "score": 1.0, + "content": "which shows the statement holds for", + "type": "text" + }, + { + "bbox": [ + 252, + 528, + 294, + 538 + ], + "score": 0.92, + "content": "d = k + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 526, + 487, + 541 + ], + "score": 1.0, + "content": "and therefore completes the proof via induction.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 506, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 366, + 567 + ], + "score": 1.0, + "content": "In order to prove Theorem 3, we note that for any norm-bounded", + "type": "text" + }, + { + "bbox": [ + 367, + 555, + 409, + 567 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and perturbation vector", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 565, + 282, + 579 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 114, + 576 + ], + "score": 0.42, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 565, + 153, + 579 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 154, + 566, + 164, + 576 + ], + "score": 0.72, + "content": "\\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 565, + 169, + 579 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 169, + 565, + 246, + 579 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 565, + 282, + 579 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 583, + 464, + 737 + ], + "lines": [ + { + "bbox": [ + 145, + 583, + 464, + 737 + ], + "spans": [ + { + "bbox": [ + 145, + 583, + 464, + 737 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\times \\mathbf { n } } , \\mathbf { r } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\quad + \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\frac { \\| \\mathbf { u } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\qquad \\times \\displaystyle ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ^ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\Big [ \\| \\mathbf { u } _ { i } \\| _ { 2 } } \\\\ & \\qquad \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\mathbf { Z } _ { i } \\| _ { 2 } + ( B + \\epsilon ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\end{array}", + "type": "interline_equation", + "image_path": "c7e04340b7ce9aac15757ad69ecbf01d745063c55f27d965482570750302c538.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 145, + 583, + 464, + 634.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 145, + 634.3333333333334, + 464, + 685.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 145, + 685.6666666666667, + 464, + 737.0000000000001 + ], + "spans": [], + "index": 24 + } + ] + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 529, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 496, + 531, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 496, + 531, + 504, + 538 + ], + "score": 0.989, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 200, + 95 + ], + "score": 1.0, + "content": "Lemma 7. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 83, + 207, + 92 + ], + "score": 0.59, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 81, + 331, + 95 + ], + "score": 1.0, + "content": "-layer neural network function", + "type": "text" + }, + { + "bbox": [ + 331, + 83, + 344, + 94 + ], + "score": 0.87, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 81, + 497, + 95 + ], + "score": 1.0, + "content": "with 1-Lipschitz, 1-smooth activation", + "type": "text" + }, + { + "bbox": [ + 497, + 87, + 504, + 92 + ], + "score": 0.71, + "content": "\\sigma", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 448, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 133, + 106 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 93, + 171, + 106 + ], + "score": 0.92, + "content": "\\sigma ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 93, + 300, + 106 + ], + "score": 1.0, + "content": ". Also, assume that training loss", + "type": "text" + }, + { + "bbox": [ + 300, + 94, + 306, + 104 + ], + "score": 0.7, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 93, + 448, + 106 + ], + "score": 1.0, + "content": "is 1-Lipschitz and 1-smooth. Then,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 504, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 111, + 450, + 147 + ], + "lines": [ + { + "bbox": [ + 162, + 111, + 450, + 147 + ], + "spans": [ + { + "bbox": [ + 162, + 111, + 450, + 147 + ], + "score": 0.94, + "content": "\\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\bigr ) \\bigr ) \\leq \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : = \\bigl ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\bigr ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } .", + "type": "interline_equation", + "image_path": "ccfff99ac86d690054dbcb7e5eee8d466598fa27a5434265ae38f1184fb1f12c.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 111, + 450, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 162, + 123.0, + 450, + 135.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 162, + 135.0, + 450, + 147.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 326, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 326, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 326, + 173 + ], + "score": 1.0, + "content": "Proof. First of all note that according to the chain rule", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 159, + 326, + 173 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 177, + 424, + 258 + ], + "lines": [ + { + "bbox": [ + 187, + 177, + 424, + 258 + ], + "spans": [ + { + "bbox": [ + 187, + 177, + 424, + 258 + ], + "score": 0.96, + "content": "\\begin{array} { r l } & { \\displaystyle \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } \\ell \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) = \\operatorname* { l i p } \\bigg ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) ( \\nabla \\ell ) \\big ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\big ) \\bigg ) } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\operatorname* { l i p } ( f _ { \\mathbf { w } } ) ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq \\operatorname* { l i p } \\big ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\big ) + \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "60522e9f0757d2290f2bebae880622c05525fd5e27a1431342a1fc567fa8614e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 187, + 177, + 424, + 193.2 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 187, + 193.2, + 424, + 209.39999999999998 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 187, + 209.39999999999998, + 424, + 225.59999999999997 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 187, + 225.59999999999997, + 424, + 241.79999999999995 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 187, + 241.79999999999995, + 424, + 257.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 429, + 277 + ], + "score": 1.0, + "content": "Considering the above result, we complete the proof by inductively proving", + "type": "text" + }, + { + "bbox": [ + 430, + 262, + 505, + 276 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } ) \\bigr ) \\ \\leq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 271, + 510, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 251, + 291 + ], + "score": 0.92, + "content": "( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i - 1 } \\| \\mathbf { W } _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 271, + 274, + 295 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 275, + 277, + 300, + 289 + ], + "score": 0.81, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 271, + 305, + 295 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 277, + 346, + 290 + ], + "score": 0.83, + "content": "\\nabla _ { \\mathbf { x } } f _ { \\mathbf { w } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 271, + 510, + 295 + ], + "score": 1.0, + "content": "is constant and hence the result holds.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 352, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 234, + 302 + ], + "score": 1.0, + "content": "Assume the statement holds for", + "type": "text" + }, + { + "bbox": [ + 235, + 291, + 259, + 300 + ], + "score": 0.89, + "content": "d = k", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 289, + 352, + 302 + ], + "score": 1.0, + "content": ". Due to the chain rule,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 262, + 510, + 302 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 306, + 451, + 322 + ], + "lines": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "spans": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "score": 0.88, + "content": "\\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k + 1 ) } ( \\mathbf { x } ) = \\nabla _ { \\mathbf { x } } \\mathbf { W } _ { k + 1 } \\sigma \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) = \\nabla _ { \\mathbf { x } } { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( { f } _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T }", + "type": "interline_equation", + "image_path": "472e9d0648034584ac6c5d561ea3806ea48c2cd4ce01585e3ea7860533a24a71.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 159, + 306, + 451, + 322 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 226, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 226, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 192, + 341 + ], + "score": 1.0, + "content": "and therefore for any", + "type": "text" + }, + { + "bbox": [ + 193, + 330, + 200, + 338 + ], + "score": 0.58, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 326, + 218, + 341 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 330, + 226, + 338 + ], + "score": 0.59, + "content": "\\mathbf { v }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 326, + 226, + 341 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 102, + 344, + 500, + 523 + ], + "lines": [ + { + "bbox": [ + 102, + 344, + 500, + 523 + ], + "spans": [ + { + "bbox": [ + 102, + 344, + 500, + 523 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\| _ { 2 } } \\\\ & { \\leq \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\nabla _ { x + 1 } ^ { F } - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T } \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\big \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\big \\| _ { 2 } } \\\\ & \\quad + \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ \\mathbf w \\end{array}", + "type": "interline_equation", + "image_path": "2fd590cd10ae9b63167fde1bc62c7d54406897c3db9c6ef8b50c7a1ff08fb33f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 102, + 344, + 500, + 403.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 102, + 403.6666666666667, + 500, + 463.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 102, + 463.33333333333337, + 500, + 523.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 527, + 488, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 487, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 252, + 541 + ], + "score": 1.0, + "content": "which shows the statement holds for", + "type": "text" + }, + { + "bbox": [ + 252, + 528, + 294, + 538 + ], + "score": 0.92, + "content": "d = k + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 526, + 487, + 541 + ], + "score": 1.0, + "content": "and therefore completes the proof via induction.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 526, + 487, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 506, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 366, + 567 + ], + "score": 1.0, + "content": "In order to prove Theorem 3, we note that for any norm-bounded", + "type": "text" + }, + { + "bbox": [ + 367, + 555, + 409, + 567 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and perturbation vector", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 565, + 282, + 579 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 114, + 576 + ], + "score": 0.42, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 565, + 153, + 579 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 154, + 566, + 164, + 576 + ], + "score": 0.72, + "content": "\\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 565, + 169, + 579 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 169, + 565, + 246, + 579 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 565, + 282, + 579 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 554, + 505, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 583, + 464, + 737 + ], + "lines": [ + { + "bbox": [ + 145, + 583, + 464, + 737 + ], + "spans": [ + { + "bbox": [ + 145, + 583, + 464, + 737 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\times \\mathbf { n } } , \\mathbf { r } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\quad + \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\frac { \\| \\mathbf { u } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\qquad \\times \\displaystyle ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ^ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\Big [ \\| \\mathbf { u } _ { i } \\| _ { 2 } } \\\\ & \\qquad \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\mathbf { Z } _ { i } \\| _ { 2 } + ( B + \\epsilon ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\end{array}", + "type": "interline_equation", + "image_path": "c7e04340b7ce9aac15757ad69ecbf01d745063c55f27d965482570750302c538.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 145, + 583, + 464, + 634.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 145, + 634.3333333333334, + 464, + 685.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 145, + 685.6666666666667, + 464, + 737.0000000000001 + ], + "spans": [], + "index": 24 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 162, + 79, + 438, + 115 + ], + "lines": [ + { + "bbox": [ + 162, + 79, + 438, + 115 + ], + "spans": [ + { + "bbox": [ + 162, + 79, + 438, + 115 + ], + "score": 0.93, + "content": "\\times \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + ( B + \\epsilon ) ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right]", + "type": "interline_equation", + "image_path": "61c3273dca94419f51c819a971f35a8ddb1318009ecf768649c89039085fc627.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 162, + 79, + 438, + 91.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 162, + 91.0, + 438, + 103.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 162, + 103.0, + 438, + 115.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 507, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 326, + 134 + ], + "score": 1.0, + "content": "The last inequality holds since as shown in Lemma", + "type": "text" + }, + { + "bbox": [ + 326, + 120, + 504, + 134 + ], + "score": 0.88, + "content": "7 \\ : \\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y ) \\bigr ) \\leq \\ : \\varlimsup _ { \\mathbf { \\theta } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) : =", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 131, + 508, + 151 + ], + "spans": [ + { + "bbox": [ + 107, + 133, + 254, + 149 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\left( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\right) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 131, + 347, + 151 + ], + "score": 1.0, + "content": ". Here the upper-bound", + "type": "text" + }, + { + "bbox": [ + 347, + 135, + 400, + 148 + ], + "score": 0.91, + "content": "\\overline { { \\mathrm { l i p } } } ( \\nabla \\ell \\circ f _ { \\widetilde { \\mathbf { w } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 131, + 416, + 151 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 416, + 136, + 426, + 146 + ], + "score": 0.53, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 131, + 508, + 151 + ], + "score": 1.0, + "content": "changes by a factor", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 148, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 138, + 165 + ], + "score": 1.0, + "content": "at most", + "type": "text" + }, + { + "bbox": [ + 139, + 150, + 157, + 161 + ], + "score": 0.89, + "content": "e ^ { 2 / r }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 148, + 223, + 165 + ], + "score": 1.0, + "content": "for w such that", + "type": "text" + }, + { + "bbox": [ + 223, + 149, + 355, + 165 + ], + "score": 0.93, + "content": "| | \\mathbf { W } _ { i } | | _ { 2 } - | | \\widetilde { \\mathbf { W } } _ { i } | | _ { 2 } \\Big | \\leq \\frac { 1 } { r _ { i } ^ { d } } | | \\widetilde { \\mathbf { W } } _ { i } | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 148, + 429, + 165 + ], + "score": 1.0, + "content": ". 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e ^ { 2 \\left( 2 \\alpha / \\kappa \\right) \\sqrt { \\operatorname* { l i p } } ( \\nabla \\ell \\circ f _ { \\widetilde { \\infty } } ) ^ { r } } } ( \\prod _ { i = 1 } ^ { d } \\Vert \\widetilde { \\mathbf { W } } _ { i } \\Vert _ { 2 } ) \\big ( 1 + \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\Vert \\widetilde { \\mathbf { W } } _ { j } \\Vert _ { 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e4a8e622ee5947f40fdbc3fd0aaf1055f9cbad3fef7b63c347425ebff3fe7906.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 124, + 195, + 527, + 205.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 124, + 205.66666666666666, + 527, + 216.33333333333331 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 124, + 216.33333333333331, + 527, + 226.99999999999997 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 509, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 215, + 249 + ], + "score": 1.0, + "content": "Then for any w satisfying", + "type": "text" + }, + { + "bbox": [ + 215, + 231, + 348, + 247 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\| _ { 2 } - \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right| \\le \\frac { 1 } { r d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 231, + 506, + 249 + ], + "score": 1.0, + "content": ", applying union bound shows that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 241, + 504, + 271 + ], + "spans": [ + { + "bbox": [ + 102, + 241, + 145, + 271 + ], + "score": 1.0, + "content": "assumptiholds for", + "type": "text" + }, + { + "bbox": [ + 154, + 241, + 205, + 271 + ], + "score": 1.0, + "content": "of Lemma 1 , and further", + "type": "text" + }, + { + "bbox": [ + 205, + 246, + 504, + 261 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\operatorname* { P r } _ { \\mathbf { u } } \\bigl ( \\operatorname* { m a x } _ { \\mathbf { x } \\in \\mathcal { X } } \\| f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { p g m } , r } ( \\mathbf { x } ) ) \\| _ { \\infty } \\leq \\frac { \\gamma } { 4 } \\bigr ) \\geq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 145, + 259, + 154, + 270 + ], + "spans": [ + { + "bbox": [ + 145, + 259, + 154, + 270 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 273, + 508, + 442 + ], + "lines": [ + { + "bbox": [ + 111, + 273, + 508, + 442 + ], + "spans": [ + { + "bbox": [ + 111, + 273, + 508, + 442 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { K L ( P _ { \\mathbf { w } + \\mathbf { u } } | | Q ) \\leq \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { E } } ^ { 2 } } { 2 \\xi _ { i } ^ { 2 } } } \\\\ & { = \\mathcal { O } \\Big ( d ^ { 2 } ( B + c ) ^ { 2 } \\widehat { l } \\mathrm { l o g } ( i a ) \\Big ) \\times } \\\\ & { \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 \\frac { 1 - 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( 2 \\alpha / \\kappa ) ^ { r } \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\Big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "97b7440ad3757a52be1c69c6019ec34bfd897c76ffb78076800818445fb923ff.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 532, + 507, + 544.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 544.0, + 507, + 556.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 556.0, + 507, + 568.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 506, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "Using a similar argument to our proof of Theorem 2, we can properly cover the spectral norms for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 127, + 595 + ], + "score": 1.0, + "content": "each", + "type": "text" + }, + { + "bbox": [ + 127, + 583, + 143, + 594 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 583, + 164, + 595 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 165, + 583, + 208, + 595 + ], + "score": 0.92, + "content": "2 r d \\log M", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 583, + 339, + 595 + ], + "score": 1.0, + "content": "points, such that for any feasible", + "type": "text" + }, + { + "bbox": [ + 340, + 583, + 370, + 595 + ], + "score": 0.92, + "content": "\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "value, satisfying the assumptions,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 167, + 609 + ], + "score": 1.0, + "content": "we have value", + "type": "text" + }, + { + "bbox": [ + 167, + 597, + 177, + 606 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 595, + 259, + 609 + ], + "score": 1.0, + "content": "in our cover where", + "type": "text" + }, + { + "bbox": [ + 259, + 594, + 350, + 608 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | \\| \\mathbf { W } _ { i } \\| _ { 2 } - a _ { i } | \\le \\frac { 1 } { r d } \\overset { \\cdot \\cdot } { a } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 595, + 506, + 609 + ], + "score": 1.0, + "content": ". Therefore, we can cover all feasible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 250, + 621 + ], + "score": 1.0, + "content": "combinations of spectral norms with", + "type": "text" + }, + { + "bbox": [ + 251, + 608, + 333, + 621 + ], + "score": 0.91, + "content": "( 2 r d \\log M ) ^ { d } d n ^ { 1 / 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 606, + 506, + 621 + ], + "score": 1.0, + "content": ", which combined with the above discussion", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 619, + 191, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 191, + 632 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 644, + 230, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 231, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 231, + 657 + ], + "score": 1.0, + "content": "C.4 PROOF OF THEOREM 4", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 449, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 449, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 449, + 677 + ], + "score": 1.0, + "content": "We first show the following lemma providing a perturbation bound for WRM attacks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 679, + 504, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 200, + 693 + ], + "score": 1.0, + "content": "Lemma 8. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 680, + 207, + 690 + ], + "score": 0.74, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 678, + 275, + 693 + ], + "score": 1.0, + "content": "-layer neural net", + "type": "text" + }, + { + "bbox": [ + 275, + 680, + 288, + 691 + ], + "score": 0.84, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "satisfying the assumptions of Lemma 3. 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e ^ { 2 } ( 2 a \\rho \\gamma ) ^ { \\top } \\widehat { \\mathbb { H } } ( \\nabla \\xi \\circ f _ { \\varphi _ { i } } ) } { 1 - e ^ { 2 } \\gamma ( 2 a \\rho \\gamma ) \\widehat { l } \\mathrm { l o g } ( \\widehat { \\mathbf { W } } \\xi \\circ f _ { \\varphi _ { i } } ) } } ^ { 2 } \\{ 1 + \\frac { \\alpha } { \\kappa } \\Big ( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\Big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } \\Big \\} ^ { 2 } \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { \\mathcal { E } } ^ { 2 } } { | \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } ^ { 2 } } \\Big ) } \\\\ & { \\leq \\mathcal { O } \\Big ( d ^ { 2 } ( B + c ) ^ { 2 } \\widehat { l } \\mathrm { l o g } ( k a ) \\Big ) \\times } \\\\ & \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 \\frac { 1 - ( 2 \\sigma \\gamma ) \\widehat { l } \\mathrm { l o g } ( \\gamma \\circ f _ { \\varphi _ { i } } ) r } { 1 - 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( 2 \\alpha / \\kappa ) ^ { r } \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\| \\mathbf { \\overline { { \\log } } } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\Big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "97b7440ad3757a52be1c69c6019ec34bfd897c76ffb78076800818445fb923ff.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 532, + 507, + 544.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 544.0, + 507, + 556.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 556.0, + 507, + 568.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 506, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "Using a similar argument to our proof of Theorem 2, we can properly cover the spectral norms for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 127, + 595 + ], + "score": 1.0, + "content": "each", + "type": "text" + }, + { + "bbox": [ + 127, + 583, + 143, + 594 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 583, + 164, + 595 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 165, + 583, + 208, + 595 + ], + "score": 0.92, + "content": "2 r d \\log M", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 583, + 339, + 595 + ], + "score": 1.0, + "content": "points, such that for any feasible", + "type": "text" + }, + { + "bbox": [ + 340, + 583, + 370, + 595 + ], + "score": 0.92, + "content": "\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "value, satisfying the assumptions,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 167, + 609 + ], + "score": 1.0, + "content": "we have value", + "type": "text" + }, + { + "bbox": [ + 167, + 597, + 177, + 606 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 595, + 259, + 609 + ], + "score": 1.0, + "content": "in our cover where", + "type": "text" + }, + { + "bbox": [ + 259, + 594, + 350, + 608 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | \\| \\mathbf { W } _ { i } \\| _ { 2 } - a _ { i } | \\le \\frac { 1 } { r d } \\overset { \\cdot \\cdot } { a } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 595, + 506, + 609 + ], + "score": 1.0, + "content": ". Therefore, we can cover all feasible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 250, + 621 + ], + "score": 1.0, + "content": "combinations of spectral norms with", + "type": "text" + }, + { + "bbox": [ + 251, + 608, + 333, + 621 + ], + "score": 0.91, + "content": "( 2 r d \\log M ) ^ { d } d n ^ { 1 / 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 606, + 506, + 621 + ], + "score": 1.0, + "content": ", which combined with the above discussion", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 619, + 191, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 191, + 632 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 572, + 506, + 632 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 644, + 230, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 231, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 231, + 657 + ], + "score": 1.0, + "content": "C.4 PROOF OF THEOREM 4", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 449, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 449, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 449, + 677 + ], + "score": 1.0, + "content": "We first show the following lemma providing a perturbation bound for WRM attacks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 664, + 449, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 679, + 504, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 200, + 693 + ], + "score": 1.0, + "content": "Lemma 8. Consider a", + "type": "text" + }, + { + "bbox": [ + 201, + 680, + 207, + 690 + ], + "score": 0.74, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 678, + 275, + 693 + ], + "score": 1.0, + "content": "-layer neural net", + "type": "text" + }, + { + "bbox": [ + 275, + 680, + 288, + 691 + ], + "score": 0.84, + "content": "f _ { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "satisfying the assumptions of Lemma 3. Then, for any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 690, + 349, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 236, + 703 + ], + "score": 1.0, + "content": "weight perturbation u such that", + "type": "text" + }, + { + "bbox": [ + 237, + 691, + 313, + 704 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 690, + 349, + 703 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 678, + 505, + 704 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 708, + 317, + 735 + ], + "lines": [ + { + "bbox": [ + 131, + 708, + 317, + 735 + ], + "spans": [ + { + "bbox": [ + 131, + 708, + 317, + 735 + ], + "score": 0.9, + "content": "\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\| _ { 2 } \\leq \\frac { e ^ { 2 } } { \\lambda - \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) }", + "type": "interline_equation", + "image_path": "40db64043a69a03d0f1ef32da4efb78c3dabfcec9b2edaa316a8888247d1ad8e.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 131, + 708, + 317, + 735 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 134, + 79, + 480, + 115 + ], + "lines": [ + { + "bbox": [ + 134, + 79, + 480, + 115 + ], + "spans": [ + { + "bbox": [ + 134, + 79, + 480, + 115 + ], + "score": 0.93, + "content": "\\times \\big ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } \\big ) \\sum _ { i = 1 } ^ { d } \\bigg [ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\big ( \\| \\mathbf { x } \\| _ { 2 } + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { j } \\| _ { 2 } } { \\lambda } \\big ) \\big ( \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\bigg ] .", + "type": "interline_equation", + "image_path": "aeae3990564f0ec40a28759b9474da665e234136b48aa1264ffe84007a35d2b6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 134, + 79, + 480, + 91.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 134, + 91.0, + 480, + 103.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 134, + 103.0, + 480, + 115.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 118, + 454, + 132 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 452, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 202, + 133 + ], + "score": 1.0, + "content": "In the above inequality,", + "type": "text" + }, + { + "bbox": [ + 202, + 119, + 255, + 131 + ], + "score": 0.89, + "content": "\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 118, + 390, + 133 + ], + "score": 1.0, + "content": "denotes the Lipschitz constant of", + "type": "text" + }, + { + "bbox": [ + 390, + 119, + 452, + 131 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf x } \\ell ( f _ { \\mathbf w } ( \\mathbf x ) , y )", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 104, + 143, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 143, + 259, + 160 + ], + "score": 1.0, + "content": "Proof. First of all note that for any", + "type": "text" + }, + { + "bbox": [ + 259, + 148, + 268, + 155 + ], + "score": 0.45, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 143, + 308, + 160 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 308, + 143, + 449, + 158 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\| _ { 2 } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 143, + 507, + 160 + ], + "score": 1.0, + "content": ", because we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 138, + 169 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 138, + 156, + 211, + 168 + ], + "score": 0.9, + "content": "\\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) < \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "implying WRM’s optimization is a convex optimization problem with the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 163, + 507, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 168, + 182 + ], + "score": 1.0, + "content": "global solution", + "type": "text" + }, + { + "bbox": [ + 168, + 168, + 203, + 179 + ], + "score": 0.89, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 163, + 245, + 182 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 246, + 167, + 395, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) = \\frac { 1 } { \\lambda } \\nabla \\ell \\circ f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 163, + 507, + 182 + ], + "score": 1.0, + "content": "which is norm-bounded by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 175, + 392, + 201 + ], + "spans": [ + { + "bbox": [ + 107, + 179, + 228, + 195 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { \\operatorname* { l i p } ( \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 175, + 392, + 201 + ], + "score": 1.0, + "content": ". Moreover, applying Lemma 3 we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 199, + 484, + 348 + ], + "lines": [ + { + "bbox": [ + 125, + 199, + 484, + 348 + ], + "spans": [ + { + "bbox": [ + 125, + 199, + 484, + 348 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { = \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\frac { \\epsilon ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\Big [ \\big \\| \\mathbf { U } _ { i } \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\frac { \\operatorname* { l i p } ( \\nabla \\ell \\cup f _ { \\mathbf { w } } ) } { \\lambda } \\big \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - 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\\frac { \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\bigr ) \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { \\leq \\frac { e ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\bigl ( \\| \\mathbf { x } \\| _ { 2 } + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { j } \\| _ { 2 } } { \\lambda } \\bigr ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "1266de337953773da1104741952e4a8b54367e0f8b7d128949cf5b96105eef5a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 126, + 366, + 484, + 386.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 126, + 386.6666666666667, + 484, + 407.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 126, + 407.33333333333337, + 484, + 428.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 506, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 389, + 467 + ], + "score": 1.0, + "content": "Combining the above lemma with Lemma 2, for any norm-bounded", + "type": "text" + }, + { + "bbox": [ + 389, + 455, + 434, + 467 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "and perturbation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 465, + 250, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 133, + 479 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 134, + 468, + 141, + 476 + ], + "score": 0.66, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 465, + 169, + 479 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 169, + 465, + 246, + 479 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 465, + 250, + 479 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 482, + 500, + 659 + ], + "lines": [ + { + "bbox": [ + 111, + 482, + 500, + 659 + ], + "spans": [ + { + "bbox": [ + 111, + 482, + 500, + 659 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\int _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - 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First of all note that for any", + "type": "text" + }, + { + "bbox": [ + 259, + 148, + 268, + 155 + ], + "score": 0.45, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 143, + 308, + 160 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 308, + 143, + 449, + 158 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\| _ { 2 } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 143, + 507, + 160 + ], + "score": 1.0, + "content": ", because we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 138, + 169 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 138, + 156, + 211, + 168 + ], + "score": 0.9, + "content": "\\mathrm { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) < \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "implying WRM’s optimization is a convex optimization problem with the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 163, + 507, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 168, + 182 + ], + "score": 1.0, + "content": "global solution", + "type": "text" + }, + { + "bbox": [ + 168, + 168, + 203, + 179 + ], + "score": 0.89, + "content": "\\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 163, + 245, + 182 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 246, + 167, + 395, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) = \\frac { 1 } { \\lambda } \\nabla \\ell \\circ f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 163, + 507, + 182 + ], + "score": 1.0, + "content": "which is norm-bounded by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 175, + 392, + 201 + ], + "spans": [ + { + "bbox": [ + 107, + 179, + 228, + 195 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { \\operatorname* { l i p } ( \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\leq ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) / \\lambda } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 175, + 392, + 201 + ], + "score": 1.0, + "content": ". Moreover, applying Lemma 3 we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 143, + 507, + 201 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 199, + 484, + 348 + ], + "lines": [ + { + "bbox": [ + 125, + 199, + 484, + 348 + ], + "spans": [ + { + "bbox": [ + 125, + 199, + 484, + 348 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { = \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\big \\| \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) - \\frac { 1 } { \\lambda } \\nabla _ { \\mathbf { x } } \\ell ( f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) ) ) \\big \\| _ { 2 } } \\\\ & { \\leq \\frac { \\epsilon ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\Big [ \\big \\| \\mathbf { U } _ { i } \\big \\| _ { 2 } } \\\\ & { \\quad \\quad + \\frac { \\operatorname* { l i p } ( \\nabla \\ell \\cup f _ { \\mathbf { w } } ) } { \\lambda } \\big \\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { s r m } } ( \\mathbf { x } ) \\big \\| _ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "5c09856ed51bcb3b14383e5b027fa549582d027dc9f57aaef98cb9f58dc51b8f.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 125, + 199, + 484, + 248.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 125, + 248.66666666666666, + 484, + 298.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 125, + 298.3333333333333, + 484, + 348.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 385, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 385, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 385, + 365 + ], + "score": 1.0, + "content": "which shows the following inequality and hence completes the proof:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 350, + 385, + 365 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 366, + 484, + 428 + ], + "lines": [ + { + "bbox": [ + 126, + 366, + 484, + 428 + ], + "spans": [ + { + "bbox": [ + 126, + 366, + 484, + 428 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\quad \\bigl ( 1 - \\frac { \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } { \\lambda } \\bigr ) \\left\\| \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) - \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) \\right\\| _ { 2 } } \\\\ & { \\leq \\frac { e ^ { 2 } } { \\lambda } ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\left[ \\frac { \\| \\mathbf { U } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + \\bigl ( \\| \\mathbf { x } \\| _ { 2 } + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { j } \\| _ { 2 } } { \\lambda } \\bigr ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } ) \\sum _ { j = 1 } ^ { i } \\frac { \\| \\mathbf { U } _ { j } \\| _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "1266de337953773da1104741952e4a8b54367e0f8b7d128949cf5b96105eef5a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 126, + 366, + 484, + 386.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 126, + 386.6666666666667, + 484, + 407.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 126, + 407.33333333333337, + 484, + 428.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 506, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 389, + 467 + ], + "score": 1.0, + "content": "Combining the above lemma with Lemma 2, for any norm-bounded", + "type": "text" + }, + { + "bbox": [ + 389, + 455, + 434, + 467 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 2 } \\leq B", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "and perturbation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 465, + 250, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 133, + 479 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 134, + 468, + 141, + 476 + ], + "score": 0.66, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 465, + 169, + 479 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 169, + 465, + 246, + 479 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\mathbf { U } _ { i } \\| _ { 2 } \\leq \\frac { 1 } { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 465, + 250, + 479 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 454, + 505, + 479 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 482, + 500, + 659 + ], + "lines": [ + { + "bbox": [ + 111, + 482, + 500, + 659 + ], + "spans": [ + { + "bbox": [ + 111, + 482, + 500, + 659 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\int _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - \\int _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s e m } } ( \\mathbf { x } ) ) \\right\\| _ { 2 } } \\\\ & { \\leq \\Big \\| \\int _ { \\mathbf { w } + \\mathbf { u } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { s e m } } ( \\mathbf { x } ) ) \\Big \\| _ { 2 } + \\Big \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathrm { m a x } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { m m } } ( \\mathbf { x } ) ) \\Big \\| _ { 2 } } \\\\ & { \\leq e ( B + \\frac { \\prod _ { j = 1 } ^ { d } | \\mathbf { w } _ { j } | } { \\lambda } ) ( \\underset { \\mathrm { L } ^ { - 1 } = 1 } { \\overset { d } { \\prod } } | \\mathbb { W } _ { 1 } | _ { 2 } ) \\underset { i = 1 } { \\overset { d } { \\sum } } \\Big \\| \\mathbb { W } _ { 1 } | _ { 2 } + \\Big ( \\underset { i = 1 } { \\overset { d } { \\prod } } | \\mathbb { W } _ { 1 } | _ { 2 } ) } \\\\ & { \\qquad \\times \\frac { e ^ { 2 } } { \\lambda - \\operatorname { l i p } ( \\nabla \\xi \\ell \\int _ { \\mathbf { w } } ) } \\underset { i = 1 } { \\overset { d } { \\sum } } \\bigg [ \\frac { \\| \\mathbf { U } _ { 1 } | _ { 2 } } { \\| \\mathbf { W } _ { 1 } \\| _ { 2 } } + \\big ( B + \\frac { \\prod _ { j = 1 } ^ { d } \\| \\mathbf { W } _ { 2 } \\| _ { 2 } } { \\lambda } \\big ) ( \\underset { j = 1 } { \\overset { i } { \\prod } } | \\mathbf { W } _ { j } | | _ { 2 } ) \\underset { j = 1 } { \\overset { i } { \\sum } } \\frac { \\| \\mathbf { U } _ { j } | _ { 2 } } { \\| \\mathbf { W } _ { j } \\| _ { 2 } } \\bigg ] } \\\\ & \\leq e \\end{array}", + "type": "interline_equation", + "image_path": "6e58575410104a826a198410cb638a01898b0e92cbdcde0ad935ddac65011fac.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 482, + 500, + 541.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 541.0, + 500, + 600.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 111, + 600.0, + 500, + 659.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 506, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 299, + 673 + ], + "score": 1.0, + "content": "Similar to the proofs of Theorems 2,3, given", + "type": "text" + }, + { + "bbox": [ + 299, + 662, + 309, + 672 + ], + "score": 0.74, + "content": "\\widetilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "we choose a zero-mean multivariate Gaussian", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 156, + 685 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 673, + 165, + 684 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 673, + 448, + 685 + ], + "score": 1.0, + "content": "e with diagonal covariance matrix for random perturbation u, with the", + "type": "text" + }, + { + "bbox": [ + 449, + 674, + 453, + 682 + ], + "score": 0.35, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 673, + 486, + 685 + ], + "score": 1.0, + "content": "th layer", + "type": "text" + }, + { + "bbox": [ + 486, + 673, + 497, + 684 + ], + "score": 0.85, + "content": "\\mathbf { u } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 683, + 308, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 225, + 701 + ], + "score": 1.0, + "content": "standard deviation parameter", + "type": "text" + }, + { + "bbox": [ + 225, + 684, + 278, + 702 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\xi _ { i } = \\frac { \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } } { \\beta _ { \\widetilde { \\mathbf { w } } } } \\xi } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 683, + 308, + 702 + ], + "score": 1.0, + "content": "kWfik2β ξ where", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 662, + 505, + 702 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 705, + 514, + 735 + ], + "lines": [ + { + "bbox": [ + 111, + 705, + 514, + 735 + ], + "spans": [ + { + "bbox": [ + 111, + 705, + 514, + 735 + ], + "score": 0.9, + "content": "\\begin{array} { r } { = \\frac { \\gamma } { 8 e ^ { 5 } d \\sqrt { 2 h \\log ( 4 h d ) } ( B + \\prod _ { j = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { j } \\| _ { 2 } / \\lambda ) \\left( \\prod _ { i = 1 } ^ { d } \\| \\widetilde { \\mathbf { W } } _ { i } \\| _ { 2 } \\right) \\left( 1 + \\frac { 1 } { \\lambda - 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\\right. } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 114, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 107, + 115, + 243, + 130 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathrm { w r m } } ( \\mathbf { x } ) ) \\| _ { \\infty } \\leq \\frac { \\gamma } { 4 } \\big ) \\geq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 114, + 284, + 132 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 284, + 117, + 293, + 128 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 114, + 412, + 132 + ], + "score": 1.0, + "content": ". 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\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "bf86ec09cd182e57c6ebca3332801aaa6ece085eba897c5c94a2bdc3d218e69c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 138, + 453, + 471, + 465.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 138, + 465.3333333333333, + 471, + 477.66666666666663 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 138, + 477.66666666666663, + 471, + 489.99999999999994 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 494, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Using a similar argument to our proofs of Theorems 2 and 3, we can cover the possible spectral", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 504, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 168, + 519 + ], + "score": 1.0, + "content": "norms for each", + "type": "text" + }, + { + "bbox": [ + 168, + 506, + 184, + 517 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 504, + 205, + 519 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 206, + 506, + 277, + 518 + ], + "score": 0.93, + "content": "O ( ( 8 d / \\tau ) \\log M )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 504, + 409, + 519 + ], + "score": 1.0, + "content": "points, such that for any feasible", + "type": "text" + }, + { + "bbox": [ + 409, + 506, + 439, + 517 + ], + "score": 0.9, + "content": "\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 504, + 506, + 519 + ], + "score": 1.0, + "content": "value satisfying", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 516, + 508, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 272, + 533 + ], + "score": 1.0, + "content": "the theorem’s assumptions, we have value", + "type": "text" + }, + { + "bbox": [ + 273, + 519, + 282, + 529 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 516, + 360, + 533 + ], + "score": 1.0, + "content": "in our cover where", + "type": "text" + }, + { + "bbox": [ + 360, + 517, + 459, + 532 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\right\\| _ { 2 } - 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\\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { W } _ { i } \\| _ { 2 } ) \\sum _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { i } \\| \\mathbf { W } _ { j } \\| _ { 2 } \\big ) \\big \\} ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\| \\mathbf { W } _ { i } \\| _ { F } ^ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "bf86ec09cd182e57c6ebca3332801aaa6ece085eba897c5c94a2bdc3d218e69c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 138, + 453, + 471, + 465.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 138, + 465.3333333333333, + 471, + 477.66666666666663 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 138, + 477.66666666666663, + 471, + 489.99999999999994 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 494, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Using a similar argument to our proofs of Theorems 2 and 3, we can cover the possible spectral", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 504, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 168, + 519 + ], + "score": 1.0, + "content": "norms for each", + "type": "text" + }, + { + "bbox": [ + 168, + 506, + 184, + 517 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 504, + 205, + 519 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 206, + 506, + 277, + 518 + ], + "score": 0.93, + "content": "O ( ( 8 d / \\tau ) \\log M )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 504, + 409, + 519 + ], + "score": 1.0, + "content": "points, such that for any feasible", + "type": "text" + }, + { + "bbox": [ + 409, + 506, + 439, + 517 + ], + "score": 0.9, + "content": "\\Vert \\mathbf { W } _ { i } \\Vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 504, + 506, + 519 + ], + "score": 1.0, + "content": "value satisfying", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 516, + 508, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 272, + 533 + ], + "score": 1.0, + "content": "the theorem’s assumptions, we have value", + "type": "text" + }, + { + "bbox": [ + 273, + 519, + 282, + 529 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 516, + 360, + 533 + ], + "score": 1.0, + "content": "in our cover where", + "type": "text" + }, + { + "bbox": [ + 360, + 517, + 459, + 532 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left| \\| \\mathbf { W } _ { i } \\right\\| _ { 2 } - a _ { i } \\Big | \\le \\frac { 1 } { 4 d / \\tau } a _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 516, + 508, + 533 + ], + "score": 1.0, + "content": ". 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Algorithm1 Convolutional power iteration
Initialize ü with a random vector matching the shape of the convolution input
for t = 0,...,T-1do
ν ← conv(W,u)/llconv(W,u)ll2
ü ← conv_transpose(W,v)/llconv_transpose(W,v)ll2
end for
σ ←v· conv(W,u)
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DatasetArchitectureTrainingTrain accTest accTrain acc (SN)Test acc (SN)
CIFAR10AlexNetERM1.000.791.000.79
CIFAR10AlexNetFGM l20.980.540.930.63
CIFAR10AlexNetFGM lo1.000.510.670.56
CIFAR10AlexNetPGM l20.990.500.920.62
CIFAR10AlexNetPGMlo0.990.440.860.54
CIFAR10AlexNetWRM1.000.610.760.65
CIFAR10ELU-AlexNetERM1.000.791.000.79
CIFAR10ELU-AlexNetFGM l20.970.520.680.60
CIFAR10ELU-AlexNetPGMl20.980.530.880.61
CIFAR10ELU-AlexNetWRM1.000.601.000.60
CIFAR10InceptionERM1.000.851.000.86
CIFAR10InceptionPGM l20.990.531.000.58
CIFAR10InceptionPGM lo0.980.480.620.56
CIFAR10InceptionWRM1.000.661.000.67
CIFAR101-layer MLPERM0.980.490.680.53
CIFAR101-layer MLPFGM l20.600.360.600.46
CIFAR101-layer MLPPGM l20.570.360.550.46
CIFAR101-layer MLPWRM0.600.410.620.50
CIFAR102-layer MLPERM0.990.510.790.56
CIFAR102-layer MLPFGM l20.570.360.660.49
CIFAR102-layer MLPPGM l20.930.350.660.48
CIFAR102-layer MLPWRM0.870.350.730.52
CIFAR10ResNetERM1.000.801.000.83
CIFAR10ResNetPGM l20.990.491.000.55
CIFAR10ResNetPGM lo0.980.440.720.53
CIFAR10ResNetWRM1.000.631.000.66
MNISTELU-NetERM1.000.991.000.99*
MNISTELU-NetFGM l20.980.971.000.97
MNISTELU-NetPGM l20.990.971.000.97
MNISTELU-NetWRM0.950.920.950.93
MNIST1-layer MLPERM1.000.981.000.98*
MNIST1-layer MLPFGM l20.880.881.000.96
MNIST1-layer MLPPGM l21.000.961.000.96
MNIST1-layer MLPWRM0.920.880.920.88
MNIST2-layer MLPERM1.000.981.000.98
MNIST2-layer MLPFGM l20.970.911.000.96
MNIST2-layer MLPPGM l21.000.961.000.97
MNIST2-layer MLPWRM0.970.880.980.90
SVHNAlexNetERM1.000.931.000.93*
SVHNAlexNetFGM l20.970.760.950.83
SVHNAlexNetPGM l21.000.780.850.81
SVHNAlexNetWRM1.000.830.870.84
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DatasetArchitectureTrainingno SN runtime SN runtimeratio
CIFAR10AlexNetERM229 s283 s1.24
CIFAR10AlexNetFGM l2407 s463 s1.14
CIFAR10AlexNetFGM loo408 s465 s1.14
CIFAR10AlexNetPGM l22917 s3077 s1.05
CIFAR10AlexNetPGM lo2896 s3048 s1.05
CIFAR10AlexNetWRM3076 s3151 s1.02
CIFAR10ELU-AlexNetERM231 s283 s1.23
CIFAR10ELU-AlexNetFGM l2410 s466 s1.14
CIFAR10ELU-AlexNetPGM l22939 s3093 s1.05
CIFAR10ELU-AlexNetWRM3094 s3150 s1.02
CIFAR10InceptionERM632 s734 s1.16
CIFAR10InceptionPGM l29994 s6082 s0.61
CIFAR10InceptionPGM lo9948 s6063 s0.61
CIFAR10InceptionWRM10247 s6356 s0.62
CIFAR101-layer MLPERM22 s31 s1.42
CIFAR101-layer MLPFGM l225 s35s1.43
CIFAR101-layer MLPPGM l279 s93 s1.18
CIFAR101-layer MLPWRM73 s86 s1.18
CIFAR102-layer MLPERM23 s37 s1.59
CIFAR102-layer MLPFGM l227 s41 s1.51
CIFAR102-layer MLPPGMl291 s108 s1.19
CIFAR102-layer MLPWRM85 s103 s1.21
CIFAR10ResNetERM315 s547 s1.73
CIFAR10ResNetPGM l22994 s3300 s1.10
CIFAR10ResNetPGM lo2980 s3300 s1.11
CIFAR10ResNetWRM3187 s3457 s1.08
MNISTELU-NetERM55 s97s1.76
MNISTELU-NetFGM l291s136 s1.49
MNISTELU-NetPGM l2614 s676 s1.10
MNISTELU-NetWRM635 s670 s1.06
MNIST1-layer MLPERM15 s24 s1.60
MNIST1-layer MLPFGM l217 s27s1.57
MNIST1-layer MLPPGM l257s71 s1.24
MNIST1-layer MLPWRM51 s63 s1.24
MNIST2-layer MLPERM17 s31 s1.84
MNIST2-layer MLPFGM l220 s35 s1.77
MNIST2-layer MLPPGM l267 s89 s1.32
MNIST2-layer MLPWRM62 s81 s1.30
SVHNAlexNetERM334 s412 s1.23
SVHNAlexNetFGM l2596 s676 s1.13
SVHNAlexNetPGM l24270 s4495 s1.05
SVHNAlexNetWRM4501 s4572 s1.02
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DatasetArchitectureTrainingproposed SN runtime Miyato SN runtime
CIFAR10AlexNetERM1.11
CIFAR10AlexNetFGM l21.06
CIFAR10AlexNetFGMloo1.11
CIFAR10AlexNetPGM l21.01
CIFAR10AlexNetPGM lo1.11
CIFAR10AlexNetWRM1.02
CIFAR10InceptionERM0.98
CIFAR10InceptionPGM l21.04
CIFAR10InceptionPGM loo1.06
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+ \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\quad + \\| f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } + \\mathbf { u } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) - f _ { \\mathbf { w } } ( \\mathbf { x } + \\delta _ { \\mathbf { w } } ^ { \\mathbf { w } \\mathbf { n } , \\mathbf { r } } ( \\mathbf { x } ) ) \\| _ { 2 } } \\\\ & { \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\frac { \\| \\mathbf { u } _ { i } \\| _ { 2 } } { \\| \\mathbf { W } _ { i } \\| _ { 2 } } + e ^ { 2 } ( 2 \\alpha / \\kappa ) \\frac { 1 - ( 2 \\alpha / \\kappa ) ^ { r } \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) ^ { r } } { 1 - ( 2 \\alpha / \\kappa ) \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } ) } } \\\\ & { \\qquad \\times \\displaystyle ( \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ^ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\Big [ \\| \\mathbf { u } _ { i } \\| _ { 2 } } \\\\ & \\qquad \\leq e ( B + \\epsilon ) ( \\displaystyle \\prod _ { i = 1 } ^ { d } \\| \\mathbf { w } _ { i } \\| _ { 2 } ) \\displaystyle \\sum _ { i = 1 } ^ { d } \\| \\mathbf { Z } _ { i } \\| _ { 2 } + ( B + \\epsilon ) ( \\displaystyle \\prod _ { j = 1 } ^ { i } \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 285, + 958, + 1391, + 958, + 1391, + 1454, + 285, + 1454 + ], + "score": 0.94, + "latex": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\| _ { 2 } } \\\\ & { \\leq \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\nabla _ { x + 1 } ^ { F } - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\mathbf { W } _ { k + 1 } ^ { T } \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\big ) \\big \\| _ { 2 } } \\\\ & { \\leq \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) \\big \\| _ { 2 } } \\\\ & \\quad + \\| \\mathbf { W } _ { k + 1 } \\| _ { 2 } \\big \\| \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\sigma ^ { \\prime } \\big ( f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } + \\mathbf { v } ) \\big ) - \\nabla _ { x } f _ { \\mathbf { w } } ^ { ( k ) } ( \\mathbf { x } ) \\sigma ^ { \\prime } \\big ( f _ \\mathbf w \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 452, + 309, + 1252, + 309, + 1252, + 409, + 452, + 409 + ], + "score": 0.94, + "latex": "\\operatorname* { l i p } \\bigl ( \\nabla _ { \\mathbf { x } } \\ell \\bigl ( f _ { \\mathbf { w } } ( \\mathbf { x } ) , y \\bigr ) \\bigr ) \\leq \\operatorname* { l i p } ( \\nabla \\ell \\circ f _ { \\mathbf { w } } 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Work? + +Samuel Stanton NYU + +Pavel Izmailov NYU + +Polina Kirichenko NYU + +Alexander A. Alemi Google Research + +Andrew Gordon Wilson NYU + +# Abstract + +Knowledge distillation is a popular technique for training a small student network to emulate a larger teacher model, such as an ensemble of networks. We show that while knowledge distillation can improve student generalization, it does not typically work as it is commonly understood: there often remains a surprisingly large discrepancy between the predictive distributions of the teacher and the student, even in cases when the student has the capacity to perfectly match the teacher. We identify difficulties in optimization as a key reason for why the student is unable to match the teacher. We also show how the details of the dataset used for distillation play a role in how closely the student matches the teacher — and that more closely matching the teacher paradoxically does not always lead to better student generalization. + +# 1 Introduction + +Large, deep networks can learn representations that generalize well. While smaller, more efficient networks lack the inductive biases to find these representations from training data alone, they may have the capacity to represent these solutions [e.g., 2, 18, 32, 45]. Influential work on knowledge distillation $\bar { \| 2 2 \| }$ argues that Bucila et al. ˘ [5] “demonstrate convincingly that the knowledge acquired by a large ensemble of models [the teacher] can be transferred to a single small model [the student]”. Indeed this quote encapsulates the conventional narrative of knowledge distillation: a student model learns a high-fidelity representation of a larger teacher, enabled by the teacher’s soft labels. + +Conversely, in Figure $^ 1$ we show that with modern architectures knowledge distillation can lead to students with very different predictions from their teachers, even when the student has the capacity to perfectly match the teacher. Indeed, it is becoming well-known that in self-distillation the student fails to match the teacher and, paradoxically, student generalization improves as a result [14, 40]. However, when the teacher is a large model (e.g. a deep ensemble) improvements in fidelity translate into improvements in generalization, as we show in Figure $\boxed { 1 } \mathbf { ( b ) }$ . For these large models there is still a significant accuracy gap between student and teacher, so fidelity is aligned with generalization. + +We will distinguish between fidelity, the ability of a student to match a teacher’s predictions, and generalization, the performance of a student in predicting unseen, in-distribution data. We show that in many cases it is surprisingly difficult to obtain good student fidelity. In Section 5 we investigate the hypothesis that low fidelity is an identifiability problem that can be solved by augmenting the distillation dataset. In Section $6$ we investigate the hypothesis that low fidelity is an optimization problem resulting in a failure of the student to match the teacher even on the original training dataset. We present a summary of our conclusions in Section 7. + +Does knowledge distillation really work? In short: Yes, in the sense that it often improves student generalization. No, in that knowledge distillation often fails to live up to its name, transferring very limited knowledge from teacher to student. + +![](images/b16f411144a106e66a8b6b067f80e336493e1572c29f9021f516793d8553eb0c.jpg) +Figure 1: Evaluating the fidelity of knowledge distillation. The effect of enlarging the CIFAR-100 distillation dataset with GAN-generated samples. (a): The student and teacher are both single ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases. (b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. Student fidelity again increases as the dataset grows, but test accuracy now slightly increases. The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials. + +# 2 Related Work + +Knowledge distillation can improve model efficiency [38, 45], unsupervised domain adaptation [37], improved object detection $\pmb { \Vert }$ , model transparency $\lVert \rVert \bigotimes \rVert$ , and adversarial robustness [15, 42]. + +Seminal work by Bucila et al. ˘ [5] showed that teacher-ensembles with thousands of simple components could be compressed into a single shallow network that matched or outperformed its teacher. Other early work proposed distilling ensembles of shallow networks into a single network [55], an idea which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53]. Recently Fakoor et al. [13] developed a data-augmentation scheme for the distillation of large ensembles of simple models for tabular data, achieving impressive results on a wide range of tabular benchmarks. Malinin et al. [35] proposed a method to model the implicit distribution over predictive distributions from which the ensemble component predictive distributions are drawn, rather than just the ensemble model average. + +Our work focuses explicitly on student fidelity, decoupling our understanding of good fidelity from good generalization. We show that achieving good fidelity is extremely difficult, even with a variety of interventions, and seek to understand, by systematically considering several hypotheses, why knowledge distillation does not produce high fidelity students for modern architectures and datasets. In contrast, the distillation literature focuses largely on improving student generalization, without particularly distinguishing between fidelity and generalization. + +For example, concurrent work by Beyer et al. [4] does not carefully distinguish generalization and fidelity metrics, but they assert that high student fidelity is conceptually desirable and apparently difficult to achieve when measured as the gap between teacher and student accuracy. As a result their work focuses most heavily on practical modifications to the distillation procedure for the best student top-1 accuracy. In this paper we investigate many of the same prescriptions, including careful treatment of data augmentation (such as showing the teacher and student the exact same input images), the addition of MixUp, and extended training duration. We also find that such interventions do improve student accuracy, but there still remains a large discrepancy between the predictive distributions of the teacher and the student. We also investigate multiple optimizers. While we do not pursue Shampoo $[ [ 1 7 , \mathbb { I } ] ]$ specifically, Beyer et al. $\mathbb { H }$ find similar qualitative results for Shampoo and Adam, besides faster convergence for Shampoo. + +# 3 Preliminaries + +We will focus on the supervised classification setting, with input space $\mathcal { X }$ and label space $\mathcal { V }$ , where $| { \mathcal { V } } | = c$ . Let $f : \mathcal { X } \times \Theta \mathbb { R } ^ { c }$ be a classifier parameterized by $\theta \in \Theta$ whose outputs define a categorical predictive distribution over $\mathcal { V }$ , $\hat { p } ( y = i | \mathbf { x } ) = \sigma _ { i } ( f ( \mathbf { x } , \theta ) )$ , where $\sigma _ { i } ( { \bf z } ) : = \dot { \exp ( z _ { i } ) } / \sum _ { j } \exp \bar { ( } z _ { j } )$ is the softmax link function. We will often refer to the outputs of a classifier $\mathbf { z } : = f ( \mathbf { x } , \theta )$ as logits. For convenience, we will use $t$ and $s$ as shorthand for $f _ { \mathrm { t e a c h e r } }$ and $f _ { \mathrm { s t u d e n t } }$ , respectively. When the teacher is an $m$ -component ensemble, the component logits $\left( \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { m } \right)$ , where $\mathbf { z } _ { i } = f _ { i } ( \mathbf { x } , \theta _ { i } )$ , are combined to form the teacher logits: $\begin{array} { r } { \mathbf { z } _ { t } = \log \bar { ( \sum _ { i = 1 } ^ { m } \sigma ( \mathbf { \bar { z } } _ { i } ) / m ) } } \end{array}$ . These combined logits correspond to the predictive distribution of the ensemble model average. The experiments in the main text consider $m \in \{ 1 , 3 , 5 \}$ , and we include results up to $m = 1 2$ in Appendix B.2.1 + +# 3.1 Knowledge Distillation + +Hinton et al. $\pmb { \mathbb { D } } 2 \mathbf { \mathbb { I } }$ proposed a simple approach to knowledge distillation. The student minimizes a weighted combination of two objectives, $\mathcal { L } _ { s } : = \alpha \mathcal { L } _ { \mathrm { N L L } } + ( 1 - \alpha ) \mathcal { L } _ { \mathrm { K D } }$ , where $\alpha \in [ 0 , 1 )$ . Specifically, + +$$ +\mathcal { L } _ { \mathrm { N L L } } ( \mathbf { z } _ { s } , \mathbf { y } ) : = - \sum _ { j = 1 } ^ { c } y _ { j } \log \sigma _ { j } ( \mathbf { z } _ { s } ) , ~ \mathcal { L } _ { \mathrm { K D } } ( \mathbf { z } _ { s } , \mathbf { z } _ { t } ) : = - \tau ^ { 2 } \sum _ { j = 1 } ^ { c } \sigma _ { j } \left( \frac { \mathbf { z } _ { t } } { \tau } \right) \log \sigma _ { j } \left( \frac { \mathbf { z } _ { s } } { \tau } \right) . +$$ + +$\mathcal { L } _ { \mathrm { N L L } }$ is the usual supervised cross-entropy between the student logits $\mathbf { z } _ { s }$ and the one-hot labels $\mathbf { y }$ . Recalling that $\begin{array} { r } { \mathrm { K L } ( p | | q ) = \sum _ { j } p _ { j } ( \log q _ { j } - \log p _ { j } ) } \end{array}$ , we see that $\mathcal { L } _ { \mathrm { N L L } }$ is equivalent (up to a constant) to the KL from the empirical data distribution to the student predictive distribution $( \hat { p } _ { s } )$ . ${ \mathcal { L } } _ { \mathrm { K D } }$ is the added knowledge distillation term that encourages the student to match the teacher. It is the cross-entropy between the teacher and student predictive distributions $\hat { p } _ { t } = \sigma ( \mathbf { z } _ { t } )$ and $\hat { p } _ { s } = \sigma ( { \bf z } _ { s } )$ , both scaled by a temperature hyperparameter $\tau > 0$ . If $\tau = 1$ then ${ \mathcal { L } } _ { \mathrm { K D } }$ is similarly equivalent to the KL from the teacher to the student, $\mathrm { K L } ( \hat { p } _ { t } | | \hat { p } _ { s } )$ . Since we focus on distillation fidelity, we choose $\alpha = 0$ for all experiments in the main text to avoid any confounding from true labels, but we also include a limited ablation of $\alpha$ in Figure $^ { 1 4 }$ in Appendix $\boxed { C . 5 }$ for the curious reader. + +As $\tau \to + \infty$ , $\nabla _ { \mathbf { z } _ { s } } \mathcal { L } _ { \mathrm { K D } } ( \mathbf { z } _ { s } , \mathbf { z } _ { t } ) \approx \mathbf { z } _ { t } - \mathbf { z } _ { s }$ , and thus in the limit $\nabla _ { \mathbf { z } _ { s } } \mathcal { L } _ { \mathrm { K D } }$ is approximately equivalent to $\nabla _ { \mathbf { z } _ { s } } | | \mathbf { z } _ { t } - \mathbf { z } _ { s } | | _ { 2 } ^ { 2 } / 2$ , assigning equal significance to every class logit, regardless of its contribution to the predictive distribution. In other words $\tau$ determines the “softness” of the teacher labels, which in turn determines the allocation of student capacity. If the student is much smaller than the teacher, the student capacity can be focused on matching the teacher’s top- $k$ predictions, rather than matching the full teacher distribution by choosing a moderate value (e.g. $\tau = 4$ ). In Appendix $\underline { { \mathbf { B . l } } }$ we include further discussion on the interplay of teacher ensemble size, teacher network capacity, and distillation temperature on the student labels. + +The teacher and student often share at least some training data. It is also common to enlarge the student training data in some way (e.g. incorporating unlabeled examples as in Ba and Caruana $\left[ \left[ 2 \right] \right]$ ). When there is a possibility of confusion, we will refer to the student’s training data as the distillation data to distinguish it from the teacher’s training data. + +# 3.2 Metrics and Evaluation + +To measure generalization, we report top-1 accuracy, negative log-likelihood (NLL) and expected calibration error (ECE) $\boxed { 1 1 6 }$ . To measure fidelity, we report the following: + +$$ +\begin{array} { r l } & { \displaystyle \mathrm { A v e r a g e ~ T o p - 1 ~ A g r e e m e n t : } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } 1 \{ \mathrm { a r g m a x } \sigma _ { j } ( \mathbf { z } _ { t , i } ) = \underset { j } { \mathrm { a r g m a x } } \sigma _ { j } ( \mathbf { z } _ { s , i } ) \} , } \\ & { \displaystyle \mathrm { A v e r a g e ~ P r e d i c t i v e ~ K L : } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathrm { K L } \left( \hat { p } _ { t } ( \mathbf { y } | \mathbf { x } _ { i } ) \parallel \hat { p } _ { s } ( \mathbf { y } | \mathbf { x } _ { i } ) \right) , } \end{array} +$$ + +Eqn. $\textcircled{2}$ is the average agreement between the student and teacher’s top-1 label. Eqn. $\textcircled{3}$ is the average KL divergence from the predictive distribution of the teacher to that of the student, a measure of fidelity sensitive to all of the labels. + +While improvements in generalization metrics are relatively easy to understand, interpreting fidelity metrics requires some care. For example, suppose we have three independent models: $f _ { 1 } , f _ { 2 }$ , and $f _ { 3 }$ that respectively achieve $55 \%$ , $7 5 \%$ , and $9 5 \%$ test accuracy. $f _ { 1 }$ and $f _ { 3 }$ can agree on at most $60 \%$ of points, whereas $f _ { 2 }$ and $f _ { 3 }$ agree on at least $70 \%$ , but it would obviously be incorrect to make any claim about $f _ { 2 }$ being a better distillation of $f _ { 3 }$ since each model was trained completely independently. To account for such confounding when evaluating the distillation of a student $s$ from a teacher $t$ , we also evaluate another student $s ^ { \prime }$ distilled through an identical procedure from an independent teacher. + +By comparing the fidelity of $( t , s )$ and $( t , s ^ { \prime } )$ we can distinguish between a generic improvement in generalization and an improvement specifically to fidelity. If $s$ and $s ^ { \prime }$ have comparable fidelity, then the students agree with the teacher at many points because they generalize well, and not the reverse. + +# 4 Knowledge Distillation Transfers Knowledge Poorly + +In this section, we present evidence that we are not able to distill large networks such as a ResNet-56 with high fidelity, and discuss why high fidelity is an important objective. + +# 4.1 When is knowledge transfer successful? + +We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as a motivating example. We train the teacher on a random subset of 200 examples from the MNIST training set for 100 epochs, resulting in a $8 4 \%$ to $8 6 \%$ teacher test accuracy across different subsets.2 We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as $2 5 \%$ , $50 \%$ , and $100 \%$ of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images. + +In Figure $2$ we see that knowledge distillation works as expected. With enough examples the student learns to make the same predictions as the teacher (over $9 9 \%$ top-1 test agreement). Notably, in this case, self-distillation does not improve generalization, since the slight difference between the teacher and student accuracy is explained by variance between trials. + +Now we consider a more challenging task: distilling a ResNet-56 teacher trained on CIFAR-100 into an identical student network (Figure $\mathbb { L } ,$ left). Since no dataset drawn from the same distribution as CIFAR-100 is publicly available, to augment the distillation data, we instead combined samples from an SN-GAN $\textcircled { \ 3 9 }$ pre-trained on CIFAR-100 with the original CIFAR-100 train dataset. Appendix A.3 details the hyperparameters and training procedure for the GAN, teacher, and student. + +Like the MNIST experiment, as we enlarge the distillation dataset the student fidelity improves. However, in this case the improvement is modest, with the fidelity reaching nowhere near $9 9 \%$ test agreement. Since a ResNet-56 has many more parameters than a LeNet-5, it is possible that the student simply has not seen enough examples to perfectly emulate the teacher, a hypothesis we discuss in more detail in Section $\underline { { \boldsymbol { \mathsf { F . 1 } } } } \big \| .$ Also, like the MNIST experiment, as the distillation dataset grows the student accuracy approaches the teacher’s. Unlike the MNIST experiment, the student test accuracy is higher than the teacher’s when the distillation dataset is small, so increasing fidelity decreases student generalization. + +![](images/346544e04a6a5855d2eed11bdd0190beeb1adbb82c1cd34a0f17fe19958b3f37.jpg) +Figure 2: LeNet-5 self-distillation on MNIST with additional distillation data. The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials. + +# 4.2 What can self-distillation tell us about knowledge distillation in general? + +We have seen in Figure $\mathbb { U } ( { \mathrm { a } } )$ that with self-distillation the student can exceed the teacher performance, in accordance with Furlanello et al. [14]. This result is only possible by virtue of failing at the distillation procedure: if the student matched the teacher perfectly then the student could not outperform the teacher. On the other hand, if the teacher generalizes significantly better than an independently trained student, we would expect the benefits of fidelity to dominate other regularization effects associated with not matching the teacher. This setting reflects the original motivation for knowledge distillation, where we wish to faithfully transfer the representation discovered by a large model or ensemble of models into a more efficient student. + +In Figure $1 ( \mathsf { b } )$ we see that if we move from self-distillation to the distillation of a 3 ResNet-56 teacher ensemble, fidelity becomes positively correlated with generalization. But there is still a significant gap in fidelity, even after the distillation set is enlarged with $5 0 k$ GAN samples. In practice, the gap remains large enough that higher fidelity students do not always have better generalization, and the regularization effects we see in self-distillation do play a role for more broadly understanding student generalization. We will indeed show in Section $\bar { 5 }$ that higher fidelity students do not always generalize better, even if the teacher generalizes much better than the student. + +![](images/a6142c0a9a06255373aeb1f38774818c84e68662d3c4d427eee5d2f1eff00a17.jpg) +Figure 3: Data augmentation and distillation: Test accuracy and teacher-student agreement when distilling a 5-component ResNet-56 teacher ensemble into a ResNet-56 student on CIFAR-100 with varying augmentation policies. The best performing policy is shown in green, results averaged over 3 runs. Additional metrics are reported in Figure $\checkmark$ in Appendix $\boxed { \mathbf { C } }$ Mixup and GAN augmentation provide the best generalization, and Mixup $\tau = 4$ ) provides the best fidelity. The baseline policy (crops and flips) with $\tau = 4$ is a surprisingly strong baseline. The error bars indicate $\pm \sigma$ . + +# 4.3 If distillation already improves generalization, why care about fidelity? + +While knowledge distillation does often improve generalization, understanding the relationship between fidelity and generalization, and how to maximize fidelity, is important for several reasons — including better generalization! + +Better generalization in distilling large teacher models and ensembles. Knowledge distillation was initially motivated as a means to deploy powerful models to small devices or low-latency controllers [e.g., 10, 21, 26, 52, 54]. While in self-distillation generalization and fidelity are in tension, there is often a significant disparity in generalization between large teacher models, including ensembles, and smaller students. We have seen this disparity in Figure $1 \bar { ( \mathbf { b } ) }$ . We additionally show in Figure 10 in Appendix $\mathbf { B . l }$ that as we increase the number of ensemble components, the generalization disparity between teacher and distilled student increases. Improving student fidelity is the most obvious way to close the generalization disparity between student and teacher in these settings. Even if one exclusively cares about student accuracy, fidelity is a key consideration outside self-distillation. + +Interpretability and reliability. Knowledge distillation has been identified as a means to transfer representations discovered by large black-box models into simpler more interpretable models, for example to provide insights into medical diagnostics, or discovering rules for understanding sentiment in text [e.g., 23, 24, 6, 33, 8]. The ability to perform this transfer could have extraordinary scientific consequences: large models can often discover structure in data that we would not have anticipated a priori. Moreover, we often want to transfer properties such as well-calibrated uncertainties or robustness, which have been well-established for larger models, so that we can safely deploy more efficient models in their place. In both cases, achieving good distillation fidelity is crucial. + +Understanding. The name knowledge distillation implies we are transferring knowledge from the teacher to the student. For this reason, improved student generalization as a consequence of a distillation procedure is sometimes conflated with fidelity. Decoupling fidelity and generalization, and explicitly studying fidelity, is foundational to understanding how knowledge distillation works and how we can make it more useful across a variety of applications. + +# 4.4 Possible causes of low distillation fidelity + +If we are able to match the student model to the teacher on a comprehensive distillation dataset, we expect it to match on the test data as well, achieving high distillation fidelity3. Possible causes of the poor distillation fidelity in our CIFAR-100 experiments include: + +![](images/8a2a6126b32fff96f219ed1e9e4115bce416ee0fae8ece2522be6c23d9e96dca.jpg) +Figure 4: Data recycling and distillation: results on subsampled CIFAR-100. Top: We fix the temperature $( \tau = 4$ ) and vary the number of ensemble components $( m )$ , comparing students distilled on the same dataset as the teacher $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 0 } )$ , a reserved dataset $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 1 } )$ , or both $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 0 } \cup \mathcal { D } _ { 1 } )$ . Distilling on both produces the best result, while distilling on $\mathcal { D } _ { 0 }$ increases accuracy and decreases fidelity, relative to $\mathcal { D } _ { 1 }$ . Bottom: We repeat the experiment, but fix $m = 3$ and vary $\tau$ . The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials. + +Student capacity – We observe low fidelity even in the self-distillation setting, so we can rule out student capacity as a primary cause, but we also confirm in Figure 12 in Appendix $\mathbb { E . l }$ that increasing the student capacity has very little effect on fidelity in the ensemble-distillation setting. + +Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2. + +Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing that our findings apply to datasets of larger scale and complexity. + +Data domain – Similarly in Section ${ \bf C . 4 }$ we observe low distillation fidelity in the context of text classification (sentiment analysis on the IMDB dataset), showing our results are relevant beyond image classification. + +Identifiability (Section $5$ ) – the distillation data is insufficient to distinguish high-fidelity and lowfidelity students. In other words, matching the teacher predictions on the distillation dataset does not lead to matching predictions on the test data. + +Optimization (Section 6) – we are unable to solve the distillation optimization problem sufficiently well. The student does not agree with the teacher on test because it does not even agree on train. + +# 5 Identifiability: Are We Using the Right Distillation Dataset? + +We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student enough of the right input-teacher label pairs to define the solution we want? + +# 5.1 Should we do more data augmentation? + +Data augmentation is a simple and practical method to increase the support of the distillation data distribution. If identifiability is a primary cause of poor distillation fidelity, using a more extensive data augmentation strategy during distillation should improve fidelity. + +To test this hypothesis, we evaluated the effect of several augmentation strategies on student fidelity and generalization. In Figure $\textcircled { 3 } ,$ the teacher is a 5-component ensemble of ResNet-56 networks trained on CIFAR-100 with the Baseline augmentation strategy: horizontal flips and random crops. + +We report the student accuracy and teacher-student agreement for each augmentation strategy, and also include results for Baseline with $\tau = 1$ and $\tau = 4$ to demonstrate the effect of logit tempering. + +We first observe that the best augmentation policies for generalization, $M i x U p$ , and $G A N \mathbb { H }$ are not the best policies for fidelity. Furthermore, although many augmentation strategies enable slightly higher distillation fidelity compared to Baseline $\tau = 1 .$ ), even the best augmentation policy, Mixup $\tau = 4 ,$ ), only achieves a modest $86 \%$ test agreement. In fact the Baseline $\tau = 4 ,$ ) policy is quite competitive, achieving $8 4 . 5 \%$ test agreement. Many of the augmentation strategies also slightly improve teacher-student KL relative to Baseline $\tau = 4$ ) (see Figure $\textcircled { 1 1 }$ + +In Figure 11 in Appendix ${ \bf B } . 3$ we report all generalization and fidelity metrics for a range of ensemble sizes, as well as the results for the independent student baseline discussed in Section $\underline { { \bar { 3 . 2 } } }$ Often these independent students, taught how to mimic a completely different model, have nearly as good test agreement with the teacher as the student explicitly trained to emulate it. See Appendix $\mathbf { \bar { A } } . 1$ for a detailed description of the augmentation procedures. + +Should data augmentation be close to the data distribution? In theory, any data augmentation should help with identifiability: if a student matches a teacher on more data, it is more likely to match the teacher elsewhere. However, the Noise and $O O D$ augmentation strategies based on noise and outof-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice, data augmentation has an effect beyond improving identifiability — it has a regularizing effect, making optimization more challenging. We explore this facet of data augmentation in Section 6. + +The slight improvements to fidelity with extensive augmentations suggest that increasing the support of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so small compared to heuristics like logit tempering (which does not modify the support at all), it is very unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity. + +# 5.2 The data recycling hypothesis + +If simply showing the student more labels does not always significantly improve fidelity, perhaps we are not showing the student the right labels. Additional data augmentation during distillation does give the student more teacher labels to match, but also introduces a distribution shift between the images the teacher was trained on and the images the student is distilling on. Even when the teacher and student have the same augmentation policy, reusing the teacher’s training data for distillation violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an independent draw from the true joint distribution over images and teacher labels. What if there was no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test distribution over images and teacher labels? + +To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100 training dataset $\mathcal { D }$ into two equal parts, $\mathcal { D } _ { 0 }$ and $\mathcal { D } _ { 1 }$ . We train teacher ResNet-56 ensembles on $\mathcal { D } _ { 0 }$ , and then compare $s _ { 0 }$ , a student distilled on the original $\mathcal { D } _ { 0 }$ , $s _ { 1 }$ , a student distilled on the unseen $\mathcal { D } _ { 1 }$ , and $s _ { 0 \cup 1 }$ , a student distilled on both: $\mathcal { D } _ { 0 } \cup \mathcal { D } _ { 1 }$ . Note that the students cannot access the true labels, only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in the top row and the logit temperature in the bottom row. + +Surprisingly, $s _ { 0 }$ attains higher test accuracy than $s _ { 1 }$ , while showing worse ECE and lower fidelity (measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis that $s _ { 1 }$ should be a higher fidelity distillation of the teacher than $s _ { 0 }$ does hold, but the gain in fidelity does not result in $s _ { 1 }$ best replicating the teacher’s accuracy. The best attributes of $s _ { 0 }$ and $s _ { 1 }$ are combined by $s _ { 0 \cup 1 }$ , which coincides with how unlabeled data is typically used in practice $\pmb { \left. 2 \right. }$ . The reason for this puzzling observation is simply that for the larger teachers fidelity has not improved enough to also improve generalization. In fact, the best teacher-student agreement is only around $8 5 \%$ , no improvement when compared to the results from extensive data augmentation in the last section. We again find that modifying the distillation data can slightly improve fidelity, but the evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data. + +![](images/ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg) +Figure 5: The train agreement for teacher ensembles $( m \in \{ 1 , 3 , 5 \} )$ ) and student on the distillation data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing the softness of the teacher labels by adding examples not in the teacher train data makes distillation more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle: agreement from subsampled CIFAR-100 experiment in Figure $4 .$ Right: agreement for some of the augmentation policies in Figure 3. The shaded region is not visible because the variance is very low. + +# 6 Optimization: Does the Student Match the Teacher on Distillation Data? + +If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data, perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does the student match the teacher on the data it is trained to match it on? + +# 6.1 More distillation data lowers train agreement + +In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved teacher-student agreement on test, but the agreement remains relatively low (below $8 0 \%$ ) even for the largest distillation dataset that we considered. In Figure $5$ (left panel), we report the teacher-student agreement for the same experiment, but now on the distillation dataset. We now observe the opposite trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match the teacher. Even when the student has identical capacity to the teacher, the student only achieves $9 5 \%$ agreement with the teacher when we use $5 0 k$ synthetic images for distillation. + +The drop in train agreement is even more pronounced when we use extensive data augmentation. In Figure ${ \bar { 5 , } }$ right panel, we report the teacher-student agreement on the train set with data augmentation for a subset of augmentation strategies presented in Section $\underline { { \boldsymbol { \mathsf { F . 1 } } } } \big \| .$ We use the CIFAR-100 dataset and the ResNet-56 model for the teachers and the students (for details, see Section $\underline { { \vert 5 . 1 \rangle } }$ . In each case, we measure agreement on the augmented training set that was used during distillation. While for the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color Jitter augmentations, the agreement is between $8 0 \%$ and $9 0 \%$ for all the considered teacher sizes. For Combined Augs, the combination of these three augmentation strategies, the agreement drops even further, to just $6 0 \%$ in self-distillation! + +Our intuition about how knowledge distillation should work largely hinges on the assumption that after distillation the student matches the teacher on the distillation set. However, the results presented in this section suggest that in practice the optimization method is unable to achieve high fidelity even on the distillation dataset when extensive data augmentation or synthetic data is used. The inability to solve the optimization problem undermines distillation: in order to find a student that would match the teacher on all inputs, we need to at least be able to find a student that would match the teacher on all of the distillation data. + +Optimization and the train-test fidelity gap. Notably, despite having the lowest train agreement, the Combined Augs policy results in better test agreement than other polices with better train agreement (Figure $3 )$ ). This result highlights a fundamental trade-off in knowledge distillation: the student needs many teacher labels match the teacher on test, but introducing examples not in the teacher train data makes matching the teacher on the distillation data very difficult. + +![](images/1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg) +Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves agreement, but it remains below $8 5 \%$ even after $5 k$ epochs. (b): Final train loss and agreement when the initialization is a convex combination of teacher and random weights, $\theta _ { s } = \lambda \theta _ { t } + \mathbf { \bar { ( } 1 - } \lambda ) \theta _ { r }$ . (c): Projections of the distillation loss surface on the plane intersecting $\theta _ { t }$ , the initial student weights, and the final student weights for different $\lambda$ . When $\lambda$ is small, the student converges to a suboptimal solution with low agreement. The uncertainty regions correspond to $\mu \pm \sigma$ , estimated over 3 trials. + +# 6.2 Why is train agreement so low? + +A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also replace the BatchNorm layers $\mathbb { \left. \overline { { 2 5 } } \right. }$ in ResNet-20 with LayerNorm $\pmb { \left[ \sqrt { 3 } \right] }$ , because we found that with BatchNorm layers even when the teacher and the student have identical weights, they can make different predictions due to differences in the activation statistics accumulated by the BatchNorm layers. Layer normalization does not collect any activation statistics, so the student will match the teacher as long as the weights coincide. + +Can we solve the optimization problem better? We verify that the distillation fidelity cannot be significantly improved by training longer or with a different optimizer. By default, in our experiments we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a weight decay value of $1 0 ^ { - 4 }$ . In Figure $\boxed { 6 }$ we report the results for the SGD and Adam $\mathbb { \left[ \left[ 2 7 \right] \right] }$ optimizers run for $1 k$ and $5 k$ epochs without weight decay. Switching from SGD to Adam only reduced fidelity. + +For both optimizers, training for more epochs does slightly improve train agreement. In particular, with SGD we achieve $8 3 . 3 \%$ agreement when training for $5 k$ epochs compared to $7 8 . 9 5 \%$ when training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train agreement could reach $1 0 0 \%$ . However, training for $5 k$ epochs is significantly longer than what is typically done in practice (100 to 500 epochs). Furthermore, the improvement from $1 k$ to $5 k$ epochs is only about $2 \%$ , suggesting that we would need to train for tens of thousands of epochs, even in the optimistic case that agreement improves linearly, in order to get close to $1 0 0 \%$ train agreement. + +The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100 with any of the interventions we have discussed so far, we now ask if there is any modification of the problem that can produce a high-fidelity student. + +In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the distillation loss — the teacher’s own weights $\theta _ { t }$ . Letting $\theta _ { r }$ be a random weight initialization, in Figure $\boxed { 6 }$ (a) we examine the effect of choosing the student initialization to be a convex combination of the teacher and random weights, $\theta _ { s } = \lambda \bar { \theta _ { t } } + ( 1 - \lambda ) \theta _ { r }$ . After being initialized in this way, the student was trained as before. In other words $\lambda = 0$ corresponds to a random initialization and $\lambda = 1$ corresponds to initializing the student weights at the final teacher weights. + +We find that if the student is initialized far from the teacher $\lambda \leq 0 . 2 5 )$ , the optimizer converges to a sub-optimal value of the distillation loss, producing a student that significantly disagrees with the teacher. However at $\lambda = 0 . 3 7 5$ there is a sudden change. The final train loss drops to the optimal value and the agreement drastically increases, and the behavior continues for $\lambda > 0 . 3 7 5$ . To further investigate, in Figure $6 ( \mathrm { c ) }$ we visualize the distillation loss surface for $\lambda \in \{ 0 , 0 . 2 5 , 0 . 3 7 5 \}$ projected on the 2D subspace intersecting $\theta _ { t }$ , the initial student weights, and the final student weights. If the student is initialized far from the teacher $( \lambda \in \{ 0 , 0 . 2 5 \} )$ , it converges to a distinct, sub-optimal basin of the loss surface. On the other hand, when initialized close to the teacher $\lambda = 0 . 3 7 5$ ), the student converges to the same basin as the teacher, achieving nearly $100 \%$ agreement. + +Table 1: We examine whether fidelity can be improved in the context of ResNet-20 self-distillation on CIFAR-100 if the teacher and student share the same weight initialization. All metrics are computed on the test set. A shared initialization does make the student slightly more similar to the teacher in activation space (measured by CKA), but in function space the results are indistinguishable from randomly initialized students. We report the mean and standard deviation, estimated from 10 trials. The average teacher accuracy was 70.522 (0.412). + +
CKA (1)
Init.Agree. (↑)KL (↓）Stage 1 Stage 2Stage 3
Rand.77.174 (0.352)0.836 (0.016)0.939 (0.017)0.925 (0.027)0.885 (0.011)
Teach.77.098 (0.238)0.838 (0.020)0.951 (0.017)0.937 (0.020)0.890 (0.015)
+ +Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained by initializing the student near the final teacher weights, it is possible that similar results could be obtained by initializing the student at the initial teacher weights. In Table $^ 1$ we compare students distilled from random initializations with those initialized at the initial teacher weights. In addition to the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA) $\left[ \left[ 2 8 \right] \right]$ of the preactivations of each of the teacher and student networks. There is a small increase in CKA, indicating that sharing an initialization between teacher and student does increase alignment in activation space, but functionally the students are identical to their randomly initialized counterparts – there is no observable change in accuracy, agreement, or predictive KL when compared to random initialization. + +To summarize, we have at last identified a root cause of the ineffectiveness of all our previous interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge to optimal student parameters, even when we know a solution and give the initialization a small head start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match the teacher on all inputs, the student has to at least match the teacher on the data used for distillation, and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to poor distillation fidelity. + +# 7 Discussion + +Our work provides several new key findings about knowledge distillation: + +• Good student accuracy does not imply good distillation fidelity: even outside of selfdistillation, the models with the best generalization do not always achieve the best fidelity. • Student fidelity is correlated with calibration when distilling ensembles: although the highest-fidelity student is not always the most accurate, it is always the best calibrated. • Optimization is challenging in knowledge distillation: even in cases when the student has sufficient capacity to match the teacher on the distillation data, it is unable to do so. • There is a trade-off between optimization complexity and distillation data quality: Enlarging the distillation dataset beyond the teacher training data makes it easier for the student to identify the correct solution, but also makes an already difficult optimization problem harder. + +In standard deep learning, we are saved by not needing to solve the optimization problem well: while it true that our training loss is highly multimodal, properties such as the flatness of good solutions, the inductive biases of the network, and the implicit biases of SGD, often enable good generalization in practice. In knowledge distillation, however, good fidelity is directly aligned with solving what turns out to be an exceptionally difficult optimization problem. + +# Acknowledgements + +The authors would like to thank Gregory Benton, Marc Finzi, Sanae Lotfi, Nate Gruver, and Ben Poole for helpful feedback. This research is supported by an Amazon Research Award, NSF I-DISRE 193471, NIH R01DA048764-01A1, NSF IIS-1910266, and NSF 1922658NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science. Samuel Stanton is also supported by a United States Department of Defense NDSEG fellowship. + +# References + +[1] Anil, R., Gupta, V., Koren, T., Regan, K., and Singer, Y. (2021). Scalable second order optimization for deep learning. arXiv preprint arXiv:2002.09018. +[2] Ba, J. and Caruana, R. (2014). Do deep nets really need to be deep? Advances in neural information processing systems, 27:2654–2662. +[3] Ba, J. L., Kiros, J. R., and Hinton, G. E. (2016). Layer normalization. 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International Conference on Learning Representations. \ No newline at end of file diff --git a/parse/train/Oa9RlXNggGy/Oa9RlXNggGy_content_list.json b/parse/train/Oa9RlXNggGy/Oa9RlXNggGy_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5c809a9514ce88193312d826a07080d06f52bd82 --- /dev/null +++ b/parse/train/Oa9RlXNggGy/Oa9RlXNggGy_content_list.json @@ -0,0 +1,1222 @@ +[ + { + "type": "text", + "text": "Does Knowledge Distillation Really Work? ", + "text_level": 1, + "bbox": [ + 238, + 122, + 759, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Samuel Stanton NYU ", + "bbox": [ + 246, + 200, + 359, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Pavel Izmailov NYU ", + "bbox": [ + 437, + 200, + 542, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Polina Kirichenko NYU ", + "bbox": [ + 622, + 200, + 751, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Alexander A. Alemi Google Research ", + "bbox": [ + 285, + 250, + 424, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andrew Gordon Wilson NYU ", + "bbox": [ + 545, + 250, + 712, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 314, + 535, + 330 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Knowledge distillation is a popular technique for training a small student network to emulate a larger teacher model, such as an ensemble of networks. We show that while knowledge distillation can improve student generalization, it does not typically work as it is commonly understood: there often remains a surprisingly large discrepancy between the predictive distributions of the teacher and the student, even in cases when the student has the capacity to perfectly match the teacher. We identify difficulties in optimization as a key reason for why the student is unable to match the teacher. We also show how the details of the dataset used for distillation play a role in how closely the student matches the teacher — and that more closely matching the teacher paradoxically does not always lead to better student generalization. ", + "bbox": [ + 232, + 344, + 766, + 497 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 520, + 310, + 536 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Large, deep networks can learn representations that generalize well. While smaller, more efficient networks lack the inductive biases to find these representations from training data alone, they may have the capacity to represent these solutions [e.g., 2, 18, 32, 45]. Influential work on knowledge distillation $\\bar { \\| 2 2 \\| }$ argues that Bucila et al. ˘ [5] “demonstrate convincingly that the knowledge acquired by a large ensemble of models [the teacher] can be transferred to a single small model [the student]”. Indeed this quote encapsulates the conventional narrative of knowledge distillation: a student model learns a high-fidelity representation of a larger teacher, enabled by the teacher’s soft labels. ", + "bbox": [ + 173, + 550, + 825, + 648 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Conversely, in Figure $^ 1$ we show that with modern architectures knowledge distillation can lead to students with very different predictions from their teachers, even when the student has the capacity to perfectly match the teacher. Indeed, it is becoming well-known that in self-distillation the student fails to match the teacher and, paradoxically, student generalization improves as a result [14, 40]. However, when the teacher is a large model (e.g. a deep ensemble) improvements in fidelity translate into improvements in generalization, as we show in Figure $\\boxed { 1 } \\mathbf { ( b ) }$ . For these large models there is still a significant accuracy gap between student and teacher, so fidelity is aligned with generalization. ", + "bbox": [ + 174, + 654, + 825, + 751 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We will distinguish between fidelity, the ability of a student to match a teacher’s predictions, and generalization, the performance of a student in predicting unseen, in-distribution data. We show that in many cases it is surprisingly difficult to obtain good student fidelity. In Section 5 we investigate the hypothesis that low fidelity is an identifiability problem that can be solved by augmenting the distillation dataset. In Section $6$ we investigate the hypothesis that low fidelity is an optimization problem resulting in a failure of the student to match the teacher even on the original training dataset. We present a summary of our conclusions in Section 7. ", + "bbox": [ + 174, + 757, + 825, + 856 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Does knowledge distillation really work? In short: Yes, in the sense that it often improves student generalization. No, in that knowledge distillation often fails to live up to its name, transferring very limited knowledge from teacher to student. ", + "bbox": [ + 176, + 861, + 823, + 902 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/b16f411144a106e66a8b6b067f80e336493e1572c29f9021f516793d8553eb0c.jpg", + "image_caption": [ + "Figure 1: Evaluating the fidelity of knowledge distillation. The effect of enlarging the CIFAR-100 distillation dataset with GAN-generated samples. (a): The student and teacher are both single ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases. (b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. Student fidelity again increases as the dataset grows, but test accuracy now slightly increases. The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. " + ], + "image_footnote": [], + "bbox": [ + 199, + 90, + 799, + 227 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 345, + 321, + 363 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Knowledge distillation can improve model efficiency [38, 45], unsupervised domain adaptation [37], improved object detection $\\pmb { \\Vert }$ , model transparency $\\lVert \\rVert \\bigotimes \\rVert$ , and adversarial robustness [15, 42]. ", + "bbox": [ + 174, + 377, + 825, + 406 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Seminal work by Bucila et al. ˘ [5] showed that teacher-ensembles with thousands of simple components could be compressed into a single shallow network that matched or outperformed its teacher. Other early work proposed distilling ensembles of shallow networks into a single network [55], an idea which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53]. Recently Fakoor et al. [13] developed a data-augmentation scheme for the distillation of large ensembles of simple models for tabular data, achieving impressive results on a wide range of tabular benchmarks. Malinin et al. [35] proposed a method to model the implicit distribution over predictive distributions from which the ensemble component predictive distributions are drawn, rather than just the ensemble model average. ", + "bbox": [ + 174, + 411, + 825, + 537 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work focuses explicitly on student fidelity, decoupling our understanding of good fidelity from good generalization. We show that achieving good fidelity is extremely difficult, even with a variety of interventions, and seek to understand, by systematically considering several hypotheses, why knowledge distillation does not produce high fidelity students for modern architectures and datasets. In contrast, the distillation literature focuses largely on improving student generalization, without particularly distinguishing between fidelity and generalization. ", + "bbox": [ + 174, + 544, + 825, + 627 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For example, concurrent work by Beyer et al. [4] does not carefully distinguish generalization and fidelity metrics, but they assert that high student fidelity is conceptually desirable and apparently difficult to achieve when measured as the gap between teacher and student accuracy. As a result their work focuses most heavily on practical modifications to the distillation procedure for the best student top-1 accuracy. In this paper we investigate many of the same prescriptions, including careful treatment of data augmentation (such as showing the teacher and student the exact same input images), the addition of MixUp, and extended training duration. We also find that such interventions do improve student accuracy, but there still remains a large discrepancy between the predictive distributions of the teacher and the student. We also investigate multiple optimizers. While we do not pursue Shampoo $[ [ 1 7 , \\mathbb { I } ] ]$ specifically, Beyer et al. $\\mathbb { H }$ find similar qualitative results for Shampoo and Adam, besides faster convergence for Shampoo. ", + "bbox": [ + 174, + 632, + 825, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 Preliminaries ", + "text_level": 1, + "bbox": [ + 174, + 806, + 318, + 824 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We will focus on the supervised classification setting, with input space $\\mathcal { X }$ and label space $\\mathcal { V }$ , where $| { \\mathcal { V } } | = c$ . Let $f : \\mathcal { X } \\times \\Theta \\mathbb { R } ^ { c }$ be a classifier parameterized by $\\theta \\in \\Theta$ whose outputs define a categorical predictive distribution over $\\mathcal { V }$ , $\\hat { p } ( y = i | \\mathbf { x } ) = \\sigma _ { i } ( f ( \\mathbf { x } , \\theta ) )$ , where $\\sigma _ { i } ( { \\bf z } ) : = \\dot { \\exp ( z _ { i } ) } / \\sum _ { j } \\exp \\bar { ( } z _ { j } )$ is the softmax link function. We will often refer to the outputs of a classifier $\\mathbf { z } : = f ( \\mathbf { x } , \\theta )$ as logits. For convenience, we will use $t$ and $s$ as shorthand for $f _ { \\mathrm { t e a c h e r } }$ and $f _ { \\mathrm { s t u d e n t } }$ , respectively. When the teacher is an $m$ -component ensemble, the component logits $\\left( \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { m } \\right)$ , where $\\mathbf { z } _ { i } = f _ { i } ( \\mathbf { x } , \\theta _ { i } )$ , are combined to form the teacher logits: $\\begin{array} { r } { \\mathbf { z } _ { t } = \\log \\bar { ( \\sum _ { i = 1 } ^ { m } \\sigma ( \\mathbf { \\bar { z } } _ { i } ) / m ) } } \\end{array}$ . These combined logits correspond to the predictive distribution of the ensemble model average. The experiments in the main text consider $m \\in \\{ 1 , 3 , 5 \\}$ , and we include results up to $m = 1 2$ in Appendix B.2.1 ", + "bbox": [ + 174, + 839, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 148 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Knowledge Distillation ", + "text_level": 1, + "bbox": [ + 174, + 162, + 370, + 178 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hinton et al. $\\pmb { \\mathbb { D } } 2 \\mathbf { \\mathbb { I } }$ proposed a simple approach to knowledge distillation. The student minimizes a weighted combination of two objectives, $\\mathcal { L } _ { s } : = \\alpha \\mathcal { L } _ { \\mathrm { N L L } } + ( 1 - \\alpha ) \\mathcal { L } _ { \\mathrm { K D } }$ , where $\\alpha \\in [ 0 , 1 )$ . Specifically, ", + "bbox": [ + 173, + 188, + 826, + 218 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cb71c932a12c9ab68952314d721b8775d82ec4a2743705ae6677175ea85b298b.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { N L L } } ( \\mathbf { z } _ { s } , \\mathbf { y } ) : = - \\sum _ { j = 1 } ^ { c } y _ { j } \\log \\sigma _ { j } ( \\mathbf { z } _ { s } ) , ~ \\mathcal { L } _ { \\mathrm { K D } } ( \\mathbf { z } _ { s } , \\mathbf { z } _ { t } ) : = - \\tau ^ { 2 } \\sum _ { j = 1 } ^ { c } \\sigma _ { j } \\left( \\frac { \\mathbf { z } _ { t } } { \\tau } \\right) \\log \\sigma _ { j } \\left( \\frac { \\mathbf { z } _ { s } } { \\tau } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 197, + 223, + 779, + 267 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "$\\mathcal { L } _ { \\mathrm { N L L } }$ is the usual supervised cross-entropy between the student logits $\\mathbf { z } _ { s }$ and the one-hot labels $\\mathbf { y }$ . Recalling that $\\begin{array} { r } { \\mathrm { K L } ( p | | q ) = \\sum _ { j } p _ { j } ( \\log q _ { j } - \\log p _ { j } ) } \\end{array}$ , we see that $\\mathcal { L } _ { \\mathrm { N L L } }$ is equivalent (up to a constant) to the KL from the empirical data distribution to the student predictive distribution $( \\hat { p } _ { s } )$ . ${ \\mathcal { L } } _ { \\mathrm { K D } }$ is the added knowledge distillation term that encourages the student to match the teacher. It is the cross-entropy between the teacher and student predictive distributions $\\hat { p } _ { t } = \\sigma ( \\mathbf { z } _ { t } )$ and $\\hat { p } _ { s } = \\sigma ( { \\bf z } _ { s } )$ , both scaled by a temperature hyperparameter $\\tau > 0$ . If $\\tau = 1$ then ${ \\mathcal { L } } _ { \\mathrm { K D } }$ is similarly equivalent to the KL from the teacher to the student, $\\mathrm { K L } ( \\hat { p } _ { t } | | \\hat { p } _ { s } )$ . Since we focus on distillation fidelity, we choose $\\alpha = 0$ for all experiments in the main text to avoid any confounding from true labels, but we also include a limited ablation of $\\alpha$ in Figure $^ { 1 4 }$ in Appendix $\\boxed { C . 5 }$ for the curious reader. ", + "bbox": [ + 173, + 271, + 826, + 400 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As $\\tau \\to + \\infty$ , $\\nabla _ { \\mathbf { z } _ { s } } \\mathcal { L } _ { \\mathrm { K D } } ( \\mathbf { z } _ { s } , \\mathbf { z } _ { t } ) \\approx \\mathbf { z } _ { t } - \\mathbf { z } _ { s }$ , and thus in the limit $\\nabla _ { \\mathbf { z } _ { s } } \\mathcal { L } _ { \\mathrm { K D } }$ is approximately equivalent to $\\nabla _ { \\mathbf { z } _ { s } } | | \\mathbf { z } _ { t } - \\mathbf { z } _ { s } | | _ { 2 } ^ { 2 } / 2$ , assigning equal significance to every class logit, regardless of its contribution to the predictive distribution. In other words $\\tau$ determines the “softness” of the teacher labels, which in turn determines the allocation of student capacity. If the student is much smaller than the teacher, the student capacity can be focused on matching the teacher’s top- $k$ predictions, rather than matching the full teacher distribution by choosing a moderate value (e.g. $\\tau = 4$ ). In Appendix $\\underline { { \\mathbf { B . l } } }$ we include further discussion on the interplay of teacher ensemble size, teacher network capacity, and distillation temperature on the student labels. ", + "bbox": [ + 173, + 404, + 826, + 515 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The teacher and student often share at least some training data. It is also common to enlarge the student training data in some way (e.g. incorporating unlabeled examples as in Ba and Caruana $\\left[ \\left[ 2 \\right] \\right]$ ). When there is a possibility of confusion, we will refer to the student’s training data as the distillation data to distinguish it from the teacher’s training data. ", + "bbox": [ + 173, + 522, + 825, + 578 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 Metrics and Evaluation ", + "text_level": 1, + "bbox": [ + 174, + 593, + 377, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To measure generalization, we report top-1 accuracy, negative log-likelihood (NLL) and expected calibration error (ECE) $\\boxed { 1 1 6 }$ . To measure fidelity, we report the following: ", + "bbox": [ + 171, + 618, + 823, + 647 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a559a29bcef38ef5367b475572a19dc8ef744e6ac20aecd0747505152909e9c2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathrm { A v e r a g e ~ T o p - 1 ~ A g r e e m e n t : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } 1 \\{ \\mathrm { a r g m a x } \\sigma _ { j } ( \\mathbf { z } _ { t , i } ) = \\underset { j } { \\mathrm { a r g m a x } } \\sigma _ { j } ( \\mathbf { z } _ { s , i } ) \\} , } \\\\ & { \\displaystyle \\mathrm { A v e r a g e ~ P r e d i c t i v e ~ K L : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathrm { K L } \\left( \\hat { p } _ { t } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\parallel \\hat { p } _ { s } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 652, + 756, + 737 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Eqn. $\\textcircled{2}$ is the average agreement between the student and teacher’s top-1 label. Eqn. $\\textcircled{3}$ is the average KL divergence from the predictive distribution of the teacher to that of the student, a measure of fidelity sensitive to all of the labels. ", + "bbox": [ + 174, + 742, + 825, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While improvements in generalization metrics are relatively easy to understand, interpreting fidelity metrics requires some care. For example, suppose we have three independent models: $f _ { 1 } , f _ { 2 }$ , and $f _ { 3 }$ that respectively achieve $55 \\%$ , $7 5 \\%$ , and $9 5 \\%$ test accuracy. $f _ { 1 }$ and $f _ { 3 }$ can agree on at most $60 \\%$ of points, whereas $f _ { 2 }$ and $f _ { 3 }$ agree on at least $70 \\%$ , but it would obviously be incorrect to make any claim about $f _ { 2 }$ being a better distillation of $f _ { 3 }$ since each model was trained completely independently. To account for such confounding when evaluating the distillation of a student $s$ from a teacher $t$ , we also evaluate another student $s ^ { \\prime }$ distilled through an identical procedure from an independent teacher. ", + "bbox": [ + 173, + 790, + 825, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By comparing the fidelity of $( t , s )$ and $( t , s ^ { \\prime } )$ we can distinguish between a generic improvement in generalization and an improvement specifically to fidelity. If $s$ and $s ^ { \\prime }$ have comparable fidelity, then the students agree with the teacher at many points because they generalize well, and not the reverse. ", + "bbox": [ + 174, + 90, + 826, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Knowledge Distillation Transfers Knowledge Poorly ", + "text_level": 1, + "bbox": [ + 173, + 151, + 637, + 170 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we present evidence that we are not able to distill large networks such as a ResNet-56 with high fidelity, and discuss why high fidelity is an important objective. ", + "bbox": [ + 173, + 184, + 823, + 212 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 When is knowledge transfer successful? ", + "text_level": 1, + "bbox": [ + 174, + 228, + 488, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as a motivating example. We train the teacher on a random subset of 200 examples from the MNIST training set for 100 epochs, resulting in a $8 4 \\%$ to $8 6 \\%$ teacher test accuracy across different subsets.2 We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as $2 5 \\%$ , $50 \\%$ , and $100 \\%$ of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images. ", + "bbox": [ + 174, + 252, + 826, + 324 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In Figure $2$ we see that knowledge distillation works as expected. With enough examples the student learns to make the same predictions as the teacher (over $9 9 \\%$ top-1 test agreement). Notably, in this case, self-distillation does not improve generalization, since the slight difference between the teacher and student accuracy is explained by variance between trials. ", + "bbox": [ + 174, + 329, + 825, + 385 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Now we consider a more challenging task: distilling a ResNet-56 teacher trained on CIFAR-100 into an identical student network (Figure $\\mathbb { L } ,$ left). Since no dataset drawn from the same distribution as CIFAR-100 is publicly available, to augment the distillation data, we instead combined samples from an SN-GAN $\\textcircled { \\ 3 9 }$ pre-trained on CIFAR-100 with the original CIFAR-100 train dataset. Appendix A.3 details the hyperparameters and training procedure for the GAN, teacher, and student. ", + "bbox": [ + 174, + 391, + 550, + 516 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Like the MNIST experiment, as we enlarge the distillation dataset the student fidelity improves. However, in this case the improvement is modest, with the fidelity reaching nowhere near $9 9 \\%$ test agreement. Since a ResNet-56 has many more parameters than a LeNet-5, it is possible that the student simply has not seen enough examples to perfectly emulate the teacher, a hypothesis we discuss in more detail in Section $\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .$ Also, like the MNIST experiment, as the distillation dataset grows the student accuracy approaches the teacher’s. Unlike the MNIST experiment, the student test accuracy is higher than the teacher’s when the distillation dataset is small, so increasing fidelity decreases student generalization. ", + "bbox": [ + 174, + 522, + 549, + 660 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/346544e04a6a5855d2eed11bdd0190beeb1adbb82c1cd34a0f17fe19958b3f37.jpg", + "image_caption": [ + "Figure 2: LeNet-5 self-distillation on MNIST with additional distillation data. The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. " + ], + "image_footnote": [], + "bbox": [ + 571, + 409, + 813, + 561 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 661, + 826, + 688 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 What can self-distillation tell us about knowledge distillation in general? ", + "text_level": 1, + "bbox": [ + 176, + 704, + 714, + 719 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We have seen in Figure $\\mathbb { U } ( { \\mathrm { a } } )$ that with self-distillation the student can exceed the teacher performance, in accordance with Furlanello et al. [14]. This result is only possible by virtue of failing at the distillation procedure: if the student matched the teacher perfectly then the student could not outperform the teacher. On the other hand, if the teacher generalizes significantly better than an independently trained student, we would expect the benefits of fidelity to dominate other regularization effects associated with not matching the teacher. This setting reflects the original motivation for knowledge distillation, where we wish to faithfully transfer the representation discovered by a large model or ensemble of models into a more efficient student. ", + "bbox": [ + 173, + 729, + 825, + 840 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In Figure $1 ( \\mathsf { b } )$ we see that if we move from self-distillation to the distillation of a 3 ResNet-56 teacher ensemble, fidelity becomes positively correlated with generalization. But there is still a significant gap in fidelity, even after the distillation set is enlarged with $5 0 k$ GAN samples. In practice, the gap remains large enough that higher fidelity students do not always have better generalization, and the regularization effects we see in self-distillation do play a role for more broadly understanding student generalization. We will indeed show in Section $\\bar { 5 }$ that higher fidelity students do not always generalize better, even if the teacher generalizes much better than the student. ", + "bbox": [ + 176, + 847, + 823, + 876 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/a6142c0a9a06255373aeb1f38774818c84e68662d3c4d427eee5d2f1eff00a17.jpg", + "image_caption": [ + "Figure 3: Data augmentation and distillation: Test accuracy and teacher-student agreement when distilling a 5-component ResNet-56 teacher ensemble into a ResNet-56 student on CIFAR-100 with varying augmentation policies. The best performing policy is shown in green, results averaged over 3 runs. Additional metrics are reported in Figure $\\checkmark$ in Appendix $\\boxed { \\mathbf { C } }$ Mixup and GAN augmentation provide the best generalization, and Mixup $\\tau = 4$ ) provides the best fidelity. The baseline policy (crops and flips) with $\\tau = 4$ is a surprisingly strong baseline. The error bars indicate $\\pm \\sigma$ . " + ], + "image_footnote": [], + "bbox": [ + 192, + 90, + 805, + 185 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 308, + 825, + 377 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 If distillation already improves generalization, why care about fidelity? ", + "text_level": 1, + "bbox": [ + 174, + 398, + 704, + 412 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While knowledge distillation does often improve generalization, understanding the relationship between fidelity and generalization, and how to maximize fidelity, is important for several reasons — including better generalization! ", + "bbox": [ + 176, + 425, + 826, + 467 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Better generalization in distilling large teacher models and ensembles. Knowledge distillation was initially motivated as a means to deploy powerful models to small devices or low-latency controllers [e.g., 10, 21, 26, 52, 54]. While in self-distillation generalization and fidelity are in tension, there is often a significant disparity in generalization between large teacher models, including ensembles, and smaller students. We have seen this disparity in Figure $1 \\bar { ( \\mathbf { b } ) }$ . We additionally show in Figure 10 in Appendix $\\mathbf { B . l }$ that as we increase the number of ensemble components, the generalization disparity between teacher and distilled student increases. Improving student fidelity is the most obvious way to close the generalization disparity between student and teacher in these settings. Even if one exclusively cares about student accuracy, fidelity is a key consideration outside self-distillation. ", + "bbox": [ + 173, + 473, + 825, + 598 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Interpretability and reliability. Knowledge distillation has been identified as a means to transfer representations discovered by large black-box models into simpler more interpretable models, for example to provide insights into medical diagnostics, or discovering rules for understanding sentiment in text [e.g., 23, 24, 6, 33, 8]. The ability to perform this transfer could have extraordinary scientific consequences: large models can often discover structure in data that we would not have anticipated a priori. Moreover, we often want to transfer properties such as well-calibrated uncertainties or robustness, which have been well-established for larger models, so that we can safely deploy more efficient models in their place. In both cases, achieving good distillation fidelity is crucial. ", + "bbox": [ + 173, + 603, + 825, + 715 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Understanding. The name knowledge distillation implies we are transferring knowledge from the teacher to the student. For this reason, improved student generalization as a consequence of a distillation procedure is sometimes conflated with fidelity. Decoupling fidelity and generalization, and explicitly studying fidelity, is foundational to understanding how knowledge distillation works and how we can make it more useful across a variety of applications. ", + "bbox": [ + 174, + 722, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.4 Possible causes of low distillation fidelity ", + "text_level": 1, + "bbox": [ + 174, + 810, + 493, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "If we are able to match the student model to the teacher on a comprehensive distillation dataset, we expect it to match on the test data as well, achieving high distillation fidelity3. Possible causes of the poor distillation fidelity in our CIFAR-100 experiments include: ", + "bbox": [ + 176, + 838, + 825, + 880 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8a2a6126b32fff96f219ed1e9e4115bce416ee0fae8ece2522be6c23d9e96dca.jpg", + "image_caption": [ + "Figure 4: Data recycling and distillation: results on subsampled CIFAR-100. Top: We fix the temperature $( \\tau = 4$ ) and vary the number of ensemble components $( m )$ , comparing students distilled on the same dataset as the teacher $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } )$ , a reserved dataset $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 1 } )$ , or both $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 } )$ . Distilling on both produces the best result, while distilling on $\\mathcal { D } _ { 0 }$ increases accuracy and decreases fidelity, relative to $\\mathcal { D } _ { 1 }$ . Bottom: We repeat the experiment, but fix $m = 3$ and vary $\\tau$ . The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. " + ], + "image_footnote": [], + "bbox": [ + 173, + 90, + 825, + 301 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Student capacity – We observe low fidelity even in the self-distillation setting, so we can rule out student capacity as a primary cause, but we also confirm in Figure 12 in Appendix $\\mathbb { E . l }$ that increasing the student capacity has very little effect on fidelity in the ensemble-distillation setting. ", + "bbox": [ + 173, + 430, + 825, + 473 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2. ", + "bbox": [ + 171, + 478, + 823, + 507 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing that our findings apply to datasets of larger scale and complexity. ", + "bbox": [ + 174, + 512, + 821, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data domain – Similarly in Section ${ \\bf C . 4 }$ we observe low distillation fidelity in the context of text classification (sentiment analysis on the IMDB dataset), showing our results are relevant beyond image classification. ", + "bbox": [ + 174, + 546, + 825, + 589 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Identifiability (Section $5$ ) – the distillation data is insufficient to distinguish high-fidelity and lowfidelity students. In other words, matching the teacher predictions on the distillation dataset does not lead to matching predictions on the test data. ", + "bbox": [ + 174, + 595, + 825, + 637 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Optimization (Section 6) – we are unable to solve the distillation optimization problem sufficiently well. The student does not agree with the teacher on test because it does not even agree on train. ", + "bbox": [ + 171, + 643, + 825, + 672 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Identifiability: Are We Using the Right Distillation Dataset? ", + "text_level": 1, + "bbox": [ + 171, + 695, + 705, + 714 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student enough of the right input-teacher label pairs to define the solution we want? ", + "bbox": [ + 174, + 729, + 825, + 772 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 Should we do more data augmentation? ", + "text_level": 1, + "bbox": [ + 173, + 794, + 488, + 809 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data augmentation is a simple and practical method to increase the support of the distillation data distribution. If identifiability is a primary cause of poor distillation fidelity, using a more extensive data augmentation strategy during distillation should improve fidelity. ", + "bbox": [ + 176, + 820, + 825, + 863 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To test this hypothesis, we evaluated the effect of several augmentation strategies on student fidelity and generalization. In Figure $\\textcircled { 3 } ,$ the teacher is a 5-component ensemble of ResNet-56 networks trained on CIFAR-100 with the Baseline augmentation strategy: horizontal flips and random crops. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We report the student accuracy and teacher-student agreement for each augmentation strategy, and also include results for Baseline with $\\tau = 1$ and $\\tau = 4$ to demonstrate the effect of logit tempering. ", + "bbox": [ + 171, + 92, + 823, + 119 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first observe that the best augmentation policies for generalization, $M i x U p$ , and $G A N \\mathbb { H }$ are not the best policies for fidelity. Furthermore, although many augmentation strategies enable slightly higher distillation fidelity compared to Baseline $\\tau = 1 .$ ), even the best augmentation policy, Mixup $\\tau = 4 ,$ ), only achieves a modest $86 \\%$ test agreement. In fact the Baseline $\\tau = 4 ,$ ) policy is quite competitive, achieving $8 4 . 5 \\%$ test agreement. Many of the augmentation strategies also slightly improve teacher-student KL relative to Baseline $\\tau = 4$ ) (see Figure $\\textcircled { 1 1 }$ ", + "bbox": [ + 173, + 126, + 825, + 209 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Figure 11 in Appendix ${ \\bf B } . 3$ we report all generalization and fidelity metrics for a range of ensemble sizes, as well as the results for the independent student baseline discussed in Section $\\underline { { \\bar { 3 . 2 } } }$ Often these independent students, taught how to mimic a completely different model, have nearly as good test agreement with the teacher as the student explicitly trained to emulate it. See Appendix $\\mathbf { \\bar { A } } . 1$ for a detailed description of the augmentation procedures. ", + "bbox": [ + 174, + 215, + 825, + 285 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Should data augmentation be close to the data distribution? In theory, any data augmentation should help with identifiability: if a student matches a teacher on more data, it is more likely to match the teacher elsewhere. However, the Noise and $O O D$ augmentation strategies based on noise and outof-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice, data augmentation has an effect beyond improving identifiability — it has a regularizing effect, making optimization more challenging. We explore this facet of data augmentation in Section 6. ", + "bbox": [ + 173, + 291, + 825, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The slight improvements to fidelity with extensive augmentations suggest that increasing the support of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so small compared to heuristics like logit tempering (which does not modify the support at all), it is very unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity. ", + "bbox": [ + 174, + 381, + 825, + 436 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 The data recycling hypothesis ", + "text_level": 1, + "bbox": [ + 174, + 468, + 418, + 483 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "If simply showing the student more labels does not always significantly improve fidelity, perhaps we are not showing the student the right labels. Additional data augmentation during distillation does give the student more teacher labels to match, but also introduces a distribution shift between the images the teacher was trained on and the images the student is distilling on. Even when the teacher and student have the same augmentation policy, reusing the teacher’s training data for distillation violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an independent draw from the true joint distribution over images and teacher labels. What if there was no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test distribution over images and teacher labels? ", + "bbox": [ + 174, + 500, + 825, + 625 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100 training dataset $\\mathcal { D }$ into two equal parts, $\\mathcal { D } _ { 0 }$ and $\\mathcal { D } _ { 1 }$ . We train teacher ResNet-56 ensembles on $\\mathcal { D } _ { 0 }$ , and then compare $s _ { 0 }$ , a student distilled on the original $\\mathcal { D } _ { 0 }$ , $s _ { 1 }$ , a student distilled on the unseen $\\mathcal { D } _ { 1 }$ , and $s _ { 0 \\cup 1 }$ , a student distilled on both: $\\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 }$ . Note that the students cannot access the true labels, only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in the top row and the logit temperature in the bottom row. ", + "bbox": [ + 174, + 631, + 825, + 714 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Surprisingly, $s _ { 0 }$ attains higher test accuracy than $s _ { 1 }$ , while showing worse ECE and lower fidelity (measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis that $s _ { 1 }$ should be a higher fidelity distillation of the teacher than $s _ { 0 }$ does hold, but the gain in fidelity does not result in $s _ { 1 }$ best replicating the teacher’s accuracy. The best attributes of $s _ { 0 }$ and $s _ { 1 }$ are combined by $s _ { 0 \\cup 1 }$ , which coincides with how unlabeled data is typically used in practice $\\pmb { \\left. 2 \\right. }$ . The reason for this puzzling observation is simply that for the larger teachers fidelity has not improved enough to also improve generalization. In fact, the best teacher-student agreement is only around $8 5 \\%$ , no improvement when compared to the results from extensive data augmentation in the last section. We again find that modifying the distillation data can slightly improve fidelity, but the evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data. ", + "bbox": [ + 173, + 720, + 825, + 859 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg", + "image_caption": [ + "Figure 5: The train agreement for teacher ensembles $( m \\in \\{ 1 , 3 , 5 \\} )$ ) and student on the distillation data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing the softness of the teacher labels by adding examples not in the teacher train data makes distillation more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle: agreement from subsampled CIFAR-100 experiment in Figure $4 .$ Right: agreement for some of the augmentation policies in Figure 3. The shaded region is not visible because the variance is very low. " + ], + "image_footnote": [], + "bbox": [ + 171, + 88, + 825, + 176 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Optimization: Does the Student Match the Teacher on Distillation Data? ", + "text_level": 1, + "bbox": [ + 171, + 311, + 808, + 330 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data, perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does the student match the teacher on the data it is trained to match it on? ", + "bbox": [ + 174, + 354, + 825, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 More distillation data lowers train agreement ", + "text_level": 1, + "bbox": [ + 174, + 444, + 529, + 459 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved teacher-student agreement on test, but the agreement remains relatively low (below $8 0 \\%$ ) even for the largest distillation dataset that we considered. In Figure $5$ (left panel), we report the teacher-student agreement for the same experiment, but now on the distillation dataset. We now observe the opposite trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match the teacher. Even when the student has identical capacity to the teacher, the student only achieves $9 5 \\%$ agreement with the teacher when we use $5 0 k$ synthetic images for distillation. ", + "bbox": [ + 174, + 476, + 825, + 588 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The drop in train agreement is even more pronounced when we use extensive data augmentation. In Figure ${ \\bar { 5 , } }$ right panel, we report the teacher-student agreement on the train set with data augmentation for a subset of augmentation strategies presented in Section $\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .$ We use the CIFAR-100 dataset and the ResNet-56 model for the teachers and the students (for details, see Section $\\underline { { \\vert 5 . 1 \\rangle } }$ . In each case, we measure agreement on the augmented training set that was used during distillation. While for the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color Jitter augmentations, the agreement is between $8 0 \\%$ and $9 0 \\%$ for all the considered teacher sizes. For Combined Augs, the combination of these three augmentation strategies, the agreement drops even further, to just $6 0 \\%$ in self-distillation! ", + "bbox": [ + 174, + 593, + 825, + 732 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our intuition about how knowledge distillation should work largely hinges on the assumption that after distillation the student matches the teacher on the distillation set. However, the results presented in this section suggest that in practice the optimization method is unable to achieve high fidelity even on the distillation dataset when extensive data augmentation or synthetic data is used. The inability to solve the optimization problem undermines distillation: in order to find a student that would match the teacher on all inputs, we need to at least be able to find a student that would match the teacher on all of the distillation data. ", + "bbox": [ + 174, + 738, + 825, + 835 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Optimization and the train-test fidelity gap. Notably, despite having the lowest train agreement, the Combined Augs policy results in better test agreement than other polices with better train agreement (Figure $3 )$ ). This result highlights a fundamental trade-off in knowledge distillation: the student needs many teacher labels match the teacher on test, but introducing examples not in the teacher train data makes matching the teacher on the distillation data very difficult. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg", + "image_caption": [ + "Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves agreement, but it remains below $8 5 \\%$ even after $5 k$ epochs. (b): Final train loss and agreement when the initialization is a convex combination of teacher and random weights, $\\theta _ { s } = \\lambda \\theta _ { t } + \\mathbf { \\bar { ( } 1 - } \\lambda ) \\theta _ { r }$ . (c): Projections of the distillation loss surface on the plane intersecting $\\theta _ { t }$ , the initial student weights, and the final student weights for different $\\lambda$ . When $\\lambda$ is small, the student converges to a suboptimal solution with low agreement. The uncertainty regions correspond to $\\mu \\pm \\sigma$ , estimated over 3 trials. " + ], + "image_footnote": [], + "bbox": [ + 169, + 88, + 813, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.2 Why is train agreement so low? ", + "text_level": 1, + "bbox": [ + 176, + 325, + 429, + 339 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also replace the BatchNorm layers $\\mathbb { \\left. \\overline { { 2 5 } } \\right. }$ in ResNet-20 with LayerNorm $\\pmb { \\left[ \\sqrt { 3 } \\right] }$ , because we found that with BatchNorm layers even when the teacher and the student have identical weights, they can make different predictions due to differences in the activation statistics accumulated by the BatchNorm layers. Layer normalization does not collect any activation statistics, so the student will match the teacher as long as the weights coincide. ", + "bbox": [ + 173, + 352, + 825, + 463 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Can we solve the optimization problem better? We verify that the distillation fidelity cannot be significantly improved by training longer or with a different optimizer. By default, in our experiments we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a weight decay value of $1 0 ^ { - 4 }$ . In Figure $\\boxed { 6 }$ we report the results for the SGD and Adam $\\mathbb { \\left[ \\left[ 2 7 \\right] \\right] }$ optimizers run for $1 k$ and $5 k$ epochs without weight decay. Switching from SGD to Adam only reduced fidelity. ", + "bbox": [ + 174, + 469, + 825, + 540 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For both optimizers, training for more epochs does slightly improve train agreement. In particular, with SGD we achieve $8 3 . 3 \\%$ agreement when training for $5 k$ epochs compared to $7 8 . 9 5 \\%$ when training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train agreement could reach $1 0 0 \\%$ . However, training for $5 k$ epochs is significantly longer than what is typically done in practice (100 to 500 epochs). Furthermore, the improvement from $1 k$ to $5 k$ epochs is only about $2 \\%$ , suggesting that we would need to train for tens of thousands of epochs, even in the optimistic case that agreement improves linearly, in order to get close to $1 0 0 \\%$ train agreement. ", + "bbox": [ + 174, + 545, + 825, + 643 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100 with any of the interventions we have discussed so far, we now ask if there is any modification of the problem that can produce a high-fidelity student. ", + "bbox": [ + 174, + 648, + 825, + 690 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the distillation loss — the teacher’s own weights $\\theta _ { t }$ . Letting $\\theta _ { r }$ be a random weight initialization, in Figure $\\boxed { 6 }$ (a) we examine the effect of choosing the student initialization to be a convex combination of the teacher and random weights, $\\theta _ { s } = \\lambda \\bar { \\theta _ { t } } + ( 1 - \\lambda ) \\theta _ { r }$ . After being initialized in this way, the student was trained as before. In other words $\\lambda = 0$ corresponds to a random initialization and $\\lambda = 1$ corresponds to initializing the student weights at the final teacher weights. ", + "bbox": [ + 174, + 696, + 825, + 780 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We find that if the student is initialized far from the teacher $\\lambda \\leq 0 . 2 5 )$ , the optimizer converges to a sub-optimal value of the distillation loss, producing a student that significantly disagrees with the teacher. However at $\\lambda = 0 . 3 7 5$ there is a sudden change. The final train loss drops to the optimal value and the agreement drastically increases, and the behavior continues for $\\lambda > 0 . 3 7 5$ . To further investigate, in Figure $6 ( \\mathrm { c ) }$ we visualize the distillation loss surface for $\\lambda \\in \\{ 0 , 0 . 2 5 , 0 . 3 7 5 \\}$ projected on the 2D subspace intersecting $\\theta _ { t }$ , the initial student weights, and the final student weights. If the student is initialized far from the teacher $( \\lambda \\in \\{ 0 , 0 . 2 5 \\} )$ , it converges to a distinct, sub-optimal basin of the loss surface. On the other hand, when initialized close to the teacher $\\lambda = 0 . 3 7 5$ ), the student converges to the same basin as the teacher, achieving nearly $100 \\%$ agreement. ", + "bbox": [ + 174, + 786, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/6261113ac0608e905470acaddd56ab5ee54a3afd4229b650ea76339020420cdb.jpg", + "table_caption": [ + "Table 1: We examine whether fidelity can be improved in the context of ResNet-20 self-distillation on CIFAR-100 if the teacher and student share the same weight initialization. All metrics are computed on the test set. A shared initialization does make the student slightly more similar to the teacher in activation space (measured by CKA), but in function space the results are indistinguishable from randomly initialized students. We report the mean and standard deviation, estimated from 10 trials. The average teacher accuracy was 70.522 (0.412). " + ], + "table_footnote": [], + "table_body": "
CKA (1)
Init.Agree. (↑)KL (↓）Stage 1 Stage 2Stage 3
Rand.77.174 (0.352)0.836 (0.016)0.939 (0.017)0.925 (0.027)0.885 (0.011)
Teach.77.098 (0.238)0.838 (0.020)0.951 (0.017)0.937 (0.020)0.890 (0.015)
", + "bbox": [ + 189, + 88, + 807, + 167 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained by initializing the student near the final teacher weights, it is possible that similar results could be obtained by initializing the student at the initial teacher weights. In Table $^ 1$ we compare students distilled from random initializations with those initialized at the initial teacher weights. In addition to the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA) $\\left[ \\left[ 2 8 \\right] \\right]$ of the preactivations of each of the teacher and student networks. There is a small increase in CKA, indicating that sharing an initialization between teacher and student does increase alignment in activation space, but functionally the students are identical to their randomly initialized counterparts – there is no observable change in accuracy, agreement, or predictive KL when compared to random initialization. ", + "bbox": [ + 173, + 284, + 825, + 421 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To summarize, we have at last identified a root cause of the ineffectiveness of all our previous interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge to optimal student parameters, even when we know a solution and give the initialization a small head start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match the teacher on all inputs, the student has to at least match the teacher on the data used for distillation, and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to poor distillation fidelity. ", + "bbox": [ + 174, + 429, + 825, + 554 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 Discussion ", + "text_level": 1, + "bbox": [ + 174, + 571, + 292, + 589 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our work provides several new key findings about knowledge distillation: ", + "bbox": [ + 174, + 603, + 656, + 618 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• Good student accuracy does not imply good distillation fidelity: even outside of selfdistillation, the models with the best generalization do not always achieve the best fidelity. • Student fidelity is correlated with calibration when distilling ensembles: although the highest-fidelity student is not always the most accurate, it is always the best calibrated. • Optimization is challenging in knowledge distillation: even in cases when the student has sufficient capacity to match the teacher on the distillation data, it is unable to do so. • There is a trade-off between optimization complexity and distillation data quality: Enlarging the distillation dataset beyond the teacher training data makes it easier for the student to identify the correct solution, but also makes an already difficult optimization problem harder. ", + "bbox": [ + 217, + 630, + 825, + 770 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In standard deep learning, we are saved by not needing to solve the optimization problem well: while it true that our training loss is highly multimodal, properties such as the flatness of good solutions, the inductive biases of the network, and the implicit biases of SGD, often enable good generalization in practice. In knowledge distillation, however, good fidelity is directly aligned with solving what turns out to be an exceptionally difficult optimization problem. ", + "bbox": [ + 174, + 780, + 825, + 851 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 89, + 338, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The authors would like to thank Gregory Benton, Marc Finzi, Sanae Lotfi, Nate Gruver, and Ben Poole for helpful feedback. This research is supported by an Amazon Research Award, NSF I-DISRE 193471, NIH R01DA048764-01A1, NSF IIS-1910266, and NSF 1922658NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science. Samuel Stanton is also supported by a United States Department of Defense NDSEG fellowship. ", + "bbox": [ + 174, + 121, + 825, + 190 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 210, + 266, + 227 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Anil, R., Gupta, V., Koren, T., Regan, K., and Singer, Y. (2021). Scalable second order optimization for deep learning. arXiv preprint arXiv:2002.09018. \n[2] Ba, J. and Caruana, R. (2014). Do deep nets really need to be deep? Advances in neural information processing systems, 27:2654–2662. \n[3] Ba, J. L., Kiros, J. R., and Hinton, G. E. (2016). Layer normalization. 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Alemi", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 182, + 209, + 253, + 222 + ], + "spans": [ + { + "bbox": [ + 182, + 209, + 253, + 222 + ], + "score": 1.0, + "content": "Google Research", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.0, + "bbox_fs": [ + 174, + 198, + 261, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 334, + 198, + 436, + 220 + ], + "lines": [ + { + "bbox": [ + 332, + 197, + 437, + 210 + ], + "spans": [ + { + "bbox": [ + 332, + 197, + 437, + 210 + ], + "score": 1.0, + "content": "Andrew Gordon Wilson", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 370, + 208, + 398, + 221 + ], + "spans": [ + { + "bbox": [ + 370, + 208, + 398, + 221 + ], + "score": 1.0, + "content": "NYU", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.0, + "bbox_fs": [ + 332, + 197, + 437, + 221 + ] + }, + { + "type": "title", + "bbox": [ + 283, + 249, + 328, + 262 + ], + "lines": [ + { + "bbox": [ + 281, + 248, + 330, + 263 + ], + "spans": [ + { + "bbox": [ + 281, + 248, + 330, + 263 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 142, + 273, + 469, + 394 + ], + "lines": [ + { + "bbox": [ + 142, + 273, + 469, + 285 + ], + "spans": [ + { + "bbox": [ + 142, + 273, + 469, + 285 + ], + "score": 1.0, + "content": "Knowledge distillation is a popular technique for training a small student network", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 284, + 469, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 469, + 296 + ], + "score": 1.0, + "content": "to emulate a larger teacher model, such as an ensemble of networks. We show", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "score": 1.0, + "content": "that while knowledge distillation can improve student generalization, it does not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "score": 1.0, + "content": "typically work as it is commonly understood: there often remains a surprisingly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 317, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 470, + 329 + ], + "score": 1.0, + "content": "large discrepancy between the predictive distributions of the teacher and the student,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 327, + 471, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 471, + 340 + ], + "score": 1.0, + "content": "even in cases when the student has the capacity to perfectly match the teacher.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 339, + 470, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 339, + 470, + 351 + ], + "score": 1.0, + "content": "We identify difficulties in optimization as a key reason for why the student is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 350, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 142, + 350, + 470, + 362 + ], + "score": 1.0, + "content": "unable to match the teacher. We also show how the details of the dataset used for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 361, + 470, + 372 + ], + "spans": [ + { + "bbox": [ + 142, + 361, + 470, + 372 + ], + "score": 1.0, + "content": "distillation play a role in how closely the student matches the teacher — and that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 372, + 470, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 470, + 383 + ], + "score": 1.0, + "content": "more closely matching the teacher paradoxically does not always lead to better", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 383, + 235, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 383, + 235, + 394 + ], + "score": 1.0, + "content": "student generalization.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 141, + 273, + 471, + 394 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 412, + 190, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 192, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 192, + 428 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "Large, deep networks can learn representations that generalize well. While smaller, more efficient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "networks lack the inductive biases to find these representations from training data alone, they may", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "have the capacity to represent these solutions [e.g., 2, 18, 32, 45]. Influential work on knowledge", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 152, + 482 + ], + "score": 1.0, + "content": "distillation", + "type": "text" + }, + { + "bbox": [ + 152, + 469, + 170, + 481 + ], + "score": 0.89, + "content": "\\bar { \\| 2 2 \\| }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "argues that Bucila et al. ˘ [5] “demonstrate convincingly that the knowledge acquired", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "by a large ensemble of models [the teacher] can be transferred to a single small model [the student]”.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "Indeed this quote encapsulates the conventional narrative of knowledge distillation: a student model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 502, + 471, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 471, + 515 + ], + "score": 1.0, + "content": "learns a high-fidelity representation of a larger teacher, enabled by the teacher’s soft labels.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 436, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 518, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 194, + 531 + ], + "score": 1.0, + "content": "Conversely, in Figure", + "type": "text" + }, + { + "bbox": [ + 194, + 518, + 204, + 531 + ], + "score": 0.62, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "we show that with modern architectures knowledge distillation can lead to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "score": 1.0, + "content": "students with very different predictions from their teachers, even when the student has the capacity to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "score": 1.0, + "content": "perfectly match the teacher. Indeed, it is becoming well-known that in self-distillation the student", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "fails to match the teacher and, paradoxically, student generalization improves as a result [14, 40].", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "However, when the teacher is a large model (e.g. a deep ensemble) improvements in fidelity translate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 336, + 585 + ], + "score": 1.0, + "content": "into improvements in generalization, as we show in Figure", + "type": "text" + }, + { + "bbox": [ + 336, + 573, + 354, + 586 + ], + "score": 0.6, + "content": "\\boxed { 1 } \\mathbf { ( b ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 573, + 506, + 585 + ], + "score": 1.0, + "content": ". For these large models there is still a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 584, + 487, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 487, + 596 + ], + "score": 1.0, + "content": "significant accuracy gap between student and teacher, so fidelity is aligned with generalization.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 518, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "We will distinguish between fidelity, the ability of a student to match a teacher’s predictions, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 623 + ], + "score": 1.0, + "content": "generalization, the performance of a student in predicting unseen, in-distribution data. 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In Section", + "type": "text" + }, + { + "bbox": [ + 231, + 644, + 240, + 657 + ], + "score": 0.56, + "content": "6", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "we investigate the hypothesis that low fidelity is an optimization", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "score": 1.0, + "content": "problem resulting in a failure of the student to match the teacher even on the original training dataset.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 326, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 326, + 678 + ], + "score": 1.0, + "content": "We present a summary of our conclusions in Section 7.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 600, + 507, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 682, + 504, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "Does knowledge distillation really work? 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The effect of enlarging the CIFAR-100", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "distillation dataset with GAN-generated samples. (a): The student and teacher are both single", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 208, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 507, + 222 + ], + "score": 1.0, + "content": "ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "(b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. 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The shaded", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 242, + 318, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 195, + 254 + ], + "score": 1.0, + "content": "region corresponds to", + "type": "text" + }, + { + "bbox": [ + 195, + 243, + 221, + 253 + ], + "score": 0.9, + "content": "\\mu \\pm \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 242, + 318, + 254 + ], + "score": 1.0, + "content": ", estimated over 3 trials.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 107, + 274, + 197, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 198, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 198, + 289 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "Knowledge distillation can improve model efficiency [38, 45], unsupervised domain adaptation [37],", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 310, + 475, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 213, + 323 + ], + "score": 1.0, + "content": "improved object detection", + "type": "text" + }, + { + "bbox": [ + 213, + 311, + 226, + 322 + ], + "score": 0.52, + "content": "\\pmb { \\Vert }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 310, + 310, + 323 + ], + "score": 1.0, + "content": ", model transparency", + "type": "text" + }, + { + "bbox": [ + 310, + 311, + 327, + 322 + ], + "score": 0.31, + "content": "\\lVert \\rVert \\bigotimes \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 310, + 475, + 323 + ], + "score": 1.0, + "content": ", and adversarial robustness [15, 42].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "Seminal work by Bucila et al. ˘ [5] showed that teacher-ensembles with thousands of simple components", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 339, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 351 + ], + "score": 1.0, + "content": "could be compressed into a single shallow network that matched or outperformed its teacher. Other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "early work proposed distilling ensembles of shallow networks into a single network [55], an idea", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53].", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "Recently Fakoor et al. 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We show that achieving good fidelity is extremely difficult, even with a variety", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "of interventions, and seek to understand, by systematically considering several hypotheses, why", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "knowledge distillation does not produce high fidelity students for modern architectures and datasets.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "In contrast, the distillation literature focuses largely on improving student generalization, without", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 358, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 358, + 498 + ], + "score": 1.0, + "content": "particularly distinguishing between fidelity and generalization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "For example, concurrent work by Beyer et al. 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The effect of enlarging the CIFAR-100", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "distillation dataset with GAN-generated samples. (a): The student and teacher are both single", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 208, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 507, + 222 + ], + "score": 1.0, + "content": "ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "(b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. 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Other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "early work proposed distilling ensembles of shallow networks into a single network [55], an idea", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53].", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "Recently Fakoor et al. [13] developed a data-augmentation scheme for the distillation of large", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "ensembles of simple models for tabular data, achieving impressive results on a wide range of tabular", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "benchmarks. Malinin et al. [35] proposed a method to model the implicit distribution over predictive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "distributions from which the ensemble component predictive distributions are drawn, rather than just", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 224, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 224, + 428 + ], + "score": 1.0, + "content": "the ensemble model average.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 326, + 506, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Our work focuses explicitly on student fidelity, decoupling our understanding of good fidelity from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "good generalization. We show that achieving good fidelity is extremely difficult, even with a variety", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "of interventions, and seek to understand, by systematically considering several hypotheses, why", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "knowledge distillation does not produce high fidelity students for modern architectures and datasets.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "In contrast, the distillation literature focuses largely on improving student generalization, without", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 358, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 358, + 498 + ], + "score": 1.0, + "content": "particularly distinguishing between fidelity and generalization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 430, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "For example, concurrent work by Beyer et al. [4] does not carefully distinguish generalization and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "fidelity metrics, but they assert that high student fidelity is conceptually desirable and apparently", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "difficult to achieve when measured as the gap between teacher and student accuracy. As a result", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "their work focuses most heavily on practical modifications to the distillation procedure for the best", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "score": 1.0, + "content": "student top-1 accuracy. In this paper we investigate many of the same prescriptions, including", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "careful treatment of data augmentation (such as showing the teacher and student the exact same input", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "images), the addition of MixUp, and extended training duration. We also find that such interventions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "score": 1.0, + "content": "do improve student accuracy, but there still remains a large discrepancy between the predictive", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "distributions of the teacher and the student. We also investigate multiple optimizers. While we do not", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 175, + 613 + ], + "score": 1.0, + "content": "pursue Shampoo", + "type": "text" + }, + { + "bbox": [ + 175, + 600, + 203, + 612 + ], + "score": 0.39, + "content": "[ [ 1 7 , \\mathbb { I } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 599, + 300, + 613 + ], + "score": 1.0, + "content": "specifically, Beyer et al.", + "type": "text" + }, + { + "bbox": [ + 301, + 600, + 314, + 612 + ], + "score": 0.59, + "content": "\\mathbb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "find similar qualitative results for Shampoo and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 300, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 300, + 624 + ], + "score": 1.0, + "content": "Adam, besides faster convergence for Shampoo.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 502, + 506, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 639, + 195, + 653 + ], + "lines": [ + { + "bbox": [ + 104, + 637, + 196, + 655 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 196, + 655 + ], + "score": 1.0, + "content": "3 Preliminaries", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 391, + 678 + ], + "score": 1.0, + "content": "We will focus on the supervised classification setting, with input space", + "type": "text" + }, + { + "bbox": [ + 391, + 666, + 401, + 675 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 664, + 465, + 678 + ], + "score": 1.0, + "content": "and label space", + "type": "text" + }, + { + "bbox": [ + 466, + 666, + 474, + 676 + ], + "score": 0.77, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 664, + 506, + 678 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 675, + 507, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 138, + 688 + ], + "score": 0.9, + "content": "| { \\mathcal { V } } | = c", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 675, + 158, + 689 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 158, + 676, + 228, + 687 + ], + "score": 0.9, + "content": "f : \\mathcal { X } \\times \\Theta \\mathbb { R } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 675, + 354, + 689 + ], + "score": 1.0, + "content": "be a classifier parameterized by", + "type": "text" + }, + { + "bbox": [ + 354, + 677, + 380, + 686 + ], + "score": 0.91, + "content": "\\theta \\in \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 675, + 507, + 689 + ], + "score": 1.0, + "content": "whose outputs define a categor-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 685, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 233, + 702 + ], + "score": 1.0, + "content": "ical predictive distribution over", + "type": "text" + }, + { + "bbox": [ + 234, + 687, + 242, + 698 + ], + "score": 0.6, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 685, + 245, + 702 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 246, + 687, + 349, + 699 + ], + "score": 0.92, + "content": "\\hat { p } ( y = i | \\mathbf { x } ) = \\sigma _ { i } ( f ( \\mathbf { x } , \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 685, + 380, + 702 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 380, + 686, + 505, + 700 + ], + "score": 0.92, + "content": "\\sigma _ { i } ( { \\bf z } ) : = \\dot { \\exp ( z _ { i } ) } / \\sum _ { j } \\exp \\bar { ( } z _ { j } )", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 415, + 713 + ], + "score": 1.0, + "content": "is the softmax link function. 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When the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 158, + 86 + ], + "score": 1.0, + "content": "teacher is an", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 159, + 75, + 168, + 83 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 169, + 71, + 346, + 86 + ], + "score": 1.0, + "content": "-component ensemble, the component logits", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 346, + 73, + 398, + 85 + ], + "score": 0.91, + "content": "\\left( \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { m } \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 399, + 71, + 429, + 86 + ], + "score": 1.0, + "content": ", where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 429, + 73, + 487, + 85 + ], + "score": 0.93, + "content": "\\mathbf { z } _ { i } = f _ { i } ( \\mathbf { x } , \\theta _ { i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 487, + 71, + 505, + 86 + ], + "score": 1.0, + "content": ", are", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 507, + 99 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 254, + 99 + ], + "score": 1.0, + "content": "combined to form the teacher logits:", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 254, + 83, + 361, + 96 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbf { z } _ { t } = \\log \\bar { ( \\sum _ { i = 1 } ^ { m } \\sigma ( \\mathbf { \\bar { z } } _ { i } ) / m ) } } \\end{array}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 361, + 82, + 507, + 99 + ], + "score": 1.0, + "content": ". 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To measure fidelity, we report the following:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 517, + 463, + 584 + ], + "lines": [ + { + "bbox": [ + 146, + 517, + 463, + 584 + ], + "spans": [ + { + "bbox": [ + 146, + 517, + 463, + 584 + ], + "score": 0.57, + "content": "\\begin{array} { r l } & { \\displaystyle \\mathrm { A v e r a g e ~ T o p - 1 ~ A g r e e m e n t : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } 1 \\{ \\mathrm { a r g m a x } \\sigma _ { j } ( \\mathbf { z } _ { t , i } ) = \\underset { j } { \\mathrm { a r g m a x } } \\sigma _ { j } ( \\mathbf { z } _ { s , i } ) \\} , } \\\\ & { \\displaystyle \\mathrm { A v e r a g e ~ P r e d i c t i v e ~ K L : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathrm { K L } \\left( \\hat { p } _ { t } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\parallel \\hat { p } _ { s } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "a559a29bcef38ef5367b475572a19dc8ef744e6ac20aecd0747505152909e9c2.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 146, + 517, + 463, + 539.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 146, + 539.3333333333334, + 463, + 561.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 146, + 561.6666666666667, + 463, + 584.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 128, + 601 + ], + "score": 1.0, + "content": "Eqn.", + "type": "text" + }, + { + "bbox": [ + 128, + 588, + 141, + 600 + ], + "score": 0.8, + "content": "\\textcircled{2}", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 587, + 465, + 601 + ], + "score": 1.0, + "content": "is the average agreement between the student and teacher’s top-1 label. 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If", + "type": "text" + }, + { + "bbox": [ + 324, + 272, + 349, + 282 + ], + "score": 0.9, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 271, + 369, + 284 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 370, + 272, + 390, + 283 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { \\mathrm { K D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "is similarly equivalent to the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 252, + 295 + ], + "score": 1.0, + "content": "KL from the teacher to the student,", + "type": "text" + }, + { + "bbox": [ + 252, + 283, + 298, + 295 + ], + "score": 0.92, + "content": "\\mathrm { K L } ( \\hat { p } _ { t } | | \\hat { p } _ { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 281, + 506, + 295 + ], + "score": 1.0, + "content": ". 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In Appendix", + "type": "text" + }, + { + "bbox": [ + 441, + 375, + 460, + 388 + ], + "score": 0.68, + "content": "\\underline { { \\mathbf { B . l } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "we include", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "further discussion on the interplay of teacher ensemble size, teacher network capacity, and distillation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 243, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 243, + 408 + ], + "score": 1.0, + "content": "temperature on the student labels.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 320, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "The teacher and student often share at least some training data. 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Eqn.", + "type": "text" + }, + { + "bbox": [ + 465, + 588, + 478, + 600 + ], + "score": 0.88, + "content": "\\textcircled{3}", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "average KL divergence from the predictive distribution of the teacher to that of the student, a measure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 610, + 262, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 262, + 622 + ], + "score": 1.0, + "content": "of fidelity sensitive to all of the labels.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 587, + 505, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "While improvements in generalization metrics are relatively easy to understand, interpreting fidelity", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 635, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 635, + 448, + 651 + ], + "score": 1.0, + "content": "metrics requires some care. 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If", + "type": "text" + }, + { + "bbox": [ + 349, + 85, + 355, + 93 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 83, + 373, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 84, + 382, + 93 + ], + "score": 0.85, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "have comparable fidelity, then", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "the students agree with the teacher at many points because they generalize well, and not the reverse.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 106, + 120, + 390, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 392, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 392, + 138 + ], + "score": 1.0, + "content": "4 Knowledge Distillation Transfers Knowledge Poorly", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 504, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 504, + 157 + ], + "score": 1.0, + "content": "In this section, we present evidence that we are not able to distill large networks such as a ResNet-56", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 401, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 401, + 169 + ], + "score": 1.0, + "content": "with high fidelity, and discuss why high fidelity is an important objective.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 181, + 299, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 300, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 300, + 194 + ], + "score": 1.0, + "content": "4.1 When is knowledge transfer successful?", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 200, + 506, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 223 + ], + "score": 1.0, + "content": "a motivating example. We train the teacher on a random subset of 200 examples from the MNIST", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 269, + 235 + ], + "score": 1.0, + "content": "training set for 100 epochs, resulting in a", + "type": "text" + }, + { + "bbox": [ + 270, + 223, + 289, + 234 + ], + "score": 0.89, + "content": "8 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 222, + 300, + 235 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 300, + 223, + 320, + 234 + ], + "score": 0.89, + "content": "8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "teacher test accuracy across different subsets.2", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 483, + 246 + ], + "score": 1.0, + "content": "We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as", + "type": "text" + }, + { + "bbox": [ + 483, + 234, + 502, + 244 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 233, + 506, + 246 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 126, + 256 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 243, + 147, + 258 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 147, + 245, + 171, + 255 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 261, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 104, + 260, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 144, + 274 + ], + "score": 1.0, + "content": "In Figure", + "type": "text" + }, + { + "bbox": [ + 144, + 261, + 154, + 273 + ], + "score": 0.41, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 260, + 506, + 274 + ], + "score": 1.0, + "content": "we see that knowledge distillation works as expected. With enough examples the student", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 330, + 284 + ], + "score": 1.0, + "content": "learns to make the same predictions as the teacher (over", + "type": "text" + }, + { + "bbox": [ + 331, + 272, + 351, + 282 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "top-1 test agreement). Notably, in this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 295 + ], + "score": 1.0, + "content": "case, self-distillation does not improve generalization, since the slight difference between the teacher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 295, + 351, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 351, + 305 + ], + "score": 1.0, + "content": "and student accuracy is explained by variance between trials.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 337, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 337, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 337, + 323 + ], + "score": 1.0, + "content": "Now we consider a more challenging task: distilling a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 321, + 337, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 337, + 333 + ], + "score": 1.0, + "content": "ResNet-56 teacher trained on CIFAR-100 into an identical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 332, + 337, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 205, + 344 + ], + "score": 1.0, + "content": "student network (Figure", + "type": "text" + }, + { + "bbox": [ + 205, + 332, + 216, + 345 + ], + "score": 0.4, + "content": "\\mathbb { L } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 332, + 337, + 344 + ], + "score": 1.0, + "content": "left). Since no dataset drawn", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 343, + 338, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 338, + 355 + ], + "score": 1.0, + "content": "from the same distribution as CIFAR-100 is publicly avail-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 338, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 338, + 365 + ], + "score": 1.0, + "content": "able, to augment the distillation data, we instead combined", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 364, + 338, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 212, + 377 + ], + "score": 1.0, + "content": "samples from an SN-GAN", + "type": "text" + }, + { + "bbox": [ + 213, + 365, + 230, + 376 + ], + "score": 0.66, + "content": "\\textcircled { \\ 3 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 364, + 338, + 377 + ], + "score": 1.0, + "content": "pre-trained on CIFAR-100", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 338, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 338, + 389 + ], + "score": 1.0, + "content": "with the original CIFAR-100 train dataset. Appendix A.3", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 386, + 337, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 337, + 399 + ], + "score": 1.0, + "content": "details the hyperparameters and training procedure for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 217, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 217, + 408 + ], + "score": 1.0, + "content": "GAN, teacher, and student.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 336, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 336, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 336, + 426 + ], + "score": 1.0, + "content": "Like the MNIST experiment, as we enlarge the distillation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 336, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 336, + 437 + ], + "score": 1.0, + "content": "dataset the student fidelity improves. However, in this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 337, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 337, + 448 + ], + "score": 1.0, + "content": "case the improvement is modest, with the fidelity reaching", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 446, + 337, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 162, + 458 + ], + "score": 1.0, + "content": "nowhere near", + "type": "text" + }, + { + "bbox": [ + 162, + 447, + 181, + 457 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 446, + 337, + 458 + ], + "score": 1.0, + "content": "test agreement. Since a ResNet-56 has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 337, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 337, + 469 + ], + "score": 1.0, + "content": "many more parameters than a LeNet-5, it is possible that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 468, + 338, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 338, + 480 + ], + "score": 1.0, + "content": "the student simply has not seen enough examples to per-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 479, + 337, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 337, + 491 + ], + "score": 1.0, + "content": "fectly emulate the teacher, a hypothesis we discuss in more", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 338, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 174, + 503 + ], + "score": 1.0, + "content": "detail in Section", + "type": "text" + }, + { + "bbox": [ + 174, + 490, + 192, + 503 + ], + "score": 0.86, + "content": "\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 490, + 338, + 503 + ], + "score": 1.0, + "content": "Also, like the MNIST experiment,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 338, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 338, + 514 + ], + "score": 1.0, + "content": "as the distillation dataset grows the student accuracy ap-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 511, + 338, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 338, + 525 + ], + "score": 1.0, + "content": "proaches the teacher’s. 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[14]. This result is only possible by virtue of failing at the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "distillation procedure: if the student matched the teacher perfectly then the student could not", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "outperform the teacher. 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This setting reflects the original motivation for", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "knowledge distillation, where we wish to faithfully transfer the representation discovered by a large", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 655, + 343, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 343, + 665 + ], + "score": 1.0, + "content": "model or ensemble of models into a more efficient student.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 108, + 671, + 504, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 143, + 683 + ], + "score": 1.0, + "content": "In Figure", + "type": "text" + }, + { + "bbox": [ + 144, + 671, + 162, + 684 + ], + "score": 0.77, + "content": "1 ( \\mathsf { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "we see that if we move from self-distillation to the distillation of a 3 ResNet-56 teacher", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "ensemble, fidelity becomes positively correlated with generalization. But there is still a significant", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "2 We took only a subset of the MNIST train set since otherwise every teacher network as well as the ensemble", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 709, + 248, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 178, + 723 + ], + "score": 1.0, + "content": "would achieve over", + "type": "text" + }, + { + "bbox": [ + 178, + 711, + 196, + 721 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 709, + 248, + 723 + ], + "score": 1.0, + "content": "test accuracy.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 222, + 86 + ], + "score": 1.0, + "content": "By comparing the fidelity of", + "type": "text" + }, + { + "bbox": [ + 223, + 73, + 244, + 85 + ], + "score": 0.92, + "content": "( t , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 72, + 262, + 86 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 73, + 286, + 85 + ], + "score": 0.92, + "content": "( t , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "we can distinguish between a generic improvement in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 349, + 96 + ], + "score": 1.0, + "content": "generalization and an improvement specifically to fidelity. If", + "type": "text" + }, + { + "bbox": [ + 349, + 85, + 355, + 93 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 83, + 373, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 84, + 382, + 93 + ], + "score": 0.85, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "have comparable fidelity, then", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "the students agree with the teacher at many points because they generalize well, and not the reverse.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 72, + 507, + 108 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 120, + 390, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 392, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 392, + 138 + ], + "score": 1.0, + "content": "4 Knowledge Distillation Transfers Knowledge Poorly", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 504, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 504, + 157 + ], + "score": 1.0, + "content": "In this section, we present evidence that we are not able to distill large networks such as a ResNet-56", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 401, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 401, + 169 + ], + "score": 1.0, + "content": "with high fidelity, and discuss why high fidelity is an important objective.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 146, + 504, + 169 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 181, + 299, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 300, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 300, + 194 + ], + "score": 1.0, + "content": "4.1 When is knowledge transfer successful?", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 200, + 506, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 223 + ], + "score": 1.0, + "content": "a motivating example. We train the teacher on a random subset of 200 examples from the MNIST", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 269, + 235 + ], + "score": 1.0, + "content": "training set for 100 epochs, resulting in a", + "type": "text" + }, + { + "bbox": [ + 270, + 223, + 289, + 234 + ], + "score": 0.89, + "content": "8 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 222, + 300, + 235 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 300, + 223, + 320, + 234 + ], + "score": 0.89, + "content": "8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "teacher test accuracy across different subsets.2", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 483, + 246 + ], + "score": 1.0, + "content": "We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as", + "type": "text" + }, + { + "bbox": [ + 483, + 234, + 502, + 244 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 233, + 506, + 246 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 126, + 256 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 243, + 147, + 258 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 147, + 245, + 171, + 255 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 201, + 506, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 261, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 104, + 260, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 144, + 274 + ], + "score": 1.0, + "content": "In Figure", + "type": "text" + }, + { + "bbox": [ + 144, + 261, + 154, + 273 + ], + "score": 0.41, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 260, + 506, + 274 + ], + "score": 1.0, + "content": "we see that knowledge distillation works as expected. With enough examples the student", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 330, + 284 + ], + "score": 1.0, + "content": "learns to make the same predictions as the teacher (over", + "type": "text" + }, + { + "bbox": [ + 331, + 272, + 351, + 282 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "top-1 test agreement). 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Knowledge distillation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "was initially motivated as a means to deploy powerful models to small devices or low-latency", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "controllers [e.g., 10, 21, 26, 52, 54]. 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We additionally show in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 195, + 443 + ], + "score": 1.0, + "content": "Figure 10 in Appendix", + "type": "text" + }, + { + "bbox": [ + 195, + 429, + 213, + 442 + ], + "score": 0.45, + "content": "\\mathbf { B . l }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "that as we increase the number of ensemble components, the generalization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "disparity between teacher and distilled student increases. 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The ability to perform this transfer could have extraordinary scientific", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "consequences: large models can often discover structure in data that we would not have anticipated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "a priori. 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In both cases, achieving good distillation fidelity is crucial.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 479, + 506, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "score": 1.0, + "content": "Understanding. The name knowledge distillation implies we are transferring knowledge from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "the teacher to the student. For this reason, improved student generalization as a consequence of a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 592, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 607 + ], + "score": 1.0, + "content": "distillation procedure is sometimes conflated with fidelity. Decoupling fidelity and generalization,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "and explicitly studying fidelity, is foundational to understanding how knowledge distillation works", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 616, + 383, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 383, + 628 + ], + "score": 1.0, + "content": "and how we can make it more useful across a variety of applications.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 570, + 506, + 628 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 642, + 302, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 303, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 303, + 657 + ], + "score": 1.0, + "content": "4.4 Possible causes of low distillation fidelity", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 664, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "If we are able to match the student model to the teacher on a comprehensive distillation dataset, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 674, + 504, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 504, + 687 + ], + "score": 1.0, + "content": "expect it to match on the test data as well, achieving high distillation fidelity3. Possible causes of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 686, + 364, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 364, + 698 + ], + "score": 1.0, + "content": "poor distillation fidelity in our CIFAR-100 experiments include:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 663, + 505, + 698 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 72, + 505, + 239 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 72, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 239 + ], + "score": 0.974, + "type": "image", + "image_path": "8a2a6126b32fff96f219ed1e9e4115bce416ee0fae8ece2522be6c23d9e96dca.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 72, + 505, + 127.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 127.66666666666666, + 505, + 183.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 183.33333333333331, + 505, + 238.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 506, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Figure 4: Data recycling and distillation: results on subsampled CIFAR-100. Top: We fix the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 158, + 274 + ], + "score": 1.0, + "content": "temperature", + "type": "text" + }, + { + "bbox": [ + 158, + 262, + 185, + 272 + ], + "score": 0.86, + "content": "( \\tau = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 261, + 372, + 274 + ], + "score": 1.0, + "content": ") and vary the number of ensemble components", + "type": "text" + }, + { + "bbox": [ + 372, + 262, + 388, + 272 + ], + "score": 0.78, + "content": "( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 261, + 506, + 274 + ], + "score": 1.0, + "content": ", comparing students distilled", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 249, + 285 + ], + "score": 1.0, + "content": "on the same dataset as the teacher", + "type": "text" + }, + { + "bbox": [ + 250, + 272, + 286, + 284 + ], + "score": 0.92, + "content": "( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 272, + 367, + 285 + ], + "score": 1.0, + "content": ", a reserved dataset", + "type": "text" + }, + { + "bbox": [ + 368, + 272, + 404, + 284 + ], + "score": 0.92, + "content": "( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 272, + 442, + 285 + ], + "score": 1.0, + "content": ", or both", + "type": "text" + }, + { + "bbox": [ + 442, + 272, + 502, + 284 + ], + "score": 0.91, + "content": "( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 272, + 506, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 283, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 354, + 296 + ], + "score": 1.0, + "content": "Distilling on both produces the best result, while distilling on", + "type": "text" + }, + { + "bbox": [ + 355, + 284, + 368, + 294 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 283, + 506, + 296 + ], + "score": 1.0, + "content": "increases accuracy and decreases", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 182, + 306 + ], + "score": 1.0, + "content": "fidelity, relative to", + "type": "text" + }, + { + "bbox": [ + 183, + 294, + 196, + 305 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 293, + 376, + 306 + ], + "score": 1.0, + "content": ". 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The shaded", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 305, + 317, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 195, + 317 + ], + "score": 1.0, + "content": "region corresponds to", + "type": "text" + }, + { + "bbox": [ + 195, + 306, + 221, + 316 + ], + "score": 0.91, + "content": "\\mu \\pm \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 305, + 317, + 317 + ], + "score": 1.0, + "content": ", estimated over 3 trials.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "Student capacity – We observe low fidelity even in the self-distillation setting, so we can rule out", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 428, + 366 + ], + "score": 1.0, + "content": "student capacity as a primary cause, but we also confirm in Figure 12 in Appendix", + "type": "text" + }, + { + "bbox": [ + 428, + 352, + 446, + 365 + ], + "score": 0.63, + "content": "\\mathbb { E . l }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "that increasing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 362, + 455, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 455, + 376 + ], + "score": 1.0, + "content": "the student capacity has very little effect on fidelity in the ensemble-distillation setting.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 379, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 484, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 484, + 403 + ], + "score": 1.0, + "content": "we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 503, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 420 + ], + "score": 1.0, + "content": "Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 416, + 367, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 367, + 431 + ], + "score": 1.0, + "content": "that our findings apply to datasets of larger scale and complexity.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 255, + 447 + ], + "score": 1.0, + "content": "Data domain – Similarly in Section", + "type": "text" + }, + { + "bbox": [ + 255, + 433, + 274, + 447 + ], + "score": 0.46, + "content": "{ \\bf C . 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "we observe low distillation fidelity in the context of text", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "classification (sentiment analysis on the IMDB dataset), showing our results are relevant beyond", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 456, + 191, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 191, + 469 + ], + "score": 1.0, + "content": "image classification.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 201, + 485 + ], + "score": 1.0, + "content": "Identifiability (Section", + "type": "text" + }, + { + "bbox": [ + 202, + 472, + 210, + 484 + ], + "score": 0.28, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 471, + 506, + 485 + ], + "score": 1.0, + "content": ") – the distillation data is insufficient to distinguish high-fidelity and low-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "fidelity students. 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The student does not agree with the teacher on test because it does not even agree on train.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 105, + 551, + 432, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 433, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 433, + 568 + ], + "score": 1.0, + "content": "5 Identifiability: Are We Using the Right Distillation Dataset?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "score": 1.0, + "content": "We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 601, + 411, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 411, + 613 + ], + "score": 1.0, + "content": "enough of the right input-teacher label pairs to define the solution we want?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 106, + 629, + 299, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 300, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 300, + 643 + ], + "score": 1.0, + "content": "5.1 Should we do more data augmentation?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 650, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "Data augmentation is a simple and practical method to increase the support of the distillation data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "distribution. 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"score": 1.0, + "content": "Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 484, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 484, + 403 + ], + "score": 1.0, + "content": "we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 378, + 505, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 503, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 420 + ], + "score": 1.0, + "content": "Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 416, + 367, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 367, + 431 + ], + 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In other words, matching the teacher predictions on the distillation dataset does not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 495, + 287, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 287, + 506 + ], + "score": 1.0, + "content": "lead to matching predictions on the test data.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 471, + 506, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "Optimization (Section 6) – we are unable to solve the distillation optimization problem sufficiently", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 492, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 492, + 535 + ], + "score": 1.0, + "content": "well. The student does not agree with the teacher on test because it does not even agree on train.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 510, + 505, + 535 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 551, + 432, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 433, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 433, + 568 + ], + "score": 1.0, + "content": "5 Identifiability: Are We Using the Right Distillation Dataset?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "score": 1.0, + "content": "We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 601, + 411, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 411, + 613 + ], + "score": 1.0, + "content": "enough of the right input-teacher label pairs to define the solution we want?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 579, + 506, + 613 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 629, + 299, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 300, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 300, + 643 + ], + "score": 1.0, + "content": "5.1 Should we do more data augmentation?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 650, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "Data augmentation is a simple and practical method to increase the support of the distillation data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "distribution. If identifiability is a primary cause of poor distillation fidelity, using a more extensive", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 672, + 387, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 387, + 686 + ], + "score": 1.0, + "content": "data augmentation strategy during distillation should improve fidelity.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 650, + 505, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "To test this hypothesis, we evaluated the effect of several augmentation strategies on student fidelity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 230, + 713 + ], + "score": 1.0, + "content": "and generalization. In Figure", + "type": "text" + }, + { + "bbox": [ + 230, + 700, + 239, + 712 + ], + "score": 0.78, + "content": "\\textcircled { 3 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "the teacher is a 5-component ensemble of ResNet-56 networks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "score": 1.0, + "content": "trained on CIFAR-100 with the Baseline augmentation strategy: horizontal flips and random crops.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 688, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "We report the student accuracy and teacher-student agreement for each augmentation strategy, and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 81, + 504, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 256, + 98 + ], + "score": 1.0, + "content": "also include results for Baseline with", + "type": "text" + }, + { + "bbox": [ + 256, + 84, + 281, + 94 + ], + "score": 0.9, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 81, + 299, + 98 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 84, + 325, + 94 + ], + "score": 0.89, + "content": "\\tau = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 81, + 504, + 98 + ], + "score": 1.0, + "content": "to demonstrate the effect of logit tempering.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 394, + 113 + ], + "score": 1.0, + "content": "We first observe that the best augmentation policies for generalization,", + "type": "text" + }, + { + "bbox": [ + 394, + 100, + 424, + 111 + ], + "score": 0.75, + "content": "M i x U p", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 99, + 445, + 113 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 445, + 97, + 474, + 113 + ], + "score": 0.89, + "content": "G A N \\mathbb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "are not", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "the best policies for fidelity. 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Many of the augmentation strategies also slightly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 394, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 302, + 167 + ], + "score": 1.0, + "content": "improve teacher-student KL relative to Baseline", + "type": "text" + }, + { + "bbox": [ + 302, + 155, + 329, + 165 + ], + "score": 0.84, + "content": "\\tau = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 154, + 378, + 167 + ], + "score": 1.0, + "content": ") (see Figure", + "type": "text" + }, + { + "bbox": [ + 379, + 154, + 394, + 167 + ], + "score": 0.27, + "content": "\\textcircled { 1 1 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 206, + 185 + ], + "score": 1.0, + "content": "In Figure 11 in Appendix", + "type": "text" + }, + { + "bbox": [ + 207, + 170, + 225, + 183 + ], + "score": 0.52, + "content": "{ \\bf B } . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 170, + 506, + 185 + ], + "score": 1.0, + "content": "we report all generalization and fidelity metrics for a range of ensemble", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 438, + 194 + ], + "score": 1.0, + "content": "sizes, as well as the results for the independent student baseline discussed in Section", + "type": "text" + }, + { + "bbox": [ + 438, + 182, + 456, + 194 + ], + "score": 0.81, + "content": "\\underline { { \\bar { 3 . 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "Often these", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "independent students, taught how to mimic a completely different model, have nearly as good test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 464, + 217 + ], + "score": 1.0, + "content": "agreement with the teacher as the student explicitly trained to emulate it. See Appendix", + "type": "text" + }, + { + "bbox": [ + 465, + 203, + 484, + 216 + ], + "score": 0.42, + "content": "\\mathbf { \\bar { A } } . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "for a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "score": 1.0, + "content": "detailed description of the augmentation procedures.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "score": 1.0, + "content": "Should data augmentation be close to the data distribution? In theory, any data augmentation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "should help with identifiability: if a student matches a teacher on more data, it is more likely to match", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 292, + 265 + ], + "score": 1.0, + "content": "the teacher elsewhere. However, the Noise and", + "type": "text" + }, + { + "bbox": [ + 293, + 253, + 315, + 263 + ], + "score": 0.3, + "content": "O O D", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "augmentation strategies based on noise and out-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "of-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 286 + ], + "score": 1.0, + "content": "data augmentation has an effect beyond improving identifiability — it has a regularizing effect,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 285, + 490, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 490, + 298 + ], + "score": 1.0, + "content": "making optimization more challenging. We explore this facet of data augmentation in Section 6.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "The slight improvements to fidelity with extensive augmentations suggest that increasing the support", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "score": 1.0, + "content": "small compared to heuristics like logit tempering (which does not modify the support at all), it is very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 477, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 477, + 348 + ], + "score": 1.0, + "content": "unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 256, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 258, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 258, + 386 + ], + "score": 1.0, + "content": "5.2 The data recycling hypothesis", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "If simply showing the student more labels does not always significantly improve fidelity, perhaps we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "are not showing the student the right labels. Additional data augmentation during distillation does", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "give the student more teacher labels to match, but also introduces a distribution shift between the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "images the teacher was trained on and the images the student is distilling on. Even when the teacher", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "and student have the same augmentation policy, reusing the teacher’s training data for distillation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "independent draw from the true joint distribution over images and teacher labels. What if there was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 283, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 283, + 496 + ], + "score": 1.0, + "content": "distribution over images and teacher labels?", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 507, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 171, + 525 + ], + "score": 1.0, + "content": "training dataset", + "type": "text" + }, + { + "bbox": [ + 171, + 512, + 181, + 521 + ], + "score": 0.77, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 510, + 267, + 525 + ], + "score": 1.0, + "content": "into two equal parts,", + "type": "text" + }, + { + "bbox": [ + 267, + 511, + 280, + 522 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 510, + 299, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 511, + 312, + 522 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 510, + 489, + 525 + ], + "score": 1.0, + "content": ". We train teacher ResNet-56 ensembles on", + "type": "text" + }, + { + "bbox": [ + 489, + 511, + 502, + 522 + ], + "score": 0.87, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 510, + 507, + 525 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 180, + 535 + ], + "score": 1.0, + "content": "and then compare", + "type": "text" + }, + { + "bbox": [ + 180, + 524, + 190, + 533 + ], + "score": 0.85, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 522, + 327, + 535 + ], + "score": 1.0, + "content": ", a student distilled on the original", + "type": "text" + }, + { + "bbox": [ + 327, + 523, + 340, + 533 + ], + "score": 0.73, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 522, + 344, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 523, + 355, + 533 + ], + "score": 0.58, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 522, + 489, + 535 + ], + "score": 1.0, + "content": ", a student distilled on the unseen", + "type": "text" + }, + { + "bbox": [ + 489, + 523, + 502, + 533 + ], + "score": 0.86, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 522, + 506, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 123, + 546 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 534, + 143, + 545 + ], + "score": 0.89, + "content": "s _ { 0 \\cup 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 532, + 256, + 546 + ], + "score": 1.0, + "content": ", a student distilled on both:", + "type": "text" + }, + { + "bbox": [ + 257, + 533, + 293, + 544 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 532, + 507, + 546 + ], + "score": 1.0, + "content": ". Note that the students cannot access the true labels,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 555, + 332, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 332, + 567 + ], + "score": 1.0, + "content": "the top row and the logit temperature in the bottom row.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 161, + 584 + ], + "score": 1.0, + "content": "Surprisingly,", + "type": "text" + }, + { + "bbox": [ + 161, + 573, + 172, + 582 + ], + "score": 0.84, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 570, + 306, + 584 + ], + "score": 1.0, + "content": "attains higher test accuracy than", + "type": "text" + }, + { + "bbox": [ + 306, + 573, + 316, + 582 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 570, + 505, + 584 + ], + "score": 1.0, + "content": ", while showing worse ECE and lower fidelity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "(measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 606 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 594, + 134, + 604 + ], + "score": 0.86, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 592, + 361, + 606 + ], + "score": 1.0, + "content": "should be a higher fidelity distillation of the teacher than", + "type": "text" + }, + { + "bbox": [ + 361, + 594, + 372, + 604 + ], + "score": 0.85, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "does hold, but the gain in fidelity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 181, + 617 + ], + "score": 1.0, + "content": "does not result in", + "type": "text" + }, + { + "bbox": [ + 181, + 605, + 192, + 615 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 603, + 447, + 617 + ], + "score": 1.0, + "content": "best replicating the teacher’s accuracy. The best attributes of", + "type": "text" + }, + { + "bbox": [ + 448, + 605, + 458, + 615 + ], + "score": 0.86, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 603, + 477, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 605, + 488, + 615 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 162, + 628 + ], + "score": 1.0, + "content": "combined by", + "type": "text" + }, + { + "bbox": [ + 162, + 617, + 181, + 626 + ], + "score": 0.89, + "content": "s _ { 0 \\cup 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 615, + 469, + 628 + ], + "score": 1.0, + "content": ", which coincides with how unlabeled data is typically used in practice", + "type": "text" + }, + { + "bbox": [ + 469, + 615, + 482, + 626 + ], + "score": 0.51, + "content": "\\pmb { \\left. 2 \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 615, + 506, + 628 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "reason for this puzzling observation is simply that for the larger teachers fidelity has not improved", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "enough to also improve generalization. In fact, the best teacher-student agreement is only around", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 126, + 659 + ], + "score": 0.87, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 648, + 505, + 659 + ], + "score": 1.0, + "content": ", no improvement when compared to the results from extensive data augmentation in the last", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "section. We again find that modifying the distillation data can slightly improve fidelity, but the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 115, + 708, + 500, + 723 + ], + "lines": [ + { + "bbox": [ + 115, + 707, + 504, + 727 + ], + "spans": [ + { + "bbox": [ + 115, + 707, + 504, + 727 + ], + "score": 1.0, + "content": "4 Unlike Figure 1, for Figure 3 we generated new GAN samples every epoch, to mimic data augmentation.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "We report the student accuracy and teacher-student agreement for each augmentation strategy, and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 81, + 504, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 256, + 98 + ], + "score": 1.0, + "content": "also include results for Baseline with", + "type": "text" + }, + { + "bbox": [ + 256, + 84, + 281, + 94 + ], + "score": 0.9, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 81, + 299, + 98 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 84, + 325, + 94 + ], + "score": 0.89, + "content": "\\tau = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 81, + 504, + 98 + ], + "score": 1.0, + "content": "to demonstrate the effect of logit tempering.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 98 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 394, + 113 + ], + "score": 1.0, + "content": "We first observe that the best augmentation policies for generalization,", + "type": "text" + }, + { + "bbox": [ + 394, + 100, + 424, + 111 + ], + "score": 0.75, + "content": "M i x U p", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 99, + 445, + 113 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 445, + 97, + 474, + 113 + ], + "score": 0.89, + "content": "G A N \\mathbb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "are not", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "the best policies for fidelity. Furthermore, although many augmentation strategies enable slightly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 303, + 135 + ], + "score": 1.0, + "content": "higher distillation fidelity compared to Baseline", + "type": "text" + }, + { + "bbox": [ + 303, + 122, + 329, + 132 + ], + "score": 0.85, + "content": "\\tau = 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "), even the best augmentation policy, Mixup", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 109, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 109, + 133, + 137, + 144 + ], + "score": 0.84, + "content": "\\tau = 4 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 132, + 240, + 145 + ], + "score": 1.0, + "content": "), only achieves a modest", + "type": "text" + }, + { + "bbox": [ + 241, + 133, + 261, + 143 + ], + "score": 0.87, + "content": "86 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 132, + 412, + 145 + ], + "score": 1.0, + "content": "test agreement. In fact the Baseline", + "type": "text" + }, + { + "bbox": [ + 413, + 133, + 440, + 144 + ], + "score": 0.85, + "content": "\\tau = 4 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 132, + 505, + 145 + ], + "score": 1.0, + "content": ") policy is quite", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 202, + 157 + ], + "score": 1.0, + "content": "competitive, achieving", + "type": "text" + }, + { + "bbox": [ + 203, + 144, + 230, + 154 + ], + "score": 0.87, + "content": "8 4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "test agreement. Many of the augmentation strategies also slightly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 394, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 302, + 167 + ], + "score": 1.0, + "content": "improve teacher-student KL relative to Baseline", + "type": "text" + }, + { + "bbox": [ + 302, + 155, + 329, + 165 + ], + "score": 0.84, + "content": "\\tau = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 154, + 378, + 167 + ], + "score": 1.0, + "content": ") (see Figure", + "type": "text" + }, + { + "bbox": [ + 379, + 154, + 394, + 167 + ], + "score": 0.27, + "content": "\\textcircled { 1 1 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 97, + 505, + 167 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 206, + 185 + ], + "score": 1.0, + "content": "In Figure 11 in Appendix", + "type": "text" + }, + { + "bbox": [ + 207, + 170, + 225, + 183 + ], + "score": 0.52, + "content": "{ \\bf B } . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 170, + 506, + 185 + ], + "score": 1.0, + "content": "we report all generalization and fidelity metrics for a range of ensemble", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 438, + 194 + ], + "score": 1.0, + "content": "sizes, as well as the results for the independent student baseline discussed in Section", + "type": "text" + }, + { + "bbox": [ + 438, + 182, + 456, + 194 + ], + "score": 0.81, + "content": "\\underline { { \\bar { 3 . 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "Often these", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "independent students, taught how to mimic a completely different model, have nearly as good test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 464, + 217 + ], + "score": 1.0, + "content": "agreement with the teacher as the student explicitly trained to emulate it. See Appendix", + "type": "text" + }, + { + "bbox": [ + 465, + 203, + 484, + 216 + ], + "score": 0.42, + "content": "\\mathbf { \\bar { A } } . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "for a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "score": 1.0, + "content": "detailed description of the augmentation procedures.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 170, + 506, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "score": 1.0, + "content": "Should data augmentation be close to the data distribution? In theory, any data augmentation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "should help with identifiability: if a student matches a teacher on more data, it is more likely to match", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 292, + 265 + ], + "score": 1.0, + "content": "the teacher elsewhere. However, the Noise and", + "type": "text" + }, + { + "bbox": [ + 293, + 253, + 315, + 263 + ], + "score": 0.3, + "content": "O O D", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "augmentation strategies based on noise and out-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "of-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 286 + ], + "score": 1.0, + "content": "data augmentation has an effect beyond improving identifiability — it has a regularizing effect,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 285, + 490, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 490, + 298 + ], + "score": 1.0, + "content": "making optimization more challenging. We explore this facet of data augmentation in Section 6.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 231, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "The slight improvements to fidelity with extensive augmentations suggest that increasing the support", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "score": 1.0, + "content": "small compared to heuristics like logit tempering (which does not modify the support at all), it is very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 477, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 477, + 348 + ], + "score": 1.0, + "content": "unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 302, + 505, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 256, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 258, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 258, + 386 + ], + "score": 1.0, + "content": "5.2 The data recycling hypothesis", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "If simply showing the student more labels does not always significantly improve fidelity, perhaps we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "are not showing the student the right labels. Additional data augmentation during distillation does", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "give the student more teacher labels to match, but also introduces a distribution shift between the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "images the teacher was trained on and the images the student is distilling on. Even when the teacher", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "and student have the same augmentation policy, reusing the teacher’s training data for distillation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "independent draw from the true joint distribution over images and teacher labels. What if there was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 283, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 283, + 496 + ], + "score": 1.0, + "content": "distribution over images and teacher labels?", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 396, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 507, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 171, + 525 + ], + "score": 1.0, + "content": "training dataset", + "type": "text" + }, + { + "bbox": [ + 171, + 512, + 181, + 521 + ], + "score": 0.77, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 510, + 267, + 525 + ], + "score": 1.0, + "content": "into two equal parts,", + "type": "text" + }, + { + "bbox": [ + 267, + 511, + 280, + 522 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 510, + 299, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 511, + 312, + 522 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 510, + 489, + 525 + ], + "score": 1.0, + "content": ". We train teacher ResNet-56 ensembles on", + "type": "text" + }, + { + "bbox": [ + 489, + 511, + 502, + 522 + ], + "score": 0.87, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 510, + 507, + 525 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 180, + 535 + ], + "score": 1.0, + "content": "and then compare", + "type": "text" + }, + { + "bbox": [ + 180, + 524, + 190, + 533 + ], + "score": 0.85, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 522, + 327, + 535 + ], + "score": 1.0, + "content": ", a student distilled on the original", + "type": "text" + }, + { + "bbox": [ + 327, + 523, + 340, + 533 + ], + "score": 0.73, + "content": "\\mathcal { D } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 522, + 344, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 523, + 355, + 533 + ], + "score": 0.58, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 522, + 489, + 535 + ], + "score": 1.0, + "content": ", a student distilled on the unseen", + "type": "text" + }, + { + "bbox": [ + 489, + 523, + 502, + 533 + ], + "score": 0.86, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 522, + 506, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 123, + 546 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 534, + 143, + 545 + ], + "score": 0.89, + "content": "s _ { 0 \\cup 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 532, + 256, + 546 + ], + "score": 1.0, + "content": ", a student distilled on both:", + "type": "text" + }, + { + "bbox": [ + 257, + 533, + 293, + 544 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 532, + 507, + 546 + ], + "score": 1.0, + "content": ". Note that the students cannot access the true labels,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 555, + 332, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 332, + 567 + ], + "score": 1.0, + "content": "the top row and the logit temperature in the bottom row.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 500, + 507, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 161, + 584 + ], + "score": 1.0, + "content": "Surprisingly,", + "type": "text" + }, + { + "bbox": [ + 161, + 573, + 172, + 582 + ], + "score": 0.84, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 570, + 306, + 584 + ], + "score": 1.0, + "content": "attains higher test accuracy than", + "type": "text" + }, + { + "bbox": [ + 306, + 573, + 316, + 582 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 570, + 505, + 584 + ], + "score": 1.0, + "content": ", while showing worse ECE and lower fidelity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "(measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 606 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 594, + 134, + 604 + ], + "score": 0.86, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 592, + 361, + 606 + ], + "score": 1.0, + "content": "should be a higher fidelity distillation of the teacher than", + "type": "text" + }, + { + "bbox": [ + 361, + 594, + 372, + 604 + ], + "score": 0.85, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "does hold, but the gain in fidelity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 181, + 617 + ], + "score": 1.0, + "content": "does not result in", + "type": "text" + }, + { + "bbox": [ + 181, + 605, + 192, + 615 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 603, + 447, + 617 + ], + "score": 1.0, + "content": "best replicating the teacher’s accuracy. The best attributes of", + "type": "text" + }, + { + "bbox": [ + 448, + 605, + 458, + 615 + ], + "score": 0.86, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 603, + 477, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 605, + 488, + 615 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 162, + 628 + ], + "score": 1.0, + "content": "combined by", + "type": "text" + }, + { + "bbox": [ + 162, + 617, + 181, + 626 + ], + "score": 0.89, + "content": "s _ { 0 \\cup 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 615, + 469, + 628 + ], + "score": 1.0, + "content": ", which coincides with how unlabeled data is typically used in practice", + "type": "text" + }, + { + "bbox": [ + 469, + 615, + 482, + 626 + ], + "score": 0.51, + "content": "\\pmb { \\left. 2 \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 615, + 506, + 628 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "reason for this puzzling observation is simply that for the larger teachers fidelity has not improved", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "enough to also improve generalization. In fact, the best teacher-student agreement is only around", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 126, + 659 + ], + "score": 0.87, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 648, + 505, + 659 + ], + "score": 1.0, + "content": ", no improvement when compared to the results from extensive data augmentation in the last", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "section. We again find that modifying the distillation data can slightly improve fidelity, but the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 570, + 506, + 682 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 70, + 505, + 140 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 70, + 505, + 140 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 505, + 140 + ], + "score": 0.96, + "type": "image", + "image_path": "ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 70, + 505, + 93.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 93.33333333333333, + 505, + 116.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 116.66666666666666, + 505, + 140.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 150, + 506, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 322, + 163 + ], + "score": 1.0, + "content": "Figure 5: The train agreement for teacher ensembles", + "type": "text" + }, + { + "bbox": [ + 322, + 150, + 380, + 162 + ], + "score": 0.92, + "content": "( m \\in \\{ 1 , 3 , 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 150, + 506, + 163 + ], + "score": 1.0, + "content": ") and student on the distillation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 174 + ], + "score": 1.0, + "content": "data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "the softness of the teacher labels by adding examples not in the teacher train data makes distillation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle:", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 193, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 356, + 207 + ], + "score": 1.0, + "content": "agreement from subsampled CIFAR-100 experiment in Figure", + "type": "text" + }, + { + "bbox": [ + 356, + 193, + 366, + 206 + ], + "score": 0.63, + "content": "4 .", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 193, + 506, + 207 + ], + "score": 1.0, + "content": "Right: agreement for some of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "score": 1.0, + "content": "augmentation policies in Figure 3. The shaded region is not visible because the variance is very low.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 105, + 247, + 495, + 262 + ], + "lines": [ + { + "bbox": [ + 104, + 247, + 497, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 247, + 497, + 263 + ], + "score": 1.0, + "content": "6 Optimization: Does the Student Match the Teacher on Distillation Data?", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 294 + ], + "score": 1.0, + "content": "If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 292, + 507, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 507, + 305 + ], + "score": 1.0, + "content": "perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 381, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 381, + 325 + ], + "score": 1.0, + "content": "the student match the teacher on the data it is trained to match it on?", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 352, + 324, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 325, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 325, + 366 + ], + "score": 1.0, + "content": "6.1 More distillation data lowers train agreement", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 432, + 412 + ], + "score": 1.0, + "content": "teacher-student agreement on test, but the agreement remains relatively low (below", + "type": "text" + }, + { + "bbox": [ + 432, + 399, + 452, + 411 + ], + "score": 0.86, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 400, + 505, + 412 + ], + "score": 1.0, + "content": ") even for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 329, + 424 + ], + "score": 1.0, + "content": "largest distillation dataset that we considered. In Figure", + "type": "text" + }, + { + "bbox": [ + 329, + 410, + 339, + 423 + ], + "score": 0.69, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "(left panel), we report the teacher-student", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "agreement for the same experiment, but now on the distillation dataset. We now observe the opposite", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "the teacher. Even when the student has identical capacity to the teacher, the student only achieves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 440, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 126, + 465 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 453, + 291, + 468 + ], + "score": 1.0, + "content": "agreement with the teacher when we use", + "type": "text" + }, + { + "bbox": [ + 291, + 455, + 308, + 465 + ], + "score": 0.84, + "content": "5 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 453, + 440, + 468 + ], + "score": 1.0, + "content": "synthetic images for distillation.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "The drop in train agreement is even more pronounced when we use extensive data augmentation. In", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 133, + 493 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 133, + 481, + 144, + 494 + ], + "score": 0.46, + "content": "{ \\bar { 5 , } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "right panel, we report the teacher-student agreement on the train set with data augmentation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 343, + 505 + ], + "score": 1.0, + "content": "for a subset of augmentation strategies presented in Section", + "type": "text" + }, + { + "bbox": [ + 344, + 492, + 361, + 505 + ], + "score": 0.83, + "content": "\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "We use the CIFAR-100 dataset and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 427, + 516 + ], + "score": 1.0, + "content": "the ResNet-56 model for the teachers and the students (for details, see Section", + "type": "text" + }, + { + "bbox": [ + 428, + 503, + 446, + 515 + ], + "score": 0.74, + "content": "\\underline { { \\vert 5 . 1 \\rangle } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 503, + 506, + 516 + ], + "score": 1.0, + "content": ". In each case,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "we measure agreement on the augmented training set that was used during distillation. While for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "score": 1.0, + "content": "heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 301, + 559 + ], + "score": 1.0, + "content": "Jitter augmentations, the agreement is between", + "type": "text" + }, + { + "bbox": [ + 301, + 547, + 321, + 558 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 547, + 339, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 547, + 360, + 558 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "for all the considered teacher sizes.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "For Combined Augs, the combination of these three augmentation strategies, the agreement drops", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 569, + 284, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 186, + 580 + ], + "score": 1.0, + "content": "even further, to just", + "type": "text" + }, + { + "bbox": [ + 186, + 569, + 206, + 580 + ], + "score": 0.88, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 569, + 284, + 580 + ], + "score": 1.0, + "content": "in self-distillation!", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "Our intuition about how knowledge distillation should work largely hinges on the assumption that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "after distillation the student matches the teacher on the distillation set. However, the results presented", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "in this section suggest that in practice the optimization method is unable to achieve high fidelity even", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "on the distillation dataset when extensive data augmentation or synthetic data is used. The inability", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 630, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 640 + ], + "score": 1.0, + "content": "to solve the optimization problem undermines distillation: in order to find a student that would match", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "the teacher on all inputs, we need to at least be able to find a student that would match the teacher on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 651, + 211, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 211, + 662 + ], + "score": 1.0, + "content": "all of the distillation data.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Optimization and the train-test fidelity gap. Notably, despite having the lowest train agreement,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "the Combined Augs policy results in better test agreement than other polices with better train", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 182, + 702 + ], + "score": 1.0, + "content": "agreement (Figure", + "type": "text" + }, + { + "bbox": [ + 182, + 689, + 191, + 702 + ], + "score": 0.31, + "content": "3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "). This result highlights a fundamental trade-off in knowledge distillation: the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "student needs many teacher labels match the teacher on test, but introducing examples not in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 438, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 438, + 723 + ], + "score": 1.0, + "content": "teacher train data makes matching the teacher on the distillation data very difficult.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 70, + 505, + 140 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 70, + 505, + 140 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 505, + 140 + ], + "score": 0.96, + "type": "image", + "image_path": "ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 70, + 505, + 93.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 93.33333333333333, + 505, + 116.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 116.66666666666666, + 505, + 140.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 150, + 506, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 322, + 163 + ], + "score": 1.0, + "content": "Figure 5: The train agreement for teacher ensembles", + "type": "text" + }, + { + "bbox": [ + 322, + 150, + 380, + 162 + ], + "score": 0.92, + "content": "( m \\in \\{ 1 , 3 , 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 150, + 506, + 163 + ], + "score": 1.0, + "content": ") and student on the distillation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 174 + ], + "score": 1.0, + "content": "data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "the softness of the teacher labels by adding examples not in the teacher train data makes distillation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle:", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 193, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 356, + 207 + ], + "score": 1.0, + "content": "agreement from subsampled CIFAR-100 experiment in Figure", + "type": "text" + }, + { + "bbox": [ + 356, + 193, + 366, + 206 + ], + "score": 0.63, + "content": "4 .", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 193, + 506, + 207 + ], + "score": 1.0, + "content": "Right: agreement for some of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "score": 1.0, + "content": "augmentation policies in Figure 3. The shaded region is not visible because the variance is very low.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 105, + 247, + 495, + 262 + ], + "lines": [ + { + "bbox": [ + 104, + 247, + 497, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 247, + 497, + 263 + ], + "score": 1.0, + "content": "6 Optimization: Does the Student Match the Teacher on Distillation Data?", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 294 + ], + "score": 1.0, + "content": "If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 292, + 507, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 507, + 305 + ], + "score": 1.0, + "content": "perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 381, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 381, + 325 + ], + "score": 1.0, + "content": "the student match the teacher on the data it is trained to match it on?", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 281, + 507, + 325 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 352, + 324, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 325, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 325, + 366 + ], + "score": 1.0, + "content": "6.1 More distillation data lowers train agreement", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 432, + 412 + ], + "score": 1.0, + "content": "teacher-student agreement on test, but the agreement remains relatively low (below", + "type": "text" + }, + { + "bbox": [ + 432, + 399, + 452, + 411 + ], + "score": 0.86, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 400, + 505, + 412 + ], + "score": 1.0, + "content": ") even for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 329, + 424 + ], + "score": 1.0, + "content": "largest distillation dataset that we considered. In Figure", + "type": "text" + }, + { + "bbox": [ + 329, + 410, + 339, + 423 + ], + "score": 0.69, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "(left panel), we report the teacher-student", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "agreement for the same experiment, but now on the distillation dataset. We now observe the opposite", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "the teacher. Even when the student has identical capacity to the teacher, the student only achieves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 440, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 126, + 465 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 453, + 291, + 468 + ], + "score": 1.0, + "content": "agreement with the teacher when we use", + "type": "text" + }, + { + "bbox": [ + 291, + 455, + 308, + 465 + ], + "score": 0.84, + "content": "5 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 453, + 440, + 468 + ], + "score": 1.0, + "content": "synthetic images for distillation.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 378, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "The drop in train agreement is even more pronounced when we use extensive data augmentation. In", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 133, + 493 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 133, + 481, + 144, + 494 + ], + "score": 0.46, + "content": "{ \\bar { 5 , } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "right panel, we report the teacher-student agreement on the train set with data augmentation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 343, + 505 + ], + "score": 1.0, + "content": "for a subset of augmentation strategies presented in Section", + "type": "text" + }, + { + "bbox": [ + 344, + 492, + 361, + 505 + ], + "score": 0.83, + "content": "\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "We use the CIFAR-100 dataset and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 427, + 516 + ], + "score": 1.0, + "content": "the ResNet-56 model for the teachers and the students (for details, see Section", + "type": "text" + }, + { + "bbox": [ + 428, + 503, + 446, + 515 + ], + "score": 0.74, + "content": "\\underline { { \\vert 5 . 1 \\rangle } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 503, + 506, + 516 + ], + "score": 1.0, + "content": ". In each case,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "we measure agreement on the augmented training set that was used during distillation. While for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "score": 1.0, + "content": "heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 301, + 559 + ], + "score": 1.0, + "content": "Jitter augmentations, the agreement is between", + "type": "text" + }, + { + "bbox": [ + 301, + 547, + 321, + 558 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 547, + 339, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 547, + 360, + 558 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "for all the considered teacher sizes.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "For Combined Augs, the combination of these three augmentation strategies, the agreement drops", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 569, + 284, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 186, + 580 + ], + "score": 1.0, + "content": "even further, to just", + "type": "text" + }, + { + "bbox": [ + 186, + 569, + 206, + 580 + ], + "score": 0.88, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 569, + 284, + 580 + ], + "score": 1.0, + "content": "in self-distillation!", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 470, + 506, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "Our intuition about how knowledge distillation should work largely hinges on the assumption that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "after distillation the student matches the teacher on the distillation set. However, the results presented", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "in this section suggest that in practice the optimization method is unable to achieve high fidelity even", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "on the distillation dataset when extensive data augmentation or synthetic data is used. The inability", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 630, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 640 + ], + "score": 1.0, + "content": "to solve the optimization problem undermines distillation: in order to find a student that would match", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "the teacher on all inputs, we need to at least be able to find a student that would match the teacher on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 651, + 211, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 211, + 662 + ], + "score": 1.0, + "content": "all of the distillation data.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 585, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Optimization and the train-test fidelity gap. 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This result highlights a fundamental trade-off in knowledge distillation: the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "student needs many teacher labels match the teacher on test, but introducing examples not in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 438, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 438, + 723 + ], + "score": 1.0, + "content": "teacher train data makes matching the teacher on the distillation data very difficult.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 666, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 70, + 498, + 149 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 70, + 498, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 70, + 498, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 498, + 149 + ], + "score": 0.967, + "type": "image", + "image_path": "1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 70, + 498, + 96.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 96.33333333333333, + 498, + 122.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 122.66666666666666, + 498, + 149.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 506, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 235, + 191 + ], + "score": 1.0, + "content": "agreement, but it remains below", + "type": "text" + }, + { + "bbox": [ + 235, + 178, + 255, + 189 + ], + "score": 0.89, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 177, + 296, + 191 + ], + "score": 1.0, + "content": "even after", + "type": "text" + }, + { + "bbox": [ + 297, + 178, + 308, + 188 + ], + "score": 0.84, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "epochs. (b): Final train loss and agreement when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 396, + 201 + ], + "score": 1.0, + "content": "the initialization is a convex combination of teacher and random weights,", + "type": "text" + }, + { + "bbox": [ + 397, + 189, + 486, + 201 + ], + "score": 0.92, + "content": "\\theta _ { s } = \\lambda \\theta _ { t } + \\mathbf { \\bar { ( } 1 - } \\lambda ) \\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 189, + 506, + 201 + ], + "score": 1.0, + "content": ". (c):", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 381, + 212 + ], + "score": 1.0, + "content": "Projections of the distillation loss surface on the plane intersecting", + "type": "text" + }, + { + "bbox": [ + 382, + 200, + 391, + 210 + ], + "score": 0.86, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 200, + 506, + 212 + ], + "score": 1.0, + "content": ", the initial student weights,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 272, + 223 + ], + "score": 1.0, + "content": "and the final student weights for different", + "type": "text" + }, + { + "bbox": [ + 273, + 211, + 280, + 221 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 211, + 310, + 223 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 310, + 211, + 317, + 221 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "is small, the student converges to a suboptimal", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 221, + 502, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 379, + 235 + ], + "score": 1.0, + "content": "solution with low agreement. The uncertainty regions correspond to", + "type": "text" + }, + { + "bbox": [ + 380, + 222, + 405, + 233 + ], + "score": 0.9, + "content": "\\mu \\pm \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 221, + 502, + 235 + ], + "score": 1.0, + "content": ", estimated over 3 trials.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 263, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 265, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 265, + 271 + ], + "score": 1.0, + "content": "6.2 Why is train agreement so low?", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 229, + 325 + ], + "score": 1.0, + "content": "replace the BatchNorm layers", + "type": "text" + }, + { + "bbox": [ + 230, + 312, + 248, + 324 + ], + "score": 0.78, + "content": "\\mathbb { \\left. \\overline { { 2 5 } } \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 313, + 375, + 325 + ], + "score": 1.0, + "content": "in ResNet-20 with LayerNorm", + "type": "text" + }, + { + "bbox": [ + 375, + 312, + 388, + 324 + ], + "score": 0.67, + "content": "\\pmb { \\left[ \\sqrt { 3 } \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ", because we found that with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "BatchNorm layers even when the teacher and the student have identical weights, they can make", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "different predictions due to differences in the activation statistics accumulated by the BatchNorm", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "layers. Layer normalization does not collect any activation statistics, so the student will match the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 357, + 266, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 266, + 368 + ], + "score": 1.0, + "content": "teacher as long as the weights coincide.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "Can we solve the optimization problem better? We verify that the distillation fidelity cannot be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "significantly improved by training longer or with a different optimizer. By default, in our experiments", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 194, + 417 + ], + "score": 1.0, + "content": "weight decay value of", + "type": "text" + }, + { + "bbox": [ + 195, + 405, + 216, + 416 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 404, + 258, + 417 + ], + "score": 1.0, + "content": ". In Figure", + "type": "text" + }, + { + "bbox": [ + 258, + 405, + 267, + 418 + ], + "score": 0.58, + "content": "\\boxed { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 404, + 442, + 417 + ], + "score": 1.0, + "content": "we report the results for the SGD and Adam", + "type": "text" + }, + { + "bbox": [ + 443, + 405, + 460, + 416 + ], + "score": 0.81, + "content": "\\mathbb { \\left[ \\left[ 2 7 \\right] \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "optimizers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 136, + 429 + ], + "score": 1.0, + "content": "run for", + "type": "text" + }, + { + "bbox": [ + 136, + 416, + 148, + 426 + ], + "score": 0.82, + "content": "1 k", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 416, + 165, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 416, + 178, + 426 + ], + "score": 0.85, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "epochs without weight decay. Switching from SGD to Adam only reduced fidelity.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 446 + ], + "score": 1.0, + "content": "For both optimizers, training for more epochs does slightly improve train agreement. In particular,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 199, + 456 + ], + "score": 1.0, + "content": "with SGD we achieve", + "type": "text" + }, + { + "bbox": [ + 199, + 443, + 227, + 454 + ], + "score": 0.88, + "content": "8 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 442, + 348, + 456 + ], + "score": 1.0, + "content": "agreement when training for", + "type": "text" + }, + { + "bbox": [ + 348, + 444, + 360, + 453 + ], + "score": 0.77, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 442, + 446, + 456 + ], + "score": 1.0, + "content": "epochs compared to", + "type": "text" + }, + { + "bbox": [ + 447, + 443, + 479, + 454 + ], + "score": 0.88, + "content": "7 8 . 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 201, + 478 + ], + "score": 1.0, + "content": "agreement could reach", + "type": "text" + }, + { + "bbox": [ + 201, + 465, + 225, + 476 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 465, + 319, + 478 + ], + "score": 1.0, + "content": ". However, training for", + "type": "text" + }, + { + "bbox": [ + 319, + 466, + 331, + 475 + ], + "score": 0.8, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "epochs is significantly longer than what is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 439, + 488 + ], + "score": 1.0, + "content": "typically done in practice (100 to 500 epochs). Furthermore, the improvement from", + "type": "text" + }, + { + "bbox": [ + 439, + 476, + 451, + 486 + ], + "score": 0.82, + "content": "1 k", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 476, + 462, + 488 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 462, + 476, + 474, + 486 + ], + "score": 0.8, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "epochs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 159, + 500 + ], + "score": 1.0, + "content": "is only about", + "type": "text" + }, + { + "bbox": [ + 160, + 487, + 174, + 498 + ], + "score": 0.88, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 487, + 505, + 500 + ], + "score": 1.0, + "content": ", suggesting that we would need to train for tens of thousands of epochs, even in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 489, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 396, + 511 + ], + "score": 1.0, + "content": "optimistic case that agreement improves linearly, in order to get close to", + "type": "text" + }, + { + "bbox": [ + 396, + 498, + 420, + 509 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 497, + 489, + 511 + ], + "score": 1.0, + "content": "train agreement.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "with any of the interventions we have discussed so far, we now ask if there is any modification of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 536, + 303, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 303, + 548 + ], + "score": 1.0, + "content": "problem that can produce a high-fidelity student.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 293, + 576 + ], + "score": 1.0, + "content": "distillation loss — the teacher’s own weights", + "type": "text" + }, + { + "bbox": [ + 294, + 564, + 303, + 574 + ], + "score": 0.86, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 562, + 341, + 576 + ], + "score": 1.0, + "content": ". Letting", + "type": "text" + }, + { + "bbox": [ + 341, + 564, + 352, + 574 + ], + "score": 0.87, + "content": "\\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "be a random weight initialization, in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 133, + 587 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 134, + 574, + 143, + 587 + ], + "score": 0.66, + "content": "\\boxed { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "(a) we examine the effect of choosing the student initialization to be a convex combination", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 252, + 598 + ], + "score": 1.0, + "content": "of the teacher and random weights,", + "type": "text" + }, + { + "bbox": [ + 253, + 585, + 343, + 597 + ], + "score": 0.91, + "content": "\\theta _ { s } = \\lambda \\bar { \\theta _ { t } } + ( 1 - \\lambda ) \\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 584, + 505, + 598 + ], + "score": 1.0, + "content": ". After being initialized in this way, the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 285, + 608 + ], + "score": 1.0, + "content": "student was trained as before. In other words", + "type": "text" + }, + { + "bbox": [ + 285, + 597, + 311, + 606 + ], + "score": 0.89, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 596, + 479, + 608 + ], + "score": 1.0, + "content": "corresponds to a random initialization and", + "type": "text" + }, + { + "bbox": [ + 479, + 596, + 505, + 606 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 403, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 403, + 619 + ], + "score": 1.0, + "content": "corresponds to initializing the student weights at the final teacher weights.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 345, + 636 + ], + "score": 1.0, + "content": "We find that if the student is initialized far from the teacher", + "type": "text" + }, + { + "bbox": [ + 346, + 624, + 386, + 635 + ], + "score": 0.86, + "content": "\\lambda \\leq 0 . 2 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 623, + 505, + 636 + ], + "score": 1.0, + "content": ", the optimizer converges to a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "sub-optimal value of the distillation loss, producing a student that significantly disagrees with the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 190, + 658 + ], + "score": 1.0, + "content": "teacher. However at", + "type": "text" + }, + { + "bbox": [ + 191, + 645, + 235, + 655 + ], + "score": 0.89, + "content": "\\lambda = 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "there is a sudden change. The final train loss drops to the optimal", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 415, + 668 + ], + "score": 1.0, + "content": "value and the agreement drastically increases, and the behavior continues for", + "type": "text" + }, + { + "bbox": [ + 416, + 656, + 459, + 666 + ], + "score": 0.89, + "content": "\\lambda > 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 655, + 506, + 668 + ], + "score": 1.0, + "content": ". 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If the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 268, + 702 + ], + "score": 1.0, + "content": "student is initialized far from the teacher", + "type": "text" + }, + { + "bbox": [ + 269, + 689, + 328, + 701 + ], + "score": 0.89, + "content": "( \\lambda \\in \\{ 0 , 0 . 2 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 688, + 505, + 702 + ], + "score": 1.0, + "content": ", it converges to a distinct, sub-optimal basin", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 408, + 712 + ], + "score": 1.0, + "content": "of the loss surface. On the other hand, when initialized close to the teacher", + "type": "text" + }, + { + "bbox": [ + 409, + 700, + 453, + 711 + ], + "score": 0.86, + "content": "\\lambda = 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "), the student", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 419, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 347, + 723 + ], + "score": 1.0, + "content": "converges to the same basin as the teacher, achieving nearly", + "type": "text" + }, + { + "bbox": [ + 348, + 711, + 372, + 721 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 711, + 419, + 723 + ], + "score": 1.0, + "content": "agreement.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 70, + 498, + 149 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 70, + 498, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 70, + 498, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 498, + 149 + ], + "score": 0.967, + "type": "image", + "image_path": "1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 70, + 498, + 96.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 96.33333333333333, + 498, + 122.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 122.66666666666666, + 498, + 149.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 506, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 235, + 191 + ], + "score": 1.0, + "content": "agreement, but it remains below", + "type": "text" + }, + { + "bbox": [ + 235, + 178, + 255, + 189 + ], + "score": 0.89, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 177, + 296, + 191 + ], + "score": 1.0, + "content": "even after", + "type": "text" + }, + { + "bbox": [ + 297, + 178, + 308, + 188 + ], + "score": 0.84, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "epochs. (b): Final train loss and agreement when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 396, + 201 + ], + "score": 1.0, + "content": "the initialization is a convex combination of teacher and random weights,", + "type": "text" + }, + { + "bbox": [ + 397, + 189, + 486, + 201 + ], + "score": 0.92, + "content": "\\theta _ { s } = \\lambda \\theta _ { t } + \\mathbf { \\bar { ( } 1 - } \\lambda ) \\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 189, + 506, + 201 + ], + "score": 1.0, + "content": ". (c):", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 381, + 212 + ], + "score": 1.0, + "content": "Projections of the distillation loss surface on the plane intersecting", + "type": "text" + }, + { + "bbox": [ + 382, + 200, + 391, + 210 + ], + "score": 0.86, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 200, + 506, + 212 + ], + "score": 1.0, + "content": ", the initial student weights,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 272, + 223 + ], + "score": 1.0, + "content": "and the final student weights for different", + "type": "text" + }, + { + "bbox": [ + 273, + 211, + 280, + 221 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 211, + 310, + 223 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 310, + 211, + 317, + 221 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "is small, the student converges to a suboptimal", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 221, + 502, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 379, + 235 + ], + "score": 1.0, + "content": "solution with low agreement. The uncertainty regions correspond to", + "type": "text" + }, + { + "bbox": [ + 380, + 222, + 405, + 233 + ], + "score": 0.9, + "content": "\\mu \\pm \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 221, + 502, + 235 + ], + "score": 1.0, + "content": ", estimated over 3 trials.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 263, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 265, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 265, + 271 + ], + "score": 1.0, + "content": "6.2 Why is train agreement so low?", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 229, + 325 + ], + "score": 1.0, + "content": "replace the BatchNorm layers", + "type": "text" + }, + { + "bbox": [ + 230, + 312, + 248, + 324 + ], + "score": 0.78, + "content": "\\mathbb { \\left. \\overline { { 2 5 } } \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 313, + 375, + 325 + ], + "score": 1.0, + "content": "in ResNet-20 with LayerNorm", + "type": "text" + }, + { + "bbox": [ + 375, + 312, + 388, + 324 + ], + "score": 0.67, + "content": "\\pmb { \\left[ \\sqrt { 3 } \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ", because we found that with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "BatchNorm layers even when the teacher and the student have identical weights, they can make", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "different predictions due to differences in the activation statistics accumulated by the BatchNorm", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "layers. Layer normalization does not collect any activation statistics, so the student will match the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 357, + 266, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 266, + 368 + ], + "score": 1.0, + "content": "teacher as long as the weights coincide.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 280, + 506, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "Can we solve the optimization problem better? We verify that the distillation fidelity cannot be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "significantly improved by training longer or with a different optimizer. By default, in our experiments", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 194, + 417 + ], + "score": 1.0, + "content": "weight decay value of", + "type": "text" + }, + { + "bbox": [ + 195, + 405, + 216, + 416 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 404, + 258, + 417 + ], + "score": 1.0, + "content": ". In Figure", + "type": "text" + }, + { + "bbox": [ + 258, + 405, + 267, + 418 + ], + "score": 0.58, + "content": "\\boxed { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 404, + 442, + 417 + ], + "score": 1.0, + "content": "we report the results for the SGD and Adam", + "type": "text" + }, + { + "bbox": [ + 443, + 405, + 460, + 416 + ], + "score": 0.81, + "content": "\\mathbb { \\left[ \\left[ 2 7 \\right] \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "optimizers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 136, + 429 + ], + "score": 1.0, + "content": "run for", + "type": "text" + }, + { + "bbox": [ + 136, + 416, + 148, + 426 + ], + "score": 0.82, + "content": "1 k", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 416, + 165, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 416, + 178, + 426 + ], + "score": 0.85, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "epochs without weight decay. Switching from SGD to Adam only reduced fidelity.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 372, + 506, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 446 + ], + "score": 1.0, + "content": "For both optimizers, training for more epochs does slightly improve train agreement. In particular,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 199, + 456 + ], + "score": 1.0, + "content": "with SGD we achieve", + "type": "text" + }, + { + "bbox": [ + 199, + 443, + 227, + 454 + ], + "score": 0.88, + "content": "8 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 442, + 348, + 456 + ], + "score": 1.0, + "content": "agreement when training for", + "type": "text" + }, + { + "bbox": [ + 348, + 444, + 360, + 453 + ], + "score": 0.77, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 442, + 446, + 456 + ], + "score": 1.0, + "content": "epochs compared to", + "type": "text" + }, + { + "bbox": [ + 447, + 443, + 479, + 454 + ], + "score": 0.88, + "content": "7 8 . 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 201, + 478 + ], + "score": 1.0, + "content": "agreement could reach", + "type": "text" + }, + { + "bbox": [ + 201, + 465, + 225, + 476 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 465, + 319, + 478 + ], + "score": 1.0, + "content": ". However, training for", + "type": "text" + }, + { + "bbox": [ + 319, + 466, + 331, + 475 + ], + "score": 0.8, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "epochs is significantly longer than what is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 439, + 488 + ], + "score": 1.0, + "content": "typically done in practice (100 to 500 epochs). Furthermore, the improvement from", + "type": "text" + }, + { + "bbox": [ + 439, + 476, + 451, + 486 + ], + "score": 0.82, + "content": "1 k", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 476, + 462, + 488 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 462, + 476, + 474, + 486 + ], + "score": 0.8, + "content": "5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "epochs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 159, + 500 + ], + "score": 1.0, + "content": "is only about", + "type": "text" + }, + { + "bbox": [ + 160, + 487, + 174, + 498 + ], + "score": 0.88, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 487, + 505, + 500 + ], + "score": 1.0, + "content": ", suggesting that we would need to train for tens of thousands of epochs, even in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 489, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 396, + 511 + ], + "score": 1.0, + "content": "optimistic case that agreement improves linearly, in order to get close to", + "type": "text" + }, + { + "bbox": [ + 396, + 498, + 420, + 509 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 497, + 489, + 511 + ], + "score": 1.0, + "content": "train agreement.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 432, + 507, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "with any of the interventions we have discussed so far, we now ask if there is any modification of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 536, + 303, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 303, + 548 + ], + "score": 1.0, + "content": "problem that can produce a high-fidelity student.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 514, + 505, + 548 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 293, + 576 + ], + "score": 1.0, + "content": "distillation loss — the teacher’s own weights", + "type": "text" + }, + { + "bbox": [ + 294, + 564, + 303, + 574 + ], + "score": 0.86, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 562, + 341, + 576 + ], + "score": 1.0, + "content": ". Letting", + "type": "text" + }, + { + "bbox": [ + 341, + 564, + 352, + 574 + ], + "score": 0.87, + "content": "\\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "be a random weight initialization, in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 133, + 587 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 134, + 574, + 143, + 587 + ], + "score": 0.66, + "content": "\\boxed { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "(a) we examine the effect of choosing the student initialization to be a convex combination", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 252, + 598 + ], + "score": 1.0, + "content": "of the teacher and random weights,", + "type": "text" + }, + { + "bbox": [ + 253, + 585, + 343, + 597 + ], + "score": 0.91, + "content": "\\theta _ { s } = \\lambda \\bar { \\theta _ { t } } + ( 1 - \\lambda ) \\theta _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 584, + 505, + 598 + ], + "score": 1.0, + "content": ". After being initialized in this way, the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 285, + 608 + ], + "score": 1.0, + "content": "student was trained as before. In other words", + "type": "text" + }, + { + "bbox": [ + 285, + 597, + 311, + 606 + ], + "score": 0.89, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 596, + 479, + 608 + ], + "score": 1.0, + "content": "corresponds to a random initialization and", + "type": "text" + }, + { + "bbox": [ + 479, + 596, + 505, + 606 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 403, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 403, + 619 + ], + "score": 1.0, + "content": "corresponds to initializing the student weights at the final teacher weights.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 553, + 505, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 345, + 636 + ], + "score": 1.0, + "content": "We find that if the student is initialized far from the teacher", + "type": "text" + }, + { + "bbox": [ + 346, + 624, + 386, + 635 + ], + "score": 0.86, + "content": "\\lambda \\leq 0 . 2 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 623, + 505, + 636 + ], + "score": 1.0, + "content": ", the optimizer converges to a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "sub-optimal value of the distillation loss, producing a student that significantly disagrees with the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 190, + 658 + ], + "score": 1.0, + "content": "teacher. However at", + "type": "text" + }, + { + "bbox": [ + 191, + 645, + 235, + 655 + ], + "score": 0.89, + "content": "\\lambda = 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "there is a sudden change. The final train loss drops to the optimal", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 415, + 668 + ], + "score": 1.0, + "content": "value and the agreement drastically increases, and the behavior continues for", + "type": "text" + }, + { + "bbox": [ + 416, + 656, + 459, + 666 + ], + "score": 0.89, + "content": "\\lambda > 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 655, + 506, + 668 + ], + "score": 1.0, + "content": ". To further", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 507, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 189, + 681 + ], + "score": 1.0, + "content": "investigate, in Figure", + "type": "text" + }, + { + "bbox": [ + 190, + 666, + 210, + 680 + ], + "score": 0.47, + "content": "6 ( \\mathrm { c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 665, + 381, + 681 + ], + "score": 1.0, + "content": "we visualize the distillation loss surface for", + "type": "text" + }, + { + "bbox": [ + 382, + 667, + 465, + 679 + ], + "score": 0.91, + "content": "\\lambda \\in \\{ 0 , 0 . 2 5 , 0 . 3 7 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 665, + 507, + 681 + ], + "score": 1.0, + "content": "projected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 237, + 691 + ], + "score": 1.0, + "content": "on the 2D subspace intersecting", + "type": "text" + }, + { + "bbox": [ + 237, + 678, + 246, + 689 + ], + "score": 0.86, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 677, + 505, + 691 + ], + "score": 1.0, + "content": ", the initial student weights, and the final student weights. If the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 268, + 702 + ], + "score": 1.0, + "content": "student is initialized far from the teacher", + "type": "text" + }, + { + "bbox": [ + 269, + 689, + 328, + 701 + ], + "score": 0.89, + "content": "( \\lambda \\in \\{ 0 , 0 . 2 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 688, + 505, + 702 + ], + "score": 1.0, + "content": ", it converges to a distinct, sub-optimal basin", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 408, + 712 + ], + "score": 1.0, + "content": "of the loss surface. On the other hand, when initialized close to the teacher", + "type": "text" + }, + { + "bbox": [ + 409, + 700, + 453, + 711 + ], + "score": 0.86, + "content": "\\lambda = 0 . 3 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "), the student", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 419, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 347, + 723 + ], + "score": 1.0, + "content": "converges to the same basin as the teacher, achieving nearly", + "type": "text" + }, + { + "bbox": [ + 348, + 711, + 372, + 721 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 711, + 419, + 723 + ], + "score": 1.0, + "content": "agreement.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 623, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 116, + 70, + 494, + 133 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 116, + 70, + 494, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 70, + 494, + 133 + ], + "spans": [ + { + "bbox": [ + 116, + 70, + 494, + 133 + ], + "score": 0.979, + "html": "
CKA (1)
Init.Agree. (↑)KL (↓）Stage 1 Stage 2Stage 3
Rand.77.174 (0.352)0.836 (0.016)0.939 (0.017)0.925 (0.027)0.885 (0.011)
Teach.77.098 (0.238)0.838 (0.020)0.951 (0.017)0.937 (0.020)0.890 (0.015)
", + "type": "table", + "image_path": "6261113ac0608e905470acaddd56ab5ee54a3afd4229b650ea76339020420cdb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 70, + 494, + 91.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 91.0, + 494, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 112.0, + 494, + 133.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 137, + 505, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "Table 1: We examine whether fidelity can be improved in the context of ResNet-20 self-distillation on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "CIFAR-100 if the teacher and student share the same weight initialization. All metrics are computed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "on the test set. A shared initialization does make the student slightly more similar to the teacher in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "activation space (measured by CKA), but in function space the results are indistinguishable from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 507, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 507, + 193 + ], + "score": 1.0, + "content": "randomly initialized students. We report the mean and standard deviation, estimated from 10 trials.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 308, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 308, + 204 + ], + "score": 1.0, + "content": "The average teacher accuracy was 70.522 (0.412).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "by initializing the student near the final teacher weights, it is possible that similar results could be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 407, + 259 + ], + "score": 1.0, + "content": "obtained by initializing the student at the initial teacher weights. In Table", + "type": "text" + }, + { + "bbox": [ + 408, + 246, + 417, + 259 + ], + "score": 0.34, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 247, + 504, + 259 + ], + "score": 1.0, + "content": "we compare students", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "distilled from random initializations with those initialized at the initial teacher weights. In addition to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 124, + 291 + ], + "score": 0.65, + "content": "\\left[ \\left[ 2 8 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "of the preactivations of each of the teacher and student networks. There is a small increase in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "CKA, indicating that sharing an initialization between teacher and student does increase alignment in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 313 + ], + "score": 1.0, + "content": "activation space, but functionally the students are identical to their randomly initialized counterparts –", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "there is no observable change in accuracy, agreement, or predictive KL when compared to random", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 162, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 162, + 335 + ], + "score": 1.0, + "content": "initialization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "To summarize, we have at last identified a root cause of the ineffectiveness of all our previous", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "to optimal student parameters, even when we know a solution and give the initialization a small head", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "the teacher on all inputs, the student has to at least match the teacher on the data used for distillation,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 205, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 205, + 440 + ], + "score": 1.0, + "content": "poor distillation fidelity.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 179, + 467 + ], + "lines": [ + { + "bbox": [ + 104, + 452, + 181, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 181, + 470 + ], + "score": 1.0, + "content": "7 Discussion", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 402, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 402, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 402, + 492 + ], + "score": 1.0, + "content": "Our work provides several new key findings about knowledge distillation:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 133, + 499, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 133, + 498, + 507, + 512 + ], + "spans": [ + { + "bbox": [ + 133, + 498, + 507, + 512 + ], + "score": 1.0, + "content": "• Good student accuracy does not imply good distillation fidelity: even outside of self-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 509, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 141, + 509, + 504, + 523 + ], + "score": 1.0, + "content": "distillation, the models with the best generalization do not always achieve the best fidelity.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 133, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "• Student fidelity is correlated with calibration when distilling ensembles: although the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 536, + 489, + 547 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 489, + 547 + ], + "score": 1.0, + "content": "highest-fidelity student is not always the most accurate, it is always the best calibrated.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "• Optimization is challenging in knowledge distillation: even in cases when the student has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 562, + 476, + 573 + ], + "spans": [ + { + "bbox": [ + 141, + 562, + 476, + 573 + ], + "score": 1.0, + "content": "sufficient capacity to match the teacher on the distillation data, it is unable to do so.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 132, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 132, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "• There is a trade-off between optimization complexity and distillation data quality: Enlarging", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 142, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "the distillation dataset beyond the teacher training data makes it easier for the student to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 599, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 142, + 599, + 506, + 610 + ], + "score": 1.0, + "content": "identify the correct solution, but also makes an already difficult optimization problem harder.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "In standard deep learning, we are saved by not needing to solve the optimization problem well: while", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "it true that our training loss is highly multimodal, properties such as the flatness of good solutions,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "the inductive biases of the network, and the implicit biases of SGD, often enable good generalization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "in practice. 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CKA (1)
Init.Agree. (↑)KL (↓）Stage 1 Stage 2Stage 3
Rand.77.174 (0.352)0.836 (0.016)0.939 (0.017)0.925 (0.027)0.885 (0.011)
Teach.77.098 (0.238)0.838 (0.020)0.951 (0.017)0.937 (0.020)0.890 (0.015)
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All metrics are computed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "on the test set. A shared initialization does make the student slightly more similar to the teacher in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "activation space (measured by CKA), but in function space the results are indistinguishable from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 507, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 507, + 193 + ], + "score": 1.0, + "content": "randomly initialized students. We report the mean and standard deviation, estimated from 10 trials.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 308, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 308, + 204 + ], + "score": 1.0, + "content": "The average teacher accuracy was 70.522 (0.412).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "by initializing the student near the final teacher weights, it is possible that similar results could be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 407, + 259 + ], + "score": 1.0, + "content": "obtained by initializing the student at the initial teacher weights. In Table", + "type": "text" + }, + { + "bbox": [ + 408, + 246, + 417, + 259 + ], + "score": 0.34, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 247, + 504, + 259 + ], + "score": 1.0, + "content": "we compare students", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "distilled from random initializations with those initialized at the initial teacher weights. In addition to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 124, + 291 + ], + "score": 0.65, + "content": "\\left[ \\left[ 2 8 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "of the preactivations of each of the teacher and student networks. There is a small increase in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "CKA, indicating that sharing an initialization between teacher and student does increase alignment in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 313 + ], + "score": 1.0, + "content": "activation space, but functionally the students are identical to their randomly initialized counterparts –", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "there is no observable change in accuracy, agreement, or predictive KL when compared to random", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 162, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 162, + 335 + ], + "score": 1.0, + "content": "initialization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 224, + 506, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "To summarize, we have at last identified a root cause of the ineffectiveness of all our previous", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "to optimal student parameters, even when we know a solution and give the initialization a small head", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "the teacher on all inputs, the student has to at least match the teacher on the data used for distillation,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 205, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 205, + 440 + ], + "score": 1.0, + "content": "poor distillation fidelity.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 340, + 506, + 440 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 179, + 467 + ], + "lines": [ + { + "bbox": [ + 104, + 452, + 181, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 181, + 470 + ], + "score": 1.0, + "content": "7 Discussion", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 402, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 402, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 402, + 492 + ], + "score": 1.0, + "content": "Our work provides several new key findings about knowledge distillation:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 477, + 402, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 499, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 133, + 498, + 507, + 512 + ], + "spans": [ + { + "bbox": [ + 133, + 498, + 507, + 512 + ], + "score": 1.0, + "content": "• Good student accuracy does not imply good distillation fidelity: even outside of self-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 509, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 141, + 509, + 504, + 523 + ], + "score": 1.0, + "content": "distillation, the models with the best generalization do not always achieve the best fidelity.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 133, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "• Student fidelity is correlated with calibration when distilling ensembles: although the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 536, + 489, + 547 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 489, + 547 + ], + "score": 1.0, + "content": "highest-fidelity student is not always the most accurate, it is always the best calibrated.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "• Optimization is challenging in knowledge distillation: even in cases when the student has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 562, + 476, + 573 + ], + "spans": [ + { + "bbox": [ + 141, + 562, + 476, + 573 + ], + "score": 1.0, + "content": "sufficient capacity to match the teacher on the distillation data, it is unable to do so.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 132, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 132, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "• There is a trade-off between optimization complexity and distillation data quality: Enlarging", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 142, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "the distillation dataset beyond the teacher training data makes it easier for the student to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 599, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 142, + 599, + 506, + 610 + ], + "score": 1.0, + "content": "identify the correct solution, but also makes an already difficult optimization problem harder.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 132, + 498, + 507, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "In standard deep learning, we are saved by not needing to solve the optimization problem well: while", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "it true that our training loss is highly multimodal, properties such as the flatness of good solutions,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "the inductive biases of the network, and the implicit biases of SGD, often enable good generalization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "in practice. In knowledge distillation, however, good fidelity is directly aligned with solving what", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 662, + 358, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 358, + 675 + ], + "score": 1.0, + "content": "turns out to be an exceptionally difficult optimization problem.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 618, + 506, + 675 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 207, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 69, + 208, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 69, + 208, + 87 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "score": 1.0, + "content": "The authors would like to thank Gregory Benton, Marc Finzi, Sanae Lotfi, Nate Gruver, and Ben", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "Poole for helpful feedback. This research is supported by an Amazon Research Award, NSF I-DISRE", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "193471, NIH R01DA048764-01A1, NSF IIS-1910266, and NSF 1922658NRT-HDR: FUTURE", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 142 + ], + "score": 1.0, + "content": "Foundations, Translation, and Responsibility for Data Science. 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CKA (1)
Init.Agree. (↑)KL (↓）Stage 1 Stage 2Stage 3
Rand.77.174 (0.352)0.836 (0.016)0.939 (0.017)0.925 (0.027)0.885 (0.011)
Teach.77.098 (0.238)0.838 (0.020)0.951 (0.017)0.937 (0.020)0.890 (0.015)
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--- /dev/null +++ b/parse/train/WEHSlH5mOk/images/f94e9505467e5df8f0016b075d3bf021c28c9651c480cede8a9e169a2c0d3ddf.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5357da7b72ef071b8b5910d4b5dae5eb6d3bd2c84a45c1eb3fdc86861dd2aba8 +size 41018 diff --git a/parse/train/ZUvaSolQZh3/ZUvaSolQZh3.md b/parse/train/ZUvaSolQZh3/ZUvaSolQZh3.md new file mode 100644 index 0000000000000000000000000000000000000000..4badbd3487e81c722c4e3ff7c71e3f99323f7dcc --- /dev/null +++ b/parse/train/ZUvaSolQZh3/ZUvaSolQZh3.md @@ -0,0 +1,253 @@ +# Uncertainty-Based Offline Reinforcement Learning with Diversified Q-Ensemble + +Gaon $\mathbf { A } \mathbf { n } ^ { * 1 2 }$ , Seungyong Moon\*1 2, Jang-Hyun $\mathbf { K i m ^ { 1 2 } }$ , Hyun Oh Song† 1 2 3 + +Seoul National University1 Neural Processing Research Center2 DeepMetrics3 {white0234,symoon11,janghyun,hyunoh}@mllab.snu.ac.kr + +# Abstract + +Offline reinforcement learning (offline RL), which aims to find an optimal policy from a previously collected static dataset, bears algorithmic difficulties due to function approximation errors from out-of-distribution (OOD) data points. To this end, offline RL algorithms adopt either a constraint or a penalty term that explicitly guides the policy to stay close to the given dataset. However, prior methods typically require accurate estimation of the behavior policy or sampling from OOD data points, which themselves can be a non-trivial problem. Moreover, these methods under-utilize the generalization ability of deep neural networks and often fall into suboptimal solutions too close to the given dataset. In this work, we propose an uncertainty-based offline RL method that takes into account the confidence of the Q-value prediction and does not require any estimation or sampling of the data distribution. We show that the clipped Q-learning, a technique widely used in online RL, can be leveraged to successfully penalize OOD data points with high prediction uncertainties. Surprisingly, we find that it is possible to substantially outperform existing offline RL methods on various tasks by simply increasing the number of Q-networks along with the clipped Q-learning. Based on this observation, we propose an ensemble-diversified actor-critic algorithm that reduces the number of required ensemble networks down to a tenth compared to the naive ensemble while achieving state-of-the-art performance on most of the D4RL benchmarks considered. + +# 1 Introduction + +Over the recent years, deep reinforcement learning (deep RL) has achieved considerable success in various domains such as robotics [20], recommendation systems [6], and strategy games [26]. However, a major drawback of RL algorithms is that they adopt an active learning procedure, where training steps require active interactions with the environment. This trial-and-error procedure can be prohibitive when scaling RL to real-world applications such as autonomous driving and healthcare, as exploratory actions can cause critical damage to the agent or the environment [19]. Offline RL, also known as batch RL, aims to overcome this problem by learning policies using only previously collected data without further interactions with the environment [2, 11, 19]. + +Even though offline RL is a promising direction to lead a more data-driven way of solving RL problems, recent works show offline RL faces new algorithmic challenges [19]. Typically, if the coverage of the dataset is not sufficient, vanilla RL algorithms suffer severely from extrapolation error, overestimating the Q-values of out-of-distribution (OOD) state-action pairs [15]. To this end, most offline RL methods apply some constraints or penalty terms on top of the existing RL algorithms to enforce the learning process to be more conservative. For example, some prior works explicitly regularize the policy to be close to the behavior policy that was used to collect the data [11, 15]. A more recent work instead penalizes the Q-values of OOD state-action pairs to enforce the Q-values to be more pessimistic [16]. + +While these methods achieve significant performance gains over vanilla RL methods, they either require an estimation of the behavior policy or explicit sampling from OOD data points, which themselves can be non-trivial to solve. Furthermore, these methods do not utilize the generalization ability of the Q-function networks and prohibit the agent from approaching any OOD state-actions without any consideration on whether they are good or bad. However, if we can identify OOD data points where we can predict their Q-values with high confidence, it is more effective not to restrain the agent from choosing those data points. + +From this intuition, we propose an uncertainty-based model-free offline RL method that effectively quantifies the uncertainty of the Q-value estimates by an ensemble of Q-function networks and does not require any estimation or sampling of the data distribution. To achieve this, we first show that a well-known technique from online RL, the clipped Q-learning [10], can be successfully leveraged as an uncertainty-based penalization term. Our experiments reveal that we can achieve state-of-the-art performance on various offline RL tasks by solely using this technique with increased ensemble size. To further improve the practical usability of the method, we develop an ensemble diversifying objective that significantly reduces the number of required ensemble networks. We evaluate our proposed method on D4RL benchmarks [9] and verify that the proposed method outperforms the previous state-of-the-art by a large margin on various types of environments and datasets. + +# 2 Preliminaries + +We consider an environment formulated as a Markov Decision Process (MDP) defined by a tuple $( S , A , T , r , d _ { 0 } , \gamma )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $T ( \mathbf { s } ^ { \prime } \mid \mathbf { s } , \mathbf { a } )$ is the transition probability distribution, $r : S \times \mathcal { A } \mathbb { R }$ is the reward function, $d _ { 0 }$ is the initial state distribution, and $\gamma \in \mathsf { \Gamma } ( 0 , 1 ]$ is the discount factor. The goal of reinforcement learning is to find an optimal policy $\pi ( \mathbf { a } \mid \mathbf { s } )$ that maximizes the cumulative discounted reward $\begin{array} { r } { \mathbb { E } _ { { \mathbf { s } } _ { t } , { \mathbf { a } } _ { t } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( { \mathbf { s } } _ { t } , { \mathbf { a } } _ { t } ) \right] } \end{array}$ , where $\mathbf { s } _ { 0 } \sim d _ { 0 } ( \cdot )$ , $\mathbf { a } _ { t } \sim \pi ( \cdot \mid \mathbf { s } _ { t } )$ , and $\mathbf { s } _ { t + 1 } \sim T ( \cdot \mid \mathbf { s } _ { t } , \mathbf { a } _ { t } )$ . + +One of the major approaches for obtaining such a policy is Q-learning [12, 20] which learns a state-action value function $Q _ { \phi } ( \mathbf { s } , \mathbf { a } )$ parameterized by a neural network that represents the expected cumulative discounted reward when starting from state s and action a. Standard actor-critic approach [14] learns this Q-function by minimizing the Bellman residual $\big ( Q _ { \phi } ( \mathbf { s } , \mathbf { a } ) - B ^ { \pi _ { \theta } } Q _ { \phi } ( \mathbf { s } , \mathbf { a } ) \big ) ^ { 2 }$ , where $B ^ { \pi _ { \theta } } Q _ { \phi } ( \mathbf { s } , \mathbf { a } ) = \mathbb { E } _ { \mathbf { s } ^ { \prime } \sim T ( \cdot | \mathbf { s } , \mathbf { a } ) } \left[ r ( \mathbf { s } , \mathbf { a } ) + \gamma \mathbb { E } _ { \mathbf { a } ^ { \prime } \sim \pi _ { \theta } ( \cdot | \mathbf { s } ^ { \prime } ) } Q _ { \phi } \left( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } \right) \right]$ is the Bellman operator. In the context of offline RL, where transitions are sampled from a static dataset $\mathcal { D }$ , the objective for the $\mathrm { Q }$ -network becomes minimizing + +$$ +J _ { q } ( Q _ { \phi } ) : = \mathbb { E } _ { ( \mathbf { s } , \mathbf { a } , \mathbf { s } ^ { \prime } ) \sim \mathcal { D } } \left[ \left( Q _ { \phi } ( \mathbf { s } , \mathbf { a } ) - \left( r ( \mathbf { s } , \mathbf { a } ) + \gamma \mathbb { E } _ { \mathbf { a } ^ { \prime } \sim \pi _ { \theta } ( \cdot | \mathbf { s } ^ { \prime } ) } \left[ Q _ { \phi ^ { \prime } } \left( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } \right) \right] \right) \right) ^ { 2 } \right] , +$$ + +where $Q _ { \phi ^ { \prime } }$ represents the target Q-network softly updated for algorithmic stability [20]. The policy, which is also parameterized by a neural network, is updated in an alternating fashion to maximize the expected $\mathrm { Q }$ -value: $J _ { p } ( \pi _ { \theta } ) : = \mathbb { E } _ { { \mathbf s } \sim \mathcal { D } , { \mathbf a } \sim \pi _ { \theta } ( \cdot | { \mathbf s } ) } \left[ Q _ { \phi } ( { \mathbf s } , { \mathbf a } ) \right] ,$ . + +However, as the policy is updated to maximize the Q-values, the actions $\mathbf { a } ^ { \prime }$ sampled from the current policy in Equation (1) can be biased towards OOD actions with erroneously high Q-values. In the offline RL setting, such errors cannot be corrected by feedback from the environment as in online RL. To handle the error propagation from these OOD actions, most offline RL algorithms regularize either the policy [11, 15] or the Q-function [16] to be biased towards the given dataset. However, the policy regularization methods typically require an accurate estimation of the behavior policy. The previous state-of-the-art method CQL [16] instead learns conservative Q-values without estimating the behavior policy by penalizing the Q-values of OOD actions by + +$$ +\operatorname* { m i n } _ { \phi } J _ { q } ( Q _ { \phi } ) + \alpha \Big ( \mathbb { E } _ { { \mathbf { s } } \sim \mathcal { D } , { \mathbf { a } } \sim \mu ( \cdot \vert \mathbf { s } ) } \left[ Q _ { \phi } \left( \mathbf { s } , { \mathbf { a } } \right) \right] - \mathbb { E } _ { ( \mathbf { s } , { \mathbf { a } } ) \sim \mathcal { D } } \left[ Q _ { \phi } \left( \mathbf { s } , { \mathbf { a } } \right) \right] \Big ) , +$$ + +where $\mu$ is an approximation of the policy that maximizes the current Q-function. While CQL does not need explicit behavior policy estimation, it requires sampling from an appropriate action distribution $\mu ( \cdot | \mathbf { \bar { s } } )$ . + +# 3 Uncertainty penalization with Q-ensemble + +![](images/5ef0099c6b2f91502d99a5bd4e4ae37be48755a4861064068d4c78cfb8b708b5.jpg) +Figure 1: Performance of SAC- $N$ on halfcheetah-medium and hopper-medium datasets while varying $N$ , compared to CQL. ‘Average Return’ denotes the undiscounted return of each policies on evaluation. Results averaged over 4 seeds. + +In this section, we turn our attention to a conventional technique from online RL, Clipped Double QLearning [10], which uses the minimum value of two parallel Q-networks as the Bellman target: $y =$ $\begin{array} { r } { r ( \mathbf { s } , \mathbf { a } ) + \gamma \mathbb { E } _ { \mathbf { a } ^ { \prime } \sim \pi _ { \theta } ( \cdot | \mathbf { s } ^ { \prime } ) } \left[ \operatorname* { m i n } _ { j = 1 , 2 } Q _ { \phi _ { j } ^ { \prime } } ( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } ) \right] } \end{array}$ . Although this technique was originally proposed in online RL to mitigate the overestimation from general prediction errors, some offline RL algorithms [11, 15, 28] also utilize this technique to enforce their $\mathrm { Q }$ -value estimates to be more pessimistic. However, the isolated effect of the clipped Q-learning in offline RL was not fully analyzed in the previous works, as they use the technique only as an auxiliary term that adds up to their core methods. + +To examine the ability of clipped Q-learning to prevent the overestimation in offline RL on its own, we modify SAC [12] by increasing the number of $\mathrm { Q }$ -ensembles from 2 to $N$ : + +$$ +\begin{array} { r l } & { \underset { \phi _ { i } } { \mathrm { n i n } } \ : \mathbb { E } _ { \mathbf { s } , \mathbf { a } , \mathbf { s } ^ { \prime } \sim \mathcal { D } } \left[ \left( Q _ { \phi _ { i } } ( \mathbf { s } , \mathbf { a } ) - \left( r ( \mathbf { s } , \mathbf { a } ) + \gamma \mathbb { E } _ { \mathbf { a } ^ { \prime } \sim \pi \theta \cdot \left( \mathbf { \cdot } \mathbf { s } ^ { \prime } \right) } \left[ \underset { j = 1 , \ldots , N } { \operatorname* { m i n } } Q _ { \phi _ { j } ^ { \prime } } \left( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } \right) - \beta \log \pi _ { \theta } \left( \mathbf { a } ^ { \prime } \mid \mathbf { s } ^ { \prime } \right) \right] \right) \right) ^ { 2 } \right] } \\ & { \underset { \theta } { \mathrm { n a x } } \ : \mathbb { E } _ { \mathbf { s } \sim \mathcal { D } , \mathbf { a } \sim \pi _ { \theta } \cdot \left( \mathbf { \cdot } \mathbf { s } \right) } \ : \left[ \underset { j = 1 , \ldots , N } { \operatorname* { m i n } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) - \beta \log \pi _ { \theta } \left( \mathbf { a } \mid \mathbf { s } \right) \right] , } \end{array} +$$ + +for $i = 1 , \ldots , N$ . We denote this modified algorithm as SAC- $N$ . + +Figure 1 shows the preliminary experiments on D4RL halfcheetah-medium and hopper-medium datasets [9] while varying $N$ . Note that these datasets are constructed from suboptimal behavior policies. Surprisingly, as we gradually increase $N$ , we can successfully find policies that outperform the previous state-of-the-art method (CQL) by a large margin. In fact, as we will present in Section 5, SAC- $N$ outperforms CQL on various types of environments and data-collection policies. + +To understand why this simple technique works so well, we can first interpret the clipping procedure (choosing the minimum value from the ensemble) as penalizing state-action pairs with high-variance Q-value estimates, which encourages the policy to favor actions that appeared in the dataset [11]. The dataset samples will naturally have lower variance compared to the OOD samples as the Bellman residual term in Equation (2) explicitly aligns the Q-value predictions for the dataset samples. More formally, we can regard this difference in variance as accounting for epistemic uncertainty [8] which refers to the uncertainty stemming from limited data and knowledge. + +Utilization of the clipped Q-value relates to methods that consider the confidence bound of the Q-value estimates [24]. Online RL methods typically utilize the Q-ensemble to form an optimistic estimate of the Q-value, by adding the standard deviation to the mean of the Q-ensembles [18]. This optimistic Q-value, also known as the upper-confidence bound (UCB), can encourage the exploration of unseen actions with high uncertainty. However, in offline RL, the dataset available during training is fixed, and we have to focus on exploiting the given data. For this purpose, it is natural to utilize the lower-confidence bound (LCB) of the Q-value estimates, for example by subtracting the standard deviation from the mean, which allows us to avoid risky state-actions. + +The clipped Q-learning algorithm, which chooses the worst-case Q-value instead to compute the pessimistic estimate, can also be interpreted as utilizing the LCB of the $\mathrm { Q }$ -value predictions. Suppose $Q ( \mathbf { s } , \mathbf { a } )$ follows a Gaussian distribution with mean $m ( \mathbf { s } , \mathbf { a } )$ and standard deviation $\sigma ( \mathbf { s } , \mathbf { a } )$ . Also, let $\{ Q _ { j } ( \mathbf { s } , \mathbf { \bar { a } } ) \} _ { j = 1 } ^ { N }$ be realizations of $Q ( \mathbf { s } , \mathbf { a } )$ . Then, we can approximate the expected minimum of the realizations following the work of Royston [23] as + +$$ +\mathbb { E } \left[ \operatorname* { m i n } _ { j = 1 , \dots , N } Q _ { j } ( \mathbf { s } , \mathbf { a } ) \right] \approx m ( \mathbf { s } , \mathbf { a } ) - \Phi ^ { - 1 } \left( \frac { N - \frac { \pi } { 8 } } { N - \frac { \pi } { 4 } + 1 } \right) \sigma ( \mathbf { s } , \mathbf { a } ) , +$$ + +where $\Phi$ is the CDF of the standard Gaussian distribution. This relation indicates that using the clipped Q-value is similar to penalizing the ensemble mean of the Q-values with the standard deviation scaled by a coefficient dependent on $N$ . + +![](images/ce40e054b643c8f22be0cd242e5ba33fb71eb6ea668533184970fad707c33dc5.jpg) +Figure 2: (a) and (b) each plots the size of the clip penalty and the standard deviation of the Qvalue estimates for in-distribution (behavior) and OOD (random) actions while training SAC-10 on halfcheetah-medium dataset. (c) plots the gap of the clip penalty between the in-distribution and OOD actions while varying $N$ . Results averaged over 4 seeds. + +We now move on to the empirical analysis of the clipped Q-learning. Figure 2a compares the strength of the uncertainty penalty on in-distribution and OOD actions. Specifically, we compare actions sampled from two types of policies: (1) the behavior policy which was used to collect the dataset, and (2) the random policy which samples actions uniformly from the action space. For each policy, we measure the size of the penalty from the clipping as $\begin{array} { r } { \mathbb { E } _ { { \mathbf s } \sim \mathcal { D } , { \mathbf a } \sim \pi ( \cdot | { \mathbf s } ) } [ \frac { 1 } { N } \sum _ { j = 1 } ^ { N } Q _ { \phi _ { j } } ( { \mathbf s } , { \mathbf a } ) - } \end{array}$ $\begin{array} { r } { \operatorname* { m i n } _ { j = 1 , \dots , N } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) ] } \end{array}$ . Figure 2a shows that the clipping term penalizes the random state-action pairs much stronger than the in-distribution pairs throughout the training. For comparison, we also measure the standard deviation of the $\mathrm { Q }$ -values for each policy. The results in Figure 2b show that as we conjectured, the Q-value predictions for the OOD actions have a higher variance. We also find that the size of the penalty and the standard deviation are highly correlated, as we noted in Equation (3). + +As we observe that OOD actions have higher variance on Q-value estimates, the effect of increasing $N$ becomes obvious: it strengthens the penalty applied to the OOD samples compared to the dataset samples. To verify this, we measured the relative penalty applied to the OOD samples in Figure 2c and found that indeed the OOD samples are penalized relatively further as $N$ increases. + +# 4 Ensemble gradient diversification + +Even though SAC- $N$ outperforms existing methods on various tasks, it sometimes requires an excessively large number of ensembles to learn stably (e.g., $N = 5 0 0$ for hopper-medium). While investigating its reason, we found that the performance of SAC- $N$ is negatively correlated with the degree to which the input gradients of Q-functions $\nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } )$ are aligned, which decreases with $N$ . Figure 4 measures the minimum cosine similarity between the gradients of the Q-functions $\begin{array} { r l } & { \operatorname* { m i n } _ { i \neq j } \langle \nabla _ { \mathbf { a } } Q _ { \phi _ { i } } ( \mathbf { s } , \mathbf { a } ) , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \rangle } \end{array}$ to examine the alignment of the gradients while varying $N$ on the D4RL hopper-medium dataset. The results imply that the performance of the learned policy degrades significantly when the Q-functions share a similar local structure. + +![](images/968b107276cad894e397102039b5ec06b011948bd1a659ae32e5bb3c98ac6b5a.jpg) +Figure 3: Illustration of the ensemble gradient diversification. The vector $\lambda _ { i } \mathbf { w } _ { i }$ represents the normalized eigenvector $\mathbf { w } _ { i }$ of $\mathrm { V a r } ( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) )$ multiplied by its eigenvalue $\lambda _ { i }$ . + +We now show that the alignment of the input gradients can induce insufficient penalization of near-distribution data points, which leads to requiring a large number of ensemble networks. Let $\nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } )$ be the gradient of the $j$ -th Q-function with respect to the behavior action a and assume the gradient is normalized for simplicity. If the gradients of the Q-functions are well-aligned as illustrated in Figure 3a, then there exists a unit vector w such that the Q-values for the OOD actions along the direction of w have a low variance. To show this, we first assume the Q-value predictions for the in-distribution state-action pairs coincide, i.e., $Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) = Q ( \mathbf { s } , \mathbf { a } )$ for $j = 1 , \ldots , N$ Note that this can be optimized by minimizing the Bellman error. Then, using the first-order Taylor approximation, the sample variance of the Q-values at an OOD action along w can be represented as + +![](images/c83721bf5936ecf612e65946d04efde71ac35cf5fc4b3dcc6fa6ee753f7522c4.jpg) +Figure 4: Plot of the minimum cosine similarity between the input gradients of Q-functions and the average return while varying the number of Q-functions. + +$$ +\begin{array} { r l } & { \mathrm { V a r } \left( Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } + k \mathbf { w } ) \right) \approx \mathrm { V a r } \left( Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) + k \left. \mathbf { w } , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. \right) } \\ & { \quad \quad \quad \quad = \mathrm { V a r } \left( Q ( \mathbf { s } , \mathbf { a } ) + k \left. \mathbf { w } , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. \right) } \\ & { \quad \quad \quad \quad = k ^ { 2 } \mathrm { V a r } \left( \left. \mathbf { w } , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. \right) } \\ & { \quad \quad \quad \quad = k ^ { 2 } \mathbf { w } ^ { \top } \mathrm { V a r } \left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right) \mathbf { w } , } \end{array} +$$ + +where $\langle \cdot , \cdot \rangle$ denotes an inner-product, $k \in \mathbb { R }$ , and Var $\left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right)$ is the sample variance matrix for the input gradients $\nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } )$ . One interesting property of the variance matrix is that its total variance, which is equivalent to the sum of its eigenvalues, can be represented as a function of the norm of the average gradients by Lemma 1. + +Lemma 1. The total variance of the matrix $\mathrm { V a r } \left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right)$ is equal to $1 - \| \bar { q } \| _ { 2 } ^ { 2 }$ , where $\bar { q } =$ $\begin{array} { r } { \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) . } \end{array}$ . + +Let $\lambda _ { \mathrm { m i n } }$ be the smallest eigenvalue of $\mathrm { V a r } \left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right)$ and $\mathbf { w } _ { \mathrm { m i n } }$ be the corresponding normalized eigenvector. Also, let $\epsilon > 0$ be the value such that $\begin{array} { r } { \operatorname* { m i n } _ { i \neq j } \left. \nabla _ { \mathbf { a } } Q _ { \phi _ { i } } ( \mathbf { s } , \mathbf { a } ) , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. = 1 - \epsilon . } \end{array}$ Then, using Lemma 1, we can prove that the variance of the $\mathrm { Q }$ -values for an OOD action along $\mathbf { w } _ { \mathrm { m i n } }$ is upper-bounded by some constant multiple of $\epsilon$ , which is given by Proposition 1. + +Proposition 1. Suppose $Q _ { \phi _ { j } } ( { \bf s } , { \bf a } ) \ = \ Q ( { \bf s } , { \bf a } )$ and $Q _ { \phi _ { j } } ( \mathbf { s } , \cdot )$ is locally linear in the neighborhood of a for all $j \in [ N ]$ . Let $\lambda _ { \mathrm { m i n } }$ and $\mathbf { w } _ { \mathrm { m i n } }$ be the smallest eigenvalue and the corresponding normalized eigenvector of the matrix Var $\left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right)$ and $\epsilon > 0$ be the value such that $\begin{array} { r } { \operatorname* { m i n } _ { i \neq j } \left. \nabla _ { \mathbf { a } } Q _ { \phi _ { i } } ( \mathbf { s } , \mathbf { a } ) , \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. = 1 - \epsilon . } \end{array}$ . Then, the variance of the $Q$ -values for an OOD action in the neighborhood along the direction of $\mathbf { w } _ { \mathrm { m i n } }$ is upper-bounded as follows: + +$$ +\mathrm { V a r } \left( Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } + k \mathbf { w } _ { \mathrm { m i n } } ) \right) \leq \frac { 1 } { | \mathcal { A } | } \frac { N - 1 } { N } k ^ { 2 } \epsilon , +$$ + +where $| { \cal A } |$ is the action space dimension. + +We provide the proofs in Appendix A.1. Proposition 1 implies that if there exists such $\epsilon > 0$ that is small, which means the gradients of Q-function are well-aligned, then the variance of the Q-values for an OOD action along a specific direction will also be small. This in turn degrades the ability of the ensembles to penalize OOD actions, which ultimately leads to requiring a large number of ensemble networks. + +To address this problem, we propose a regularizer that effectively increases the variance of the Qvalues for near-distribution OOD actions. Note that the variance is lower-bounded by some constant multiple of the smallest eigenvalue $\lambda _ { \mathrm { m i n } }$ : + +$$ +\begin{array} { r l } & { \mathrm { V a r } \left( Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } + k \mathbf { w } ) \right) \approx k ^ { 2 } \mathbf { w } ^ { \top } \mathrm { V a r } \left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right) \mathbf { w } } \\ & { \quad \quad \quad \quad \geq k ^ { 2 } \mathbf { w } _ { \operatorname* { m i n } } ^ { \top } \mathrm { V a r } \left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right) \mathbf { w } _ { \operatorname* { m i n } } } \\ & { \quad \quad \quad = k ^ { 2 } \lambda _ { \operatorname* { m i n } } . } \end{array} +$$ + +Therefore, an obvious way to increase this variance is to maximize the smallest eigenvalue of Var $\left( \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right)$ , which can be formulated as + +$$ +\begin{array} { r l } & { \underset { \phi } { \mathrm { m a x i m i z e } } ~ \mathbb { E } _ { { \mathbf s } , { \mathbf a } \sim \mathcal { D } } \left[ \lambda _ { \mathrm { m i n } } \left( { \mathrm { V a r } \left( { \nabla _ { { \mathbf a } } { Q _ { \phi _ { j } } } \left( { \mathbf s } , { \mathbf a } \right) } \right) } \right) \right] , } \end{array} +$$ + +where $\phi$ denotes the collection of the parameters $\{ \phi _ { j } \} _ { j = 1 } ^ { N }$ . There are several methods to compute the smallest eigenvalue, such as the power method or the QR algorithm [27]. However, these iterative methods require constructing huge computation graphs, which makes optimizing the eigenvalue using back-propagation inefficient. Instead, we aim to maximize the sum of all eigenvalues, which is equal to the total variance. By Lemma 1, it is equivalent to minimizing the norm of the average gradients: + +$$ +\underset { \phi } { \mathrm { m i n i m i z e } } \ \mathbb { E } _ { \mathbf { s } , \mathbf { a } \sim \mathcal { D } } \left[ \left. \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \nabla _ { \mathbf { a } } Q _ { \phi _ { i } } ( \mathbf { s } , \mathbf { a } ) , \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \nabla _ { \mathbf { a } } Q _ { \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right. \right] . +$$ + +With simple modification, we can reformulate Equation (4) as diversifying the gradients of each Q-function network for in-distribution actions: + +$$ +\operatorname* { m i n i m i z e } J _ { \mathrm { E S } } ( Q _ { \phi } ) : = \mathbb { E } _ { { \bf s } , { \bf a } \sim \mathcal { D } } \left[ \frac { 1 } { N - 1 } \sum _ { 1 \leq i \neq j \leq N } \underbrace { \left. \nabla _ { { \bf a } } Q _ { \phi _ { i } } ( { \bf s } , { \bf a } ) , \nabla _ { { \bf a } } Q _ { \phi _ { j } } ( { \bf s } , { \bf a } ) \right. } _ { \mathrm { E S } _ { \phi _ { i } , \phi _ { j } } ( { \bf s } , { \bf a } ) } \right] . +$$ + +Concretely, our final objective can be interpreted as measuring the pairwise alignment of the gradients using cosine similarity, which we denote as the Ensemble Similarity (ES) metric $\mathrm { E S } _ { \phi _ { i } , \phi _ { j } } ( \mathbf { s } , \mathbf { a } )$ , and minimizing the ES values for every pair in the Q-ensemble with regard to the dataset state-actions. The illustration of the ensemble gradient diversification is shown in Figure 3b. Note that we instead maximize the total variance to reduce the computational burden. Nevertheless, the modified objective is closely related to maximizing the smallest eigenvalue. The detailed explanation can be found in Appendix A.2. + +We name the resulting actor-critic algorithm as Ensemble-Diversified Actor Critic (EDAC) and present the detailed procedure in Algorithm 1 (differences with the original SAC algorithm marked in blue). Note that Algorithm 1 reduces to SAC- $N$ when $\eta = 0$ , and further reduces to vanilla SAC when also $N = 2$ . + +# 5 Experiments + +We evaluate our proposed methods against the previous offline RL algorithms on the standard D4RL benchmark [9] . Concretely, we perform our evaluation on MuJoCo Gym (Section 5.1) and Adroit + +1: Initialize policy parameters $\theta$ , Q-function parameters $\{ \phi _ { j } \} _ { j = 1 } ^ { N }$ , target Q-function parameters $\{ \phi _ { j } ^ { \prime } \} _ { j = 1 } ^ { N }$ , and offline data replay buffer $\mathcal { D }$ + +# 2: repeat + +3: Sample a mini-batch $B = \{ ( \mathbf { s } , \mathbf { a } , r , \mathbf { s } ^ { \prime } ) \}$ from $\mathcal { D }$ + +4: Compute target Q-values (shared by all Q-functions): + +$$ +y ( r , \mathbf { s } ^ { \prime } ) = r + \gamma \left( \operatorname* { m i n } _ { j = 1 , \dots , N } Q _ { \phi _ { j } ^ { \prime } } \left( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } \right) - \beta \log \pi _ { \theta } \left( \mathbf { a } ^ { \prime } \mid \mathbf { s } ^ { \prime } \right) \right) , \quad \mathbf { a } ^ { \prime } \sim \pi _ { \theta } \left( \cdot \mid \mathbf { s } ^ { \prime } \right) +$$ + +5: Update each Q-function $Q _ { \phi _ { i } }$ with gradient descent using + +$$ +\nabla _ { \phi _ { i } } \frac { 1 } { \left| B \right| } \sum _ { ( \mathbf { s } , \mathbf { a } , r , \mathbf { s } ^ { \prime } ) \in B } \left( \left( Q _ { \phi _ { i } } \left( \mathbf { s } , \mathbf { a } \right) - y \left( r , \mathbf { s } ^ { \prime } \right) \right) ^ { 2 } + \frac { \eta } { N - 1 } \sum _ { 1 \leq i \neq j \leq N } \mathrm { E S } _ { \phi _ { i } , \phi _ { j } } ( \mathbf { s } , \mathbf { a } ) \right) +$$ + +6: Update policy with gradient ascent using + +$$ +\nabla _ { \boldsymbol { \theta } } \frac { 1 } { \left| \boldsymbol { B } \right| } \sum _ { \mathbf { s } \in \boldsymbol { B } } \left( \operatorname* { m i n } _ { j = 1 , \dots , N } Q _ { \phi _ { j } } \left( \mathbf { s } , \tilde { \mathbf { a } } _ { \boldsymbol { \theta } } ( \mathbf { s } ) \right) - \beta \log \pi _ { \boldsymbol { \theta } } \left( \tilde { \mathbf { a } } _ { \boldsymbol { \theta } } ( \mathbf { s } ) \mid \mathbf { s } \right) \right) , +$$ + +where $\tilde { \mathbf { a } } _ { \theta } ( \mathbf { s } )$ is a sample from $\pi _ { \boldsymbol { \theta } } ( \cdot \mid \mathbf { s } )$ which is differentiable w.r.t. $\theta$ via the reparametrization trick. + +7: Update target networks with $\phi _ { i } ^ { \prime } \rho \phi _ { i } ^ { \prime } + ( 1 - \rho ) \phi _ { i }$ (Section 5.2) domains. We consider the following baselines: SAC, the backbone algorithm of our method, CQL, the previous state-of-the-art on the D4RL benchmark, REM [2], an offline RL method which utilized Q-network ensemble on discrete control environments, and BC, the behavior cloning method. We evaluate each method under the normalized average return metric where the average return is scaled such that 0 and 100 each equals the performance of a random policy and an online expert policy. In addition to the performance evaluation, we compare the computational cost of each method (Section 5.3). For the implementation details of our algorithm and the baselines, please refer to Appendix B and C. Also, we provide more experiments such as comparison with more baselines, CQL with $N$ Q-networks, and hyperparameter sensitivity from Appendix E to H. The code is available online3. + +# 5.1 Evaluation on D4RL MuJoCo Gym tasks + +We first evaluate each method on D4RL MuJoCo Gym tasks which consist of three environments, halfcheetah, hopper, and walker2d, each with six datasets from different data-collecting policies. In detail, the considered policies are random: a uniform random policy, expert: a fully trained online expert, medium: a suboptimal policy with approximately 1/3 the performance of the expert, medium-expert: a mixture of medium and expert policies, medium-replay: the replay buffer of a policy trained up to the performance of the medium agent, and full-replay: the final replay buffer of the expert policy. Each dataset consists of 1M transitions except for medium-expert and medium-replay. + +The experiment results in Table 1 show EDAC and SAC- $N$ both outperform or are competitive with the previous state-of-the-art on all of the tasks considered. Notably, the performance gap is especially high for random, medium, and medium-replay datasets, where the performances of the previous works are relatively low. Both the proposed methods achieve average normalized scores over 80, reducing the gap with the online expert by $40 \%$ compared to CQL. While the performance of EDAC is marginally better than the performance of SAC- $N$ , EDAC achieves this result with a much smaller Q-ensemble size. As noted in Figure 5, on hopper tasks, SAC- $N$ requires 200 to $5 0 0 \mathrm { Q }$ -networks, while EDAC requires less than 50. + +Figure 6 compares the distance between the actions chosen by each method and the dataset actions. Concretely, we measure $\mathbb { E } _ { ( \mathbf { s } , \mathbf { a } ) \sim \mathcal { D } , \hat { \mathbf { a } } \sim \pi _ { \boldsymbol { \theta } } ( \cdot | \mathbf { s } ) } [ | \hat { \mathbf { a } } - \mathbf { a } | | _ { 2 } ^ { 2 } ]$ for EDAC, SAC- $N$ , CQL, SAC-2, and a random policy on $^ *$ -medium datasets. We find that our proposed methods choose from a more diverse range of actions compared to CQL. This shows the advantage of the uncertainty-based penalization which considers the prediction confidence other than penalizing all OOD actions. + +Table 1: Normalized average returns on D4RL Gym tasks, averaged over 4 random seeds. CQL (Paper) denotes the results reported in the original paper. + +
Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
halfcheetah-random2.2±0.029.7±1.4-0.8±1.135.431.3±3.528.0±0.928.4±1.0
halfcheetah-medium43.2±0.655.2±27.8-0.8±1.344.446.9±0.467.5±1.265.9±0.6
halfcheetah-expert91.8±1.5-0.8±1.84.1±5.7104.897.3±1.1105.2±2.6106.8±3.4
halfcheetah-medium-expert44.0±1.628.4±19.40.7±3.762.495.0±1.4107.1±2.0106.3±1.9
halfcheetah-medium-replay37.6±2.10.8±1.06.6±11.046.245.3±0.363.9±0.861.3±1.9
halfcheetah-full-replay62.9±0.886.8±1.027.8±35.4-76.9±0.984.5±1.284.6±0.9
hopper-random3.7±0.69.9±1.53.4±2.210.85.3±0.631.3±0.025.3±10.4
hopper-medium54.1±3.80.8±0.00.7±0.086.661.9±6.4100.3±0.3101.6±0.6
hopper-expert107.7±9.70.7±0.00.8±0.0109.9106.5±9.1110.3±0.3110.1±0.1
hopper-medium-expert53.9±4.70.7±0.00.8±0.0111.096.9±15.1110.1±0.3110.7±0.1
hopper-medium-replay16.6±4.87.4±0.527.5±15.248.686.3±7.3101.8±0.5101.0±0.5
hopper-full-replay19.9±12.941.1±17.919.7±24.6-101.9±0.6102.9±0.3105.4±0.7
walker2d-random1.3±0.10.9±0.86.9±8.37.05.4±1.721.7±0.016.6±7.0
walker2d-medium70.9±11.0-0.3±0.20.2±0.774.579.5±3.287.9±0.292.5±0.8
walker2d-expert108.7±0.20.7±0.31.0±2.3121.6109.3±0.1107.4±2.4115.1±1.9
walker2d-medium-expert90.1±13.21.9±3.9-0.1±0.098.7109.1±0.2116.7±0.4114.7±0.9
walker2d-medium-replay20.3±9.8-0.4±0.312.5±6.232.676.8±10.078.7±0.787.1±2.3
walker2d-full-replay68.8±17.727.9±47.3-0.2±0.3-94.2±1.994.6±0.599.8±0.7
Average49.916.26.2-73.784.585.2
+ +![](images/8cc13d52b0b3edcc798dadfe4273b9dde139c1cec3b4cc0c697de114ee27b598.jpg) +Figure 5: Minimum number of Q-ensembles $( N )$ required to achieve the performance reported in Table 1. M-E denotes medium-expert. We omit the results of medium-replay and full-replay as SAC- $. N$ already works well with a small number of ensembles (less than or equal to 5). For more details of the experiment, please refer to Appendix C. + +![](images/1116de177c49343d5cd95d790cc9fd29db3d3f213b6f83a11e54c2cb38f4d694.jpg) +Figure 6: Histograms of the distances between the actions from each methods (EDAC, SAC- $N$ , CQL, SAC-2, and a random policy) and the actions from the dataset. For more details of the experiment, please refer to Appendix C. + +# 5.2 Evaluation on D4RL Adroit tasks + +We also experiment on the more complex D4RL Adroit tasks that require controlling a 24-DoF robotic hand to perform tasks such as aligning a pen, hammering a nail, opening a door, or relocating a ball. We use two types of datasets for each environment: human, containing 25 trajectories of human demonstrations, and cloned, a 50-50 mixture between the demonstration data and the behavioral cloned policy on the demonstrations. Note that for the Adroit tasks, we could not reproduce the CQL results from the paper completely. For the detailed procedure of reproducing the results of CQL, please refer to Appendix D. + +Table 2: Normalized average returns on D4RL Adroit tasks, averaged over 4 random seeds. + +
Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
pen-human25.8±8.84.3±3.85.4±4.355.835.2±6.69.5±1.152.1±8.6
hammer-human3.1±3.20.2±0.00.3±0.02.10.6±0.50.3±0.00.8±0.4
door-human2.8±0.7-0.3±0.0-0.3±0.09.11.2±1.8-0.3±0.010.7±6.8
relocate-human0.0±0.0-0.3±0.0-0.3±0.00.350.0±0.0-0.1±0.10.1±0.1
pen-cloned38.3±11.9-0.8±3.2-1.0±0.140.327.2±11.364.1±8.768.2±7.3
hammer-cloned0.7±0.30.1±0.1-0.3±0.05.71.4±2.10.2±0.20.3±0.0
door-cloned0.0±0.0-0.3±0.1-0.3±0.03.52.4±2.4-0.3±0.09.6±8.3
relocate-cloned0.1±0.0-0.1±0.1-0.2±0.2-0.10.0±0.00.0±0.00.0±0.0
+ +The evaluation results are summarized in Table 2. For pen- $^ { \ast }$ tasks, where the considered algorithms achieve meaningful performance, EDAC outperforms or matches with the previous state-of-the-art. Especially, for pen-cloned, both EDAC and SAC- $. N$ achieve $7 5 \%$ higher score compared to CQL. Unlike the results from the Gym tasks, we find that SAC- $N$ falls behind in some datasets, for example, pen-human, which could in part due to the size of the dataset being exceptionally small (5000 transitions). However, our method with ensemble diversification successfully overcomes this difficulty. + +# 5.3 Computational cost comparison + +We compared the computational cost of our methods with vanilla SAC and CQL on hopper-medium-v2, where our methods require the largest number of Q-networks. For each method, we measure the runtime per training epoch (1000 gradient steps) along with GPU memory consumption. We run our experiments on a single machine with one RTX 3090 GPU and provide the results in Table 3. + +Table 3: Computational costs of each method. + +
Runtime (s/epoch)GPU Mem. (GB)
SAC21.41.3
CQL38.21.4
SAC-50044.15.1
EDAC30.81.8
+ +As the result shows, our method EDAC runs faster than CQL with comparable memory consumption. Note that CQL is about twice as slower than vanilla SAC due to the additional computations for Q-value regularization (e.g., dual update and approximate logsumexp via sampling). Meanwhile, the inference to the Q-network ensemble in SAC- $N$ and EDAC is embarrassingly parallelizable, minimizing the runtime increase with the number of Q-networks. Also, we emphasize that our gradient diversification term in Equation (4) has linear computational complexity, as we can reformulate the term using the sum of the gradients. + +# 6 Related Works + +Model-free offline RL A popular approach for offline RL is to regularize the learned policy to be close to the behavior policy where the offline dataset was collected. BCQ [11] uses a generative model to produce actions with high similarity to the dataset and trains a restricted policy to choose the best action from the neighborhood of the generated actions. Another line of work, such as BEAR [15] or BRAC [28], stabilizes policy learning by penalizing the divergence from the dataset measured by KL divergence or MMD. While these policy-constraint methods demonstrate high performance on datasets from expert behavior policies, they fail to find optimal policies from datasets with suboptimal policies due to the strict policy constraints [9]. Also, these methods require an accurate estimation of the behavior policy, which might be difficult in complex settings with multiple behavior sources or high-dimensional environments. To address these issues, CQL [16] directly regularizes Q-functions by introducing a term that minimizes the Q-values for out-of-distribution actions and maximizes the Q-values for in-distribution actions. Without such explicit regularizations, REM [2] proposes to use a random convex combination of Q-network ensembles on environments with discrete action spaces [4]. + +Estimation bias in Q-learning While Q-learning is one of the most popular algorithms in reinforcement learning, it suffers from overestimation bias due to the maximum operation $\mathrm { m a x } _ { \mathbf { a } ^ { \prime } \in \mathcal { A } } Q ( \mathbf { s } ^ { \prime } , \mathbf { a } ^ { \prime } )$ used during Q-function updates [10, 25]. This overestimation bias, together with the bootstrapping, can lead to a catastrophic build-up of errors during the Q-learning process. To resolve this issue, TD3 [10] introduces a clipped version of Double Q-learning [25] that takes the minimum value of two critics. Subsequently, Maxmin Q-learning [17] theoretically shows that the overestimation bias can be controlled by the number of ensembles in the clipped Q-learning. The overestimation problem in Q-learning can be exacerbated in the offline setting since the extrapolation error cannot be corrected with further interactions with the environment, and existing offline RL algorithms handle the bias by introducing constrained policy optimization [11, 15] or conservative Q-learning frameworks [16]. + +Uncertainty measures in RL Uncertainty estimates have been widely used in RL for various purposes including exploration, Q-learning, and planning. Bootstrapped DQN [21] leverages an ensemble of Q-functions to quantify the uncertainty of the Q-value, and utilizes it for efficient exploration. Following this work, the UCB exploration algorithm [5] constructs an upper confidence bound [3] of the Q-values using the empirical mean and standard deviation of Q-ensembles, which is used to promote efficient exploration by applying the principle of optimism in the face of uncertainty [7]. Osband et al. [22] proposes a randomly initialized Q-ensemble that reflects the concept of prior functions in Bayesian inference and Abbas et al. [1] introduces an uncertainty incorporated planning with imperfect models. The notion of uncertainty has also been considered in offline RL, mostly in the framework of model-based offline RL. Especially, MOPO [29] and MOReL [13] measure the uncertainty of the model’s prediction to formulate an uncertainty-penalized policy optimization problem in the offline RL setting. These methods introduce an ensemble of dynamics models for the quantification of the uncertainty, whereas our work adopts an ensemble of Q-functions for uncertainty-aware Q-learning. + +# 7 Conclusion + +We have shown that clipped Q-learning can be efficiently leveraged to construct an uncertaintybased offline RL method that outperforms previous methods on various datasets. Based on this observation, we proposed Ensemble-Diversifying Actor-Critic (EDAC) that effectively reduces the required number of ensemble networks for quantifying and penalizing the epistemic uncertainty. Our method does not require any explicit estimation of the data collecting policy or sampling from the out-of-distribution data and respects the epistemic uncertainty of each data point during penalization. EDAC, while requiring up to $90 \%$ less number of ensemble networks compared to the vanilla Q-ensemble, exhibits state-of-the-art performance on various datasets. + +# Acknowledgements + +This work was supported in part by Samsung Advanced Institute of Technology, Samsung Electronics Co., Ltd., Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00882, (SW STAR LAB) Development of deployable learning intelligence via self-sustainable and trustworthy machine learning and No. 2019- 0-01371, Development of brain-inspired AI with human-like intelligence), and Research Resettlement Fund for the new faculty of Seoul National University. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA2386-20-1-4043. Hyun Oh Song is the corresponding author. + +# References + +[1] Zaheer Abbas, Samuel Sokota, Erin Talvitie, and Martha White. Selective dyna-style planning under limited model capacity. In ICML, 2020. +[2] Rishabh Agarwal, Dale Schuurmans, and Mohammad Norouzi. An optimistic perspective on offline reinforcement learning. In ICML, 2020. +[3] Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Exploration–exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science, 410(19): 1876–1902, 2009. +[4] Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013. +[5] Richard Y Chen, Szymon Sidor, Pieter Abbeel, and John Schulman. Ucb exploration via q-ensembles. arXiv preprint arXiv:1706.01502, 2017. +[6] Xinshi Chen, Shuang Li, Hui Li, Shaohua Jiang, and Le Song. Generative adversarial user model for reinforcement learning based recommendation system. In ICML, 2019. +[7] Kamil Ciosek, Quan Vuong, Robert Loftin, and Katja Hofmann. Better exploration with optimistic actor-critic. In NeurIPS, 2019. +[8] William R Clements, Bastien Van Delft, Benoît-Marie Robaglia, Reda Bahi Slaoui, and Sébastien Toth. Estimating risk and uncertainty in deep reinforcement learning. arXiv preprint arXiv:1905.09638, 2019. +[9] Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020. +[10] Scott Fujimoto, Herke Hoof, and David Meger. Addressing function approximation error in actor-critic methods. In ICML, 2018. +[11] Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. In ICML, 2019. +[12] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In ICML, 2018. +[13] Rahul Kidambi, Aravind Rajeswaran, Praneeth Netrapalli, and Thorsten Joachims. Morel: Model-based offline reinforcement learning. In NeurIPS, 2020. +[14] Vijay R Konda and John N Tsitsiklis. Actor-critic algorithms. In NeurIPS, 2000. +[15] Aviral Kumar, Justin Fu, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. In NeurIPS, 2019. +[16] Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative q-learning for offline reinforcement learning. In NeurIPS, 2020. +[17] Qingfeng Lan, Yangchen Pan, Alona Fyshe, and Martha White. Maxmin q-learning: Controlling the estimation bias of q-learning. In ICLR, 2020. +[18] Kimin Lee, Michael Laskin, Aravind Srinivas, and Pieter Abbeel. Sunrise: A simple unified framework for ensemble learning in deep reinforcement learning. In ICML, 2021. +[19] Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020. +[20] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. nature, 518(7540):529–533, 2015. +[21] Ian Osband, Charles Blundell, Alexander Pritzel, and Benjamin Van Roy. Deep exploration via bootstrapped dqn. In NeurIPS, 2016. +[22] Ian Osband, John Aslanides, and Albin Cassirer. Randomized prior functions for deep reinforcement learning. In NeurIPS, 2018. +[23] JP Royston. Expected normal order statistics(exact and approximate). Applied Statistics, 31(2): 161–5, 1982. +[24] Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. arXiv preprint arXiv:1206.2944, 2012. +[25] Hado Van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double q-learning. In AAAI, 2016. +[26] Oriol Vinyals, Igor Babuschkin, Junyoung Chung, Michael Mathieu, Max Jaderberg, Wojciech M Czarnecki, Andrew Dudzik, Aja Huang, Petko Georgiev, Richard Powell, et al. Alphastar: Mastering the real-time strategy game starcraft ii. DeepMind blog, 2, 2019. +[27] David S Watkins. Understanding the qr algorithm. SIAM review, 24(4):427–440, 1982. +[28] Yifan Wu, George Tucker, and Ofir Nachum. Behavior regularized offline reinforcement learning. arXiv preprint arXiv:1911.11361, 2019. +[29] Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon, James Zou, Sergey Levine, Chelsea Finn, and Tengyu Ma. Mopo: Model-based offline policy optimization. In NeurIPS, 2020. \ No newline at end of file diff --git a/parse/train/ZUvaSolQZh3/ZUvaSolQZh3_content_list.json b/parse/train/ZUvaSolQZh3/ZUvaSolQZh3_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fa6d31e69176ee77a0b564945e6e6e888a46e3e8 --- /dev/null +++ b/parse/train/ZUvaSolQZh3/ZUvaSolQZh3_content_list.json @@ -0,0 +1,1204 @@ +[ + { + "type": "text", + "text": "Uncertainty-Based Offline Reinforcement Learning with Diversified Q-Ensemble ", + "text_level": 1, + "bbox": [ + 189, + 122, + 812, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gaon $\\mathbf { A } \\mathbf { n } ^ { * 1 2 }$ , Seungyong Moon\\*1 2, Jang-Hyun $\\mathbf { K i m ^ { 1 2 } }$ , Hyun Oh Song† 1 2 3 ", + "bbox": [ + 225, + 223, + 771, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Seoul National University1 Neural Processing Research Center2 DeepMetrics3 {white0234,symoon11,janghyun,hyunoh}@mllab.snu.ac.kr ", + "bbox": [ + 276, + 241, + 722, + 297 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 332, + 535, + 348 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Offline reinforcement learning (offline RL), which aims to find an optimal policy from a previously collected static dataset, bears algorithmic difficulties due to function approximation errors from out-of-distribution (OOD) data points. To this end, offline RL algorithms adopt either a constraint or a penalty term that explicitly guides the policy to stay close to the given dataset. However, prior methods typically require accurate estimation of the behavior policy or sampling from OOD data points, which themselves can be a non-trivial problem. Moreover, these methods under-utilize the generalization ability of deep neural networks and often fall into suboptimal solutions too close to the given dataset. In this work, we propose an uncertainty-based offline RL method that takes into account the confidence of the Q-value prediction and does not require any estimation or sampling of the data distribution. We show that the clipped Q-learning, a technique widely used in online RL, can be leveraged to successfully penalize OOD data points with high prediction uncertainties. Surprisingly, we find that it is possible to substantially outperform existing offline RL methods on various tasks by simply increasing the number of Q-networks along with the clipped Q-learning. Based on this observation, we propose an ensemble-diversified actor-critic algorithm that reduces the number of required ensemble networks down to a tenth compared to the naive ensemble while achieving state-of-the-art performance on most of the D4RL benchmarks considered. ", + "bbox": [ + 233, + 363, + 766, + 641 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 176, + 667, + 310, + 685 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Over the recent years, deep reinforcement learning (deep RL) has achieved considerable success in various domains such as robotics [20], recommendation systems [6], and strategy games [26]. However, a major drawback of RL algorithms is that they adopt an active learning procedure, where training steps require active interactions with the environment. This trial-and-error procedure can be prohibitive when scaling RL to real-world applications such as autonomous driving and healthcare, as exploratory actions can cause critical damage to the agent or the environment [19]. Offline RL, also known as batch RL, aims to overcome this problem by learning policies using only previously collected data without further interactions with the environment [2, 11, 19]. ", + "bbox": [ + 174, + 700, + 826, + 811 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Even though offline RL is a promising direction to lead a more data-driven way of solving RL problems, recent works show offline RL faces new algorithmic challenges [19]. Typically, if the coverage of the dataset is not sufficient, vanilla RL algorithms suffer severely from extrapolation error, overestimating the Q-values of out-of-distribution (OOD) state-action pairs [15]. To this end, most offline RL methods apply some constraints or penalty terms on top of the existing RL algorithms to enforce the learning process to be more conservative. For example, some prior works explicitly regularize the policy to be close to the behavior policy that was used to collect the data [11, 15]. A more recent work instead penalizes the Q-values of OOD state-action pairs to enforce the Q-values to be more pessimistic [16]. ", + "bbox": [ + 178, + 819, + 823, + 859 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 175 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While these methods achieve significant performance gains over vanilla RL methods, they either require an estimation of the behavior policy or explicit sampling from OOD data points, which themselves can be non-trivial to solve. Furthermore, these methods do not utilize the generalization ability of the Q-function networks and prohibit the agent from approaching any OOD state-actions without any consideration on whether they are good or bad. However, if we can identify OOD data points where we can predict their Q-values with high confidence, it is more effective not to restrain the agent from choosing those data points. ", + "bbox": [ + 173, + 180, + 825, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "From this intuition, we propose an uncertainty-based model-free offline RL method that effectively quantifies the uncertainty of the Q-value estimates by an ensemble of Q-function networks and does not require any estimation or sampling of the data distribution. To achieve this, we first show that a well-known technique from online RL, the clipped Q-learning [10], can be successfully leveraged as an uncertainty-based penalization term. Our experiments reveal that we can achieve state-of-the-art performance on various offline RL tasks by solely using this technique with increased ensemble size. To further improve the practical usability of the method, we develop an ensemble diversifying objective that significantly reduces the number of required ensemble networks. We evaluate our proposed method on D4RL benchmarks [9] and verify that the proposed method outperforms the previous state-of-the-art by a large margin on various types of environments and datasets. ", + "bbox": [ + 173, + 284, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Preliminaries ", + "text_level": 1, + "bbox": [ + 174, + 441, + 318, + 459 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider an environment formulated as a Markov Decision Process (MDP) defined by a tuple $( S , A , T , r , d _ { 0 } , \\gamma )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $T ( \\mathbf { s } ^ { \\prime } \\mid \\mathbf { s } , \\mathbf { a } )$ is the transition probability distribution, $r : S \\times \\mathcal { A } \\mathbb { R }$ is the reward function, $d _ { 0 }$ is the initial state distribution, and $\\gamma \\in \\mathsf { \\Gamma } ( 0 , 1 ]$ is the discount factor. The goal of reinforcement learning is to find an optimal policy $\\pi ( \\mathbf { a } \\mid \\mathbf { s } )$ that maximizes the cumulative discounted reward $\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } ) \\right] } \\end{array}$ , where $\\mathbf { s } _ { 0 } \\sim d _ { 0 } ( \\cdot )$ , $\\mathbf { a } _ { t } \\sim \\pi ( \\cdot \\mid \\mathbf { s } _ { t } )$ , and $\\mathbf { s } _ { t + 1 } \\sim T ( \\cdot \\mid \\mathbf { s } _ { t } , \\mathbf { a } _ { t } )$ . ", + "bbox": [ + 173, + 473, + 825, + 559 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "One of the major approaches for obtaining such a policy is Q-learning [12, 20] which learns a state-action value function $Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } )$ parameterized by a neural network that represents the expected cumulative discounted reward when starting from state s and action a. Standard actor-critic approach [14] learns this Q-function by minimizing the Bellman residual $\\big ( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) \\big ) ^ { 2 }$ , where $B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) = \\mathbb { E } _ { \\mathbf { s } ^ { \\prime } \\sim T ( \\cdot | \\mathbf { s } , \\mathbf { a } ) } \\left[ r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } Q _ { \\phi } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right]$ is the Bellman operator. In the context of offline RL, where transitions are sampled from a static dataset $\\mathcal { D }$ , the objective for the $\\mathrm { Q }$ -network becomes minimizing ", + "bbox": [ + 173, + 563, + 825, + 665 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/dd72bed5b712c6f643e4a71f2f1a93fde3332a7b416d74c86a8f6308169989dc.jpg", + "text": "$$\nJ _ { q } ( Q _ { \\phi } ) : = \\mathbb { E } _ { ( \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ Q _ { \\phi ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 227, + 670, + 767, + 705 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $Q _ { \\phi ^ { \\prime } }$ represents the target Q-network softly updated for algorithmic stability [20]. The policy, which is also parameterized by a neural network, is updated in an alternating fashion to maximize the expected $\\mathrm { Q }$ -value: $J _ { p } ( \\pi _ { \\theta } ) : = \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi _ { \\theta } ( \\cdot | { \\mathbf s } ) } \\left[ Q _ { \\phi } ( { \\mathbf s } , { \\mathbf a } ) \\right] ,$ . ", + "bbox": [ + 173, + 712, + 825, + 756 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, as the policy is updated to maximize the Q-values, the actions $\\mathbf { a } ^ { \\prime }$ sampled from the current policy in Equation (1) can be biased towards OOD actions with erroneously high Q-values. In the offline RL setting, such errors cannot be corrected by feedback from the environment as in online RL. To handle the error propagation from these OOD actions, most offline RL algorithms regularize either the policy [11, 15] or the Q-function [16] to be biased towards the given dataset. However, the policy regularization methods typically require an accurate estimation of the behavior policy. The previous state-of-the-art method CQL [16] instead learns conservative Q-values without estimating the behavior policy by penalizing the Q-values of OOD actions by ", + "bbox": [ + 173, + 761, + 825, + 873 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5200b1ab6df96c14f926c8a3dd66849ac370e617d49cbd25d3e463c66cab5da6.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\phi } J _ { q } ( Q _ { \\phi } ) + \\alpha \\Big ( \\mathbb { E } _ { { \\mathbf { s } } \\sim \\mathcal { D } , { \\mathbf { a } } \\sim \\mu ( \\cdot \\vert \\mathbf { s } ) } \\left[ Q _ { \\phi } \\left( \\mathbf { s } , { \\mathbf { a } } \\right) \\right] - \\mathbb { E } _ { ( \\mathbf { s } , { \\mathbf { a } } ) \\sim \\mathcal { D } } \\left[ Q _ { \\phi } \\left( \\mathbf { s } , { \\mathbf { a } } \\right) \\right] \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 266, + 880, + 727, + 909 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mu$ is an approximation of the policy that maximizes the current Q-function. While CQL does not need explicit behavior policy estimation, it requires sampling from an appropriate action distribution $\\mu ( \\cdot | \\mathbf { \\bar { s } } )$ . ", + "bbox": [ + 173, + 90, + 825, + 133 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Uncertainty penalization with Q-ensemble ", + "text_level": 1, + "bbox": [ + 173, + 152, + 558, + 170 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/5ef0099c6b2f91502d99a5bd4e4ae37be48755a4861064068d4c78cfb8b708b5.jpg", + "image_caption": [ + "Figure 1: Performance of SAC- $N$ on halfcheetah-medium and hopper-medium datasets while varying $N$ , compared to CQL. ‘Average Return’ denotes the undiscounted return of each policies on evaluation. Results averaged over 4 seeds. " + ], + "image_footnote": [], + "bbox": [ + 199, + 191, + 794, + 369 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we turn our attention to a conventional technique from online RL, Clipped Double QLearning [10], which uses the minimum value of two parallel Q-networks as the Bellman target: $y =$ $\\begin{array} { r } { r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ \\operatorname* { m i n } _ { j = 1 , 2 } Q _ { \\phi _ { j } ^ { \\prime } } ( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } ) \\right] } \\end{array}$ . Although this technique was originally proposed in online RL to mitigate the overestimation from general prediction errors, some offline RL algorithms [11, 15, 28] also utilize this technique to enforce their $\\mathrm { Q }$ -value estimates to be more pessimistic. However, the isolated effect of the clipped Q-learning in offline RL was not fully analyzed in the previous works, as they use the technique only as an auxiliary term that adds up to their core methods. ", + "bbox": [ + 173, + 436, + 826, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To examine the ability of clipped Q-learning to prevent the overestimation in offline RL on its own, we modify SAC [12] by increasing the number of $\\mathrm { Q }$ -ensembles from 2 to $N$ : ", + "bbox": [ + 173, + 547, + 823, + 575 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2774320fff9c4e1fa2ee39692e9d78ca73bb14e4e792e8943040295899bad1a6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\phi _ { i } } { \\mathrm { n i n } } \\ : \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi \\theta \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } ^ { \\prime } \\right) } \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } ^ { \\prime } \\mid \\mathbf { s } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] } \\\\ & { \\underset { \\theta } { \\mathrm { n a x } } \\ : \\mathbb { E } _ { \\mathbf { s } \\sim \\mathcal { D } , \\mathbf { a } \\sim \\pi _ { \\theta } \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } \\right) } \\ : \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } \\mid \\mathbf { s } \\right) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 580, + 826, + 652 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "for $i = 1 , \\ldots , N$ . We denote this modified algorithm as SAC- $N$ . ", + "bbox": [ + 173, + 655, + 598, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows the preliminary experiments on D4RL halfcheetah-medium and hopper-medium datasets [9] while varying $N$ . Note that these datasets are constructed from suboptimal behavior policies. Surprisingly, as we gradually increase $N$ , we can successfully find policies that outperform the previous state-of-the-art method (CQL) by a large margin. In fact, as we will present in Section 5, SAC- $N$ outperforms CQL on various types of environments and data-collection policies. ", + "bbox": [ + 173, + 676, + 825, + 746 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To understand why this simple technique works so well, we can first interpret the clipping procedure (choosing the minimum value from the ensemble) as penalizing state-action pairs with high-variance Q-value estimates, which encourages the policy to favor actions that appeared in the dataset [11]. The dataset samples will naturally have lower variance compared to the OOD samples as the Bellman residual term in Equation (2) explicitly aligns the Q-value predictions for the dataset samples. More formally, we can regard this difference in variance as accounting for epistemic uncertainty [8] which refers to the uncertainty stemming from limited data and knowledge. ", + "bbox": [ + 173, + 752, + 825, + 849 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Utilization of the clipped Q-value relates to methods that consider the confidence bound of the Q-value estimates [24]. Online RL methods typically utilize the Q-ensemble to form an optimistic estimate of the Q-value, by adding the standard deviation to the mean of the Q-ensembles [18]. This optimistic Q-value, also known as the upper-confidence bound (UCB), can encourage the exploration of unseen actions with high uncertainty. However, in offline RL, the dataset available during training is fixed, and we have to focus on exploiting the given data. For this purpose, it is natural to utilize the lower-confidence bound (LCB) of the Q-value estimates, for example by subtracting the standard deviation from the mean, which allows us to avoid risky state-actions. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The clipped Q-learning algorithm, which chooses the worst-case Q-value instead to compute the pessimistic estimate, can also be interpreted as utilizing the LCB of the $\\mathrm { Q }$ -value predictions. Suppose $Q ( \\mathbf { s } , \\mathbf { a } )$ follows a Gaussian distribution with mean $m ( \\mathbf { s } , \\mathbf { a } )$ and standard deviation $\\sigma ( \\mathbf { s } , \\mathbf { a } )$ . Also, let $\\{ Q _ { j } ( \\mathbf { s } , \\mathbf { \\bar { a } } ) \\} _ { j = 1 } ^ { N }$ be realizations of $Q ( \\mathbf { s } , \\mathbf { a } )$ . Then, we can approximate the expected minimum of the realizations following the work of Royston [23] as ", + "bbox": [ + 174, + 152, + 825, + 223 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ca4f68505ad91174a21804ab1d2728e4d21634a54371811adf093c2e43133fa2.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { j } ( \\mathbf { s } , \\mathbf { a } ) \\right] \\approx m ( \\mathbf { s } , \\mathbf { a } ) - \\Phi ^ { - 1 } \\left( \\frac { N - \\frac { \\pi } { 8 } } { N - \\frac { \\pi } { 4 } + 1 } \\right) \\sigma ( \\mathbf { s } , \\mathbf { a } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 228, + 712, + 263 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\Phi$ is the CDF of the standard Gaussian distribution. This relation indicates that using the clipped Q-value is similar to penalizing the ensemble mean of the Q-values with the standard deviation scaled by a coefficient dependent on $N$ . ", + "bbox": [ + 174, + 270, + 825, + 311 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/ce40e054b643c8f22be0cd242e5ba33fb71eb6ea668533184970fad707c33dc5.jpg", + "image_caption": [ + "Figure 2: (a) and (b) each plots the size of the clip penalty and the standard deviation of the Qvalue estimates for in-distribution (behavior) and OOD (random) actions while training SAC-10 on halfcheetah-medium dataset. (c) plots the gap of the clip penalty between the in-distribution and OOD actions while varying $N$ . Results averaged over 4 seeds. " + ], + "image_footnote": [], + "bbox": [ + 186, + 327, + 810, + 465 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now move on to the empirical analysis of the clipped Q-learning. Figure 2a compares the strength of the uncertainty penalty on in-distribution and OOD actions. Specifically, we compare actions sampled from two types of policies: (1) the behavior policy which was used to collect the dataset, and (2) the random policy which samples actions uniformly from the action space. For each policy, we measure the size of the penalty from the clipping as $\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi ( \\cdot | { \\mathbf s } ) } [ \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } Q _ { \\phi _ { j } } ( { \\mathbf s } , { \\mathbf a } ) - } \\end{array}$ $\\begin{array} { r } { \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) ] } \\end{array}$ . Figure 2a shows that the clipping term penalizes the random state-action pairs much stronger than the in-distribution pairs throughout the training. For comparison, we also measure the standard deviation of the $\\mathrm { Q }$ -values for each policy. The results in Figure 2b show that as we conjectured, the Q-value predictions for the OOD actions have a higher variance. We also find that the size of the penalty and the standard deviation are highly correlated, as we noted in Equation (3). ", + "bbox": [ + 173, + 544, + 825, + 688 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As we observe that OOD actions have higher variance on Q-value estimates, the effect of increasing $N$ becomes obvious: it strengthens the penalty applied to the OOD samples compared to the dataset samples. To verify this, we measured the relative penalty applied to the OOD samples in Figure 2c and found that indeed the OOD samples are penalized relatively further as $N$ increases. ", + "bbox": [ + 174, + 694, + 825, + 751 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Ensemble gradient diversification ", + "text_level": 1, + "bbox": [ + 174, + 768, + 485, + 786 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Even though SAC- $N$ outperforms existing methods on various tasks, it sometimes requires an excessively large number of ensembles to learn stably (e.g., $N = 5 0 0$ for hopper-medium). While investigating its reason, we found that the performance of SAC- $N$ is negatively correlated with the degree to which the input gradients of Q-functions $\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )$ are aligned, which decreases with $N$ . Figure 4 measures the minimum cosine similarity between the gradients of the Q-functions $\\begin{array} { r l } & { \\operatorname* { m i n } _ { i \\neq j } \\langle \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\rangle } \\end{array}$ to examine the alignment of the gradients while varying $N$ on the D4RL hopper-medium dataset. The results imply that the performance of the learned policy degrades significantly when the Q-functions share a similar local structure. ", + "bbox": [ + 173, + 800, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/968b107276cad894e397102039b5ec06b011948bd1a659ae32e5bb3c98ac6b5a.jpg", + "image_caption": [ + "Figure 3: Illustration of the ensemble gradient diversification. The vector $\\lambda _ { i } \\mathbf { w } _ { i }$ represents the normalized eigenvector $\\mathbf { w } _ { i }$ of $\\mathrm { V a r } ( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) )$ multiplied by its eigenvalue $\\lambda _ { i }$ . " + ], + "image_footnote": [], + "bbox": [ + 202, + 89, + 807, + 300 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now show that the alignment of the input gradients can induce insufficient penalization of near-distribution data points, which leads to requiring a large number of ensemble networks. Let $\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )$ be the gradient of the $j$ -th Q-function with respect to the behavior action a and assume the gradient is normalized for simplicity. If the gradients of the Q-functions are well-aligned as illustrated in Figure 3a, then there exists a unit vector w such that the Q-values for the OOD actions along the direction of w have a low variance. To show this, we first assume the Q-value predictions for the in-distribution state-action pairs coincide, i.e., $Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) = Q ( \\mathbf { s } , \\mathbf { a } )$ for $j = 1 , \\ldots , N$ Note that this can be optimized by minimizing the Bellman error. Then, using the first-order Taylor approximation, the sample variance of the Q-values at an OOD action along w can be represented as ", + "bbox": [ + 174, + 359, + 555, + 580 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/c83721bf5936ecf612e65946d04efde71ac35cf5fc4b3dcc6fa6ee753f7522c4.jpg", + "image_caption": [ + "Figure 4: Plot of the minimum cosine similarity between the input gradients of Q-functions and the average return while varying the number of Q-functions. " + ], + "image_footnote": [], + "bbox": [ + 563, + 367, + 831, + 502 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c516ccb12b1c7cec2cadc3c22c4e3c9c00941050f096e5a6804326e4b666a370.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } ) \\right) \\approx \\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) + k \\left. \\mathbf { w } , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right) } \\\\ & { \\quad \\quad \\quad \\quad = \\mathrm { V a r } \\left( Q ( \\mathbf { s } , \\mathbf { a } ) + k \\left. \\mathbf { w } , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right) } \\\\ & { \\quad \\quad \\quad \\quad = k ^ { 2 } \\mathrm { V a r } \\left( \\left. \\mathbf { w } , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right) } \\\\ & { \\quad \\quad \\quad \\quad = k ^ { 2 } \\mathbf { w } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 606, + 717, + 689 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\langle \\cdot , \\cdot \\rangle$ denotes an inner-product, $k \\in \\mathbb { R }$ , and Var $\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)$ is the sample variance matrix for the input gradients $\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )$ . One interesting property of the variance matrix is that its total variance, which is equivalent to the sum of its eigenvalues, can be represented as a function of the norm of the average gradients by Lemma 1. ", + "bbox": [ + 173, + 694, + 826, + 753 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 1. The total variance of the matrix $\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)$ is equal to $1 - \\| \\bar { q } \\| _ { 2 } ^ { 2 }$ , where $\\bar { q } =$ $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) . } \\end{array}$ . ", + "bbox": [ + 171, + 756, + 821, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let $\\lambda _ { \\mathrm { m i n } }$ be the smallest eigenvalue of $\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)$ and $\\mathbf { w } _ { \\mathrm { m i n } }$ be the corresponding normalized eigenvector. Also, let $\\epsilon > 0$ be the value such that $\\begin{array} { r } { \\operatorname* { m i n } _ { i \\neq j } \\left. \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. = 1 - \\epsilon . } \\end{array}$ Then, using Lemma 1, we can prove that the variance of the $\\mathrm { Q }$ -values for an OOD action along $\\mathbf { w } _ { \\mathrm { m i n } }$ is upper-bounded by some constant multiple of $\\epsilon$ , which is given by Proposition 1. ", + "bbox": [ + 173, + 803, + 825, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. Suppose $Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\ = \\ Q ( { \\bf s } , { \\bf a } )$ and $Q _ { \\phi _ { j } } ( \\mathbf { s } , \\cdot )$ is locally linear in the neighborhood of a for all $j \\in [ N ]$ . Let $\\lambda _ { \\mathrm { m i n } }$ and $\\mathbf { w } _ { \\mathrm { m i n } }$ be the smallest eigenvalue and the corresponding normalized eigenvector of the matrix Var $\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)$ and $\\epsilon > 0$ be the value such that $\\begin{array} { r } { \\operatorname* { m i n } _ { i \\neq j } \\left. \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. = 1 - \\epsilon . } \\end{array}$ . Then, the variance of the $Q$ -values for an OOD action in the neighborhood along the direction of $\\mathbf { w } _ { \\mathrm { m i n } }$ is upper-bounded as follows: ", + "bbox": [ + 174, + 866, + 825, + 914 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 823, + 121 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ee43964a993eee20e022a5c8c485989210b8ae0300a2d8436215b0db442b3d11.jpg", + "text": "$$\n\\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } _ { \\mathrm { m i n } } ) \\right) \\leq \\frac { 1 } { | \\mathcal { A } | } \\frac { N - 1 } { N } k ^ { 2 } \\epsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 125, + 647, + 157 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $| { \\cal A } |$ is the action space dimension. ", + "bbox": [ + 176, + 162, + 439, + 179 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We provide the proofs in Appendix A.1. Proposition 1 implies that if there exists such $\\epsilon > 0$ that is small, which means the gradients of Q-function are well-aligned, then the variance of the Q-values for an OOD action along a specific direction will also be small. This in turn degrades the ability of the ensembles to penalize OOD actions, which ultimately leads to requiring a large number of ensemble networks. ", + "bbox": [ + 173, + 188, + 825, + 258 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To address this problem, we propose a regularizer that effectively increases the variance of the Qvalues for near-distribution OOD actions. Note that the variance is lower-bounded by some constant multiple of the smallest eigenvalue $\\lambda _ { \\mathrm { m i n } }$ : ", + "bbox": [ + 174, + 263, + 825, + 306 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/811413a540f1df9255ac01cd613a36f965755918dbbfb18243bb3a4f2c812718.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } ) \\right) \\approx k ^ { 2 } \\mathbf { w } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } } \\\\ & { \\quad \\quad \\quad \\quad \\geq k ^ { 2 } \\mathbf { w } _ { \\operatorname* { m i n } } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } _ { \\operatorname* { m i n } } } \\\\ & { \\quad \\quad \\quad = k ^ { 2 } \\lambda _ { \\operatorname* { m i n } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 309, + 696, + 371 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Therefore, an obvious way to increase this variance is to maximize the smallest eigenvalue of Var $\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)$ , which can be formulated as ", + "bbox": [ + 173, + 375, + 825, + 404 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d430e1e503420db590a293421a6ff423942fda889c72749104eed69bbfbf1637.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\phi } { \\mathrm { m a x i m i z e } } ~ \\mathbb { E } _ { { \\mathbf s } , { \\mathbf a } \\sim \\mathcal { D } } \\left[ \\lambda _ { \\mathrm { m i n } } \\left( { \\mathrm { V a r } \\left( { \\nabla _ { { \\mathbf a } } { Q _ { \\phi _ { j } } } \\left( { \\mathbf s } , { \\mathbf a } \\right) } \\right) } \\right) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 409, + 663, + 436 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\phi$ denotes the collection of the parameters $\\{ \\phi _ { j } \\} _ { j = 1 } ^ { N }$ . There are several methods to compute the smallest eigenvalue, such as the power method or the QR algorithm [27]. However, these iterative methods require constructing huge computation graphs, which makes optimizing the eigenvalue using back-propagation inefficient. Instead, we aim to maximize the sum of all eigenvalues, which is equal to the total variance. By Lemma 1, it is equivalent to minimizing the norm of the average gradients: ", + "bbox": [ + 174, + 441, + 825, + 513 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a2ca5fc1796f265830d4bb41d4f48e96f8e49c295e79a3291b4b822c1b02d1a5.jpg", + "text": "$$\n\\underset { \\phi } { \\mathrm { m i n i m i z e } } \\ \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } \\left[ \\left. \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 517, + 725, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "With simple modification, we can reformulate Equation (4) as diversifying the gradients of each Q-function network for in-distribution actions: ", + "bbox": [ + 174, + 571, + 823, + 599 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0bb016ef1444d7ea26e6acd18f0bbb0053254181f6d9f0db8777ddf9750ab08f.jpg", + "text": "$$\n\\operatorname* { m i n i m i z e } J _ { \\mathrm { E S } } ( Q _ { \\phi } ) : = \\mathbb { E } _ { { \\bf s } , { \\bf a } \\sim \\mathcal { D } } \\left[ \\frac { 1 } { N - 1 } \\sum _ { 1 \\leq i \\neq j \\leq N } \\underbrace { \\left. \\nabla _ { { \\bf a } } Q _ { \\phi _ { i } } ( { \\bf s } , { \\bf a } ) , \\nabla _ { { \\bf a } } Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\right. } _ { \\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( { \\bf s } , { \\bf a } ) } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 603, + 772, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Concretely, our final objective can be interpreted as measuring the pairwise alignment of the gradients using cosine similarity, which we denote as the Ensemble Similarity (ES) metric $\\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )$ , and minimizing the ES values for every pair in the Q-ensemble with regard to the dataset state-actions. The illustration of the ensemble gradient diversification is shown in Figure 3b. Note that we instead maximize the total variance to reduce the computational burden. Nevertheless, the modified objective is closely related to maximizing the smallest eigenvalue. The detailed explanation can be found in Appendix A.2. ", + "bbox": [ + 173, + 672, + 826, + 771 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We name the resulting actor-critic algorithm as Ensemble-Diversified Actor Critic (EDAC) and present the detailed procedure in Algorithm 1 (differences with the original SAC algorithm marked in blue). Note that Algorithm 1 reduces to SAC- $N$ when $\\eta = 0$ , and further reduces to vanilla SAC when also $N = 2$ . ", + "bbox": [ + 173, + 776, + 825, + 833 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 852, + 312, + 869 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate our proposed methods against the previous offline RL algorithms on the standard D4RL benchmark [9] . Concretely, we perform our evaluation on MuJoCo Gym (Section 5.1) and Adroit ", + "bbox": [ + 173, + 882, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: Initialize policy parameters $\\theta$ , Q-function parameters $\\{ \\phi _ { j } \\} _ { j = 1 } ^ { N }$ , target Q-function parameters $\\{ \\phi _ { j } ^ { \\prime } \\} _ { j = 1 } ^ { N }$ , and offline data replay buffer $\\mathcal { D }$ ", + "bbox": [ + 179, + 111, + 823, + 143 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2: repeat ", + "text_level": 1, + "bbox": [ + 181, + 145, + 248, + 156 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3: Sample a mini-batch $B = \\{ ( \\mathbf { s } , \\mathbf { a } , r , \\mathbf { s } ^ { \\prime } ) \\}$ from $\\mathcal { D }$ ", + "bbox": [ + 181, + 156, + 540, + 171 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4: Compute target Q-values (shared by all Q-functions): ", + "bbox": [ + 181, + 171, + 576, + 185 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f264f5f645761a2e88dccd0131b38c54447c995b1d5999ab75da53b1cd57ea94.jpg", + "text": "$$\ny ( r , \\mathbf { s } ^ { \\prime } ) = r + \\gamma \\left( \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { \\phi _ { j } ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } ^ { \\prime } \\mid \\mathbf { s } ^ { \\prime } \\right) \\right) , \\quad \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } \\left( \\cdot \\mid \\mathbf { s } ^ { \\prime } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 251, + 191, + 772, + 227 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5: Update each Q-function $Q _ { \\phi _ { i } }$ with gradient descent using ", + "bbox": [ + 179, + 232, + 599, + 247 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/b833b643b112d74a084ffa5b751d8c9bebebf7e2a36916e2630f4ee6f0f9cfcf.jpg", + "text": "$$\n\\nabla _ { \\phi _ { i } } \\frac { 1 } { \\left| B \\right| } \\sum _ { ( \\mathbf { s } , \\mathbf { a } , r , \\mathbf { s } ^ { \\prime } ) \\in B } \\left( \\left( Q _ { \\phi _ { i } } \\left( \\mathbf { s } , \\mathbf { a } \\right) - y \\left( r , \\mathbf { s } ^ { \\prime } \\right) \\right) ^ { 2 } + \\frac { \\eta } { N - 1 } \\sum _ { 1 \\leq i \\neq j \\leq N } \\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 240, + 253, + 782, + 304 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6: Update policy with gradient ascent using ", + "bbox": [ + 178, + 309, + 496, + 324 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/50634f0dba1636e83fac269bb89a7471b5b1f3e86cfe253f4a7ca489768ac4d4.jpg", + "text": "$$\n\\nabla _ { \\boldsymbol { \\theta } } \\frac { 1 } { \\left| \\boldsymbol { B } \\right| } \\sum _ { \\mathbf { s } \\in \\boldsymbol { B } } \\left( \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { \\phi _ { j } } \\left( \\mathbf { s } , \\tilde { \\mathbf { a } } _ { \\boldsymbol { \\theta } } ( \\mathbf { s } ) \\right) - \\beta \\log \\pi _ { \\boldsymbol { \\theta } } \\left( \\tilde { \\mathbf { a } } _ { \\boldsymbol { \\theta } } ( \\mathbf { s } ) \\mid \\mathbf { s } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 330, + 728, + 369 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\tilde { \\mathbf { a } } _ { \\theta } ( \\mathbf { s } )$ is a sample from $\\pi _ { \\boldsymbol { \\theta } } ( \\cdot \\mid \\mathbf { s } )$ which is differentiable w.r.t. $\\theta$ via the reparametrization trick. ", + "bbox": [ + 220, + 376, + 823, + 405 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7: Update target networks with $\\phi _ { i } ^ { \\prime } \\rho \\phi _ { i } ^ { \\prime } + ( 1 - \\rho ) \\phi _ { i }$ (Section 5.2) domains. We consider the following baselines: SAC, the backbone algorithm of our method, CQL, the previous state-of-the-art on the D4RL benchmark, REM [2], an offline RL method which utilized Q-network ensemble on discrete control environments, and BC, the behavior cloning method. We evaluate each method under the normalized average return metric where the average return is scaled such that 0 and 100 each equals the performance of a random policy and an online expert policy. In addition to the performance evaluation, we compare the computational cost of each method (Section 5.3). For the implementation details of our algorithm and the baselines, please refer to Appendix B and C. Also, we provide more experiments such as comparison with more baselines, CQL with $N$ Q-networks, and hyperparameter sensitivity from Appendix E to H. The code is available online3. ", + "bbox": [ + 181, + 406, + 566, + 422 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 454, + 825, + 592 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 Evaluation on D4RL MuJoCo Gym tasks ", + "text_level": 1, + "bbox": [ + 173, + 609, + 500, + 625 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first evaluate each method on D4RL MuJoCo Gym tasks which consist of three environments, halfcheetah, hopper, and walker2d, each with six datasets from different data-collecting policies. In detail, the considered policies are random: a uniform random policy, expert: a fully trained online expert, medium: a suboptimal policy with approximately 1/3 the performance of the expert, medium-expert: a mixture of medium and expert policies, medium-replay: the replay buffer of a policy trained up to the performance of the medium agent, and full-replay: the final replay buffer of the expert policy. Each dataset consists of 1M transitions except for medium-expert and medium-replay. ", + "bbox": [ + 174, + 635, + 826, + 733 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The experiment results in Table 1 show EDAC and SAC- $N$ both outperform or are competitive with the previous state-of-the-art on all of the tasks considered. Notably, the performance gap is especially high for random, medium, and medium-replay datasets, where the performances of the previous works are relatively low. Both the proposed methods achieve average normalized scores over 80, reducing the gap with the online expert by $40 \\%$ compared to CQL. While the performance of EDAC is marginally better than the performance of SAC- $N$ , EDAC achieves this result with a much smaller Q-ensemble size. As noted in Figure 5, on hopper tasks, SAC- $N$ requires 200 to $5 0 0 \\mathrm { Q }$ -networks, while EDAC requires less than 50. ", + "bbox": [ + 174, + 738, + 826, + 849 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 6 compares the distance between the actions chosen by each method and the dataset actions. Concretely, we measure $\\mathbb { E } _ { ( \\mathbf { s } , \\mathbf { a } ) \\sim \\mathcal { D } , \\hat { \\mathbf { a } } \\sim \\pi _ { \\boldsymbol { \\theta } } ( \\cdot | \\mathbf { s } ) } [ | \\hat { \\mathbf { a } } - \\mathbf { a } | | _ { 2 } ^ { 2 } ]$ for EDAC, SAC- $N$ , CQL, SAC-2, and a random policy on $^ *$ -medium datasets. We find that our proposed methods choose from a more diverse range of actions compared to CQL. This shows the advantage of the uncertainty-based penalization which considers the prediction confidence other than penalizing all OOD actions. ", + "bbox": [ + 174, + 856, + 821, + 886 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 823, + 133 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/dd4937790ed8c5038c622f13f8b529536577c928f7394753ebe1dca88c2e80f1.jpg", + "table_caption": [ + "Table 1: Normalized average returns on D4RL Gym tasks, averaged over 4 random seeds. CQL (Paper) denotes the results reported in the original paper. " + ], + "table_footnote": [], + "table_body": "
Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
halfcheetah-random2.2±0.029.7±1.4-0.8±1.135.431.3±3.528.0±0.928.4±1.0
halfcheetah-medium43.2±0.655.2±27.8-0.8±1.344.446.9±0.467.5±1.265.9±0.6
halfcheetah-expert91.8±1.5-0.8±1.84.1±5.7104.897.3±1.1105.2±2.6106.8±3.4
halfcheetah-medium-expert44.0±1.628.4±19.40.7±3.762.495.0±1.4107.1±2.0106.3±1.9
halfcheetah-medium-replay37.6±2.10.8±1.06.6±11.046.245.3±0.363.9±0.861.3±1.9
halfcheetah-full-replay62.9±0.886.8±1.027.8±35.4-76.9±0.984.5±1.284.6±0.9
hopper-random3.7±0.69.9±1.53.4±2.210.85.3±0.631.3±0.025.3±10.4
hopper-medium54.1±3.80.8±0.00.7±0.086.661.9±6.4100.3±0.3101.6±0.6
hopper-expert107.7±9.70.7±0.00.8±0.0109.9106.5±9.1110.3±0.3110.1±0.1
hopper-medium-expert53.9±4.70.7±0.00.8±0.0111.096.9±15.1110.1±0.3110.7±0.1
hopper-medium-replay16.6±4.87.4±0.527.5±15.248.686.3±7.3101.8±0.5101.0±0.5
hopper-full-replay19.9±12.941.1±17.919.7±24.6-101.9±0.6102.9±0.3105.4±0.7
walker2d-random1.3±0.10.9±0.86.9±8.37.05.4±1.721.7±0.016.6±7.0
walker2d-medium70.9±11.0-0.3±0.20.2±0.774.579.5±3.287.9±0.292.5±0.8
walker2d-expert108.7±0.20.7±0.31.0±2.3121.6109.3±0.1107.4±2.4115.1±1.9
walker2d-medium-expert90.1±13.21.9±3.9-0.1±0.098.7109.1±0.2116.7±0.4114.7±0.9
walker2d-medium-replay20.3±9.8-0.4±0.312.5±6.232.676.8±10.078.7±0.787.1±2.3
walker2d-full-replay68.8±17.727.9±47.3-0.2±0.3-94.2±1.994.6±0.599.8±0.7
Average49.916.26.2-73.784.585.2
", + "bbox": [ + 173, + 185, + 826, + 440 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/8cc13d52b0b3edcc798dadfe4273b9dde139c1cec3b4cc0c697de114ee27b598.jpg", + "image_caption": [ + "Figure 5: Minimum number of Q-ensembles $( N )$ required to achieve the performance reported in Table 1. M-E denotes medium-expert. We omit the results of medium-replay and full-replay as SAC- $. N$ already works well with a small number of ensembles (less than or equal to 5). For more details of the experiment, please refer to Appendix C. " + ], + "image_footnote": [], + "bbox": [ + 178, + 473, + 823, + 613 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1116de177c49343d5cd95d790cc9fd29db3d3f213b6f83a11e54c2cb38f4d694.jpg", + "image_caption": [ + "Figure 6: Histograms of the distances between the actions from each methods (EDAC, SAC- $N$ , CQL, SAC-2, and a random policy) and the actions from the dataset. For more details of the experiment, please refer to Appendix C. " + ], + "image_footnote": [], + "bbox": [ + 184, + 713, + 813, + 848 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 Evaluation on D4RL Adroit tasks ", + "text_level": 1, + "bbox": [ + 176, + 90, + 446, + 106 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also experiment on the more complex D4RL Adroit tasks that require controlling a 24-DoF robotic hand to perform tasks such as aligning a pen, hammering a nail, opening a door, or relocating a ball. We use two types of datasets for each environment: human, containing 25 trajectories of human demonstrations, and cloned, a 50-50 mixture between the demonstration data and the behavioral cloned policy on the demonstrations. Note that for the Adroit tasks, we could not reproduce the CQL results from the paper completely. For the detailed procedure of reproducing the results of CQL, please refer to Appendix D. ", + "bbox": [ + 173, + 116, + 826, + 214 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/3a3b89599957f84a92ce6ed692ce6916ae6000d7157a0a6ffbc708641159dd62.jpg", + "table_caption": [ + "Table 2: Normalized average returns on D4RL Adroit tasks, averaged over 4 random seeds. " + ], + "table_footnote": [], + "table_body": "
Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
pen-human25.8±8.84.3±3.85.4±4.355.835.2±6.69.5±1.152.1±8.6
hammer-human3.1±3.20.2±0.00.3±0.02.10.6±0.50.3±0.00.8±0.4
door-human2.8±0.7-0.3±0.0-0.3±0.09.11.2±1.8-0.3±0.010.7±6.8
relocate-human0.0±0.0-0.3±0.0-0.3±0.00.350.0±0.0-0.1±0.10.1±0.1
pen-cloned38.3±11.9-0.8±3.2-1.0±0.140.327.2±11.364.1±8.768.2±7.3
hammer-cloned0.7±0.30.1±0.1-0.3±0.05.71.4±2.10.2±0.20.3±0.0
door-cloned0.0±0.0-0.3±0.1-0.3±0.03.52.4±2.4-0.3±0.09.6±8.3
relocate-cloned0.1±0.0-0.1±0.1-0.2±0.2-0.10.0±0.00.0±0.00.0±0.0
", + "bbox": [ + 173, + 256, + 825, + 401 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The evaluation results are summarized in Table 2. For pen- $^ { \\ast }$ tasks, where the considered algorithms achieve meaningful performance, EDAC outperforms or matches with the previous state-of-the-art. Especially, for pen-cloned, both EDAC and SAC- $. N$ achieve $7 5 \\%$ higher score compared to CQL. Unlike the results from the Gym tasks, we find that SAC- $N$ falls behind in some datasets, for example, pen-human, which could in part due to the size of the dataset being exceptionally small (5000 transitions). However, our method with ensemble diversification successfully overcomes this difficulty. ", + "bbox": [ + 173, + 424, + 825, + 522 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 Computational cost comparison ", + "text_level": 1, + "bbox": [ + 176, + 537, + 433, + 553 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We compared the computational cost of our methods with vanilla SAC and CQL on hopper-medium-v2, where our methods require the largest number of Q-networks. For each method, we measure the runtime per training epoch (1000 gradient steps) along with GPU memory consumption. We run our experiments on a single machine with one RTX 3090 GPU and provide the results in Table 3. ", + "bbox": [ + 174, + 563, + 549, + 660 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/f1c501c86effbba26588f9524287241b6686582cec6f2ed75bb579342e8318c4.jpg", + "table_caption": [ + "Table 3: Computational costs of each method. " + ], + "table_footnote": [], + "table_body": "
Runtime (s/epoch)GPU Mem. (GB)
SAC21.41.3
CQL38.21.4
SAC-50044.15.1
EDAC30.81.8
", + "bbox": [ + 568, + 601, + 813, + 699 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As the result shows, our method EDAC runs faster than CQL with comparable memory consumption. Note that CQL is about twice as slower than vanilla SAC due to the additional computations for Q-value regularization (e.g., dual update and approximate logsumexp via sampling). Meanwhile, the inference to the Q-network ensemble in SAC- $N$ and EDAC is embarrassingly parallelizable, minimizing the runtime increase with the number of Q-networks. Also, we emphasize that our gradient diversification term in Equation (4) has linear computational complexity, as we can reformulate the term using the sum of the gradients. ", + "bbox": [ + 174, + 666, + 549, + 708 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 708, + 825, + 779 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Related Works ", + "text_level": 1, + "bbox": [ + 174, + 796, + 330, + 814 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Model-free offline RL A popular approach for offline RL is to regularize the learned policy to be close to the behavior policy where the offline dataset was collected. BCQ [11] uses a generative model to produce actions with high similarity to the dataset and trains a restricted policy to choose the best action from the neighborhood of the generated actions. Another line of work, such as BEAR [15] or BRAC [28], stabilizes policy learning by penalizing the divergence from the dataset measured by KL divergence or MMD. While these policy-constraint methods demonstrate high performance on datasets from expert behavior policies, they fail to find optimal policies from datasets with suboptimal policies due to the strict policy constraints [9]. Also, these methods require an accurate estimation of the behavior policy, which might be difficult in complex settings with multiple behavior sources or high-dimensional environments. To address these issues, CQL [16] directly regularizes Q-functions by introducing a term that minimizes the Q-values for out-of-distribution actions and maximizes the Q-values for in-distribution actions. Without such explicit regularizations, REM [2] proposes to use a random convex combination of Q-network ensembles on environments with discrete action spaces [4]. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 202 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Estimation bias in Q-learning While Q-learning is one of the most popular algorithms in reinforcement learning, it suffers from overestimation bias due to the maximum operation $\\mathrm { m a x } _ { \\mathbf { a } ^ { \\prime } \\in \\mathcal { A } } Q ( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } )$ used during Q-function updates [10, 25]. This overestimation bias, together with the bootstrapping, can lead to a catastrophic build-up of errors during the Q-learning process. To resolve this issue, TD3 [10] introduces a clipped version of Double Q-learning [25] that takes the minimum value of two critics. Subsequently, Maxmin Q-learning [17] theoretically shows that the overestimation bias can be controlled by the number of ensembles in the clipped Q-learning. The overestimation problem in Q-learning can be exacerbated in the offline setting since the extrapolation error cannot be corrected with further interactions with the environment, and existing offline RL algorithms handle the bias by introducing constrained policy optimization [11, 15] or conservative Q-learning frameworks [16]. ", + "bbox": [ + 174, + 222, + 825, + 359 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Uncertainty measures in RL Uncertainty estimates have been widely used in RL for various purposes including exploration, Q-learning, and planning. Bootstrapped DQN [21] leverages an ensemble of Q-functions to quantify the uncertainty of the Q-value, and utilizes it for efficient exploration. Following this work, the UCB exploration algorithm [5] constructs an upper confidence bound [3] of the Q-values using the empirical mean and standard deviation of Q-ensembles, which is used to promote efficient exploration by applying the principle of optimism in the face of uncertainty [7]. Osband et al. [22] proposes a randomly initialized Q-ensemble that reflects the concept of prior functions in Bayesian inference and Abbas et al. [1] introduces an uncertainty incorporated planning with imperfect models. The notion of uncertainty has also been considered in offline RL, mostly in the framework of model-based offline RL. Especially, MOPO [29] and MOReL [13] measure the uncertainty of the model’s prediction to formulate an uncertainty-penalized policy optimization problem in the offline RL setting. These methods introduce an ensemble of dynamics models for the quantification of the uncertainty, whereas our work adopts an ensemble of Q-functions for uncertainty-aware Q-learning. ", + "bbox": [ + 174, + 380, + 825, + 574 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 597, + 297, + 613 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have shown that clipped Q-learning can be efficiently leveraged to construct an uncertaintybased offline RL method that outperforms previous methods on various datasets. Based on this observation, we proposed Ensemble-Diversifying Actor-Critic (EDAC) that effectively reduces the required number of ensemble networks for quantifying and penalizing the epistemic uncertainty. Our method does not require any explicit estimation of the data collecting policy or sampling from the out-of-distribution data and respects the epistemic uncertainty of each data point during penalization. EDAC, while requiring up to $90 \\%$ less number of ensemble networks compared to the vanilla Q-ensemble, exhibits state-of-the-art performance on various datasets. ", + "bbox": [ + 174, + 631, + 825, + 742 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 766, + 338, + 782 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported in part by Samsung Advanced Institute of Technology, Samsung Electronics Co., Ltd., Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00882, (SW STAR LAB) Development of deployable learning intelligence via self-sustainable and trustworthy machine learning and No. 2019- 0-01371, Development of brain-inspired AI with human-like intelligence), and Research Resettlement Fund for the new faculty of Seoul National University. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA2386-20-1-4043. Hyun Oh Song is the corresponding author. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 90, + 266, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Zaheer Abbas, Samuel Sokota, Erin Talvitie, and Martha White. Selective dyna-style planning under limited model capacity. In ICML, 2020. \n[2] Rishabh Agarwal, Dale Schuurmans, and Mohammad Norouzi. An optimistic perspective on offline reinforcement learning. In ICML, 2020. \n[3] Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Exploration–exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science, 410(19): 1876–1902, 2009. \n[4] Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013. \n[5] Richard Y Chen, Szymon Sidor, Pieter Abbeel, and John Schulman. 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This trial-and-error procedure can be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 507, + 612 + ], + "score": 1.0, + "content": "prohibitive when scaling RL to real-world applications such as autonomous driving and healthcare,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "as exploratory actions can cause critical damage to the agent or the environment [19]. 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To", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "score": 1.0, + "content": "this end, offline RL algorithms adopt either a constraint or a penalty term that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 333, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 346 + ], + "score": 1.0, + "content": "explicitly guides the policy to stay close to the given dataset. However, prior", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 343, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 469, + 357 + ], + "score": 1.0, + "content": "methods typically require accurate estimation of the behavior policy or sampling", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 355, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 470, + 367 + ], + "score": 1.0, + "content": "from OOD data points, which themselves can be a non-trivial problem. Moreover,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "these methods under-utilize the generalization ability of deep neural networks", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 377, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 377, + 469, + 389 + ], + "score": 1.0, + "content": "and often fall into suboptimal solutions too close to the given dataset. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 388, + 469, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 400 + ], + "score": 1.0, + "content": "work, we propose an uncertainty-based offline RL method that takes into account", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "the confidence of the Q-value prediction and does not require any estimation or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 409, + 470, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 470, + 423 + ], + "score": 1.0, + "content": "sampling of the data distribution. We show that the clipped Q-learning, a technique", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "widely used in online RL, can be leveraged to successfully penalize OOD data", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 431, + 470, + 444 + ], + "spans": [ + { + "bbox": [ + 141, + 431, + 470, + 444 + ], + "score": 1.0, + "content": "points with high prediction uncertainties. Surprisingly, we find that it is possible to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 441, + 470, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 441, + 470, + 455 + ], + "score": 1.0, + "content": "substantially outperform existing offline RL methods on various tasks by simply", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 453, + 470, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 470, + 465 + ], + "score": 1.0, + "content": "increasing the number of Q-networks along with the clipped Q-learning. Based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 465, + 470, + 477 + ], + "spans": [ + { + "bbox": [ + 142, + 465, + 470, + 477 + ], + "score": 1.0, + "content": "on this observation, we propose an ensemble-diversified actor-critic algorithm that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 475, + 469, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 469, + 487 + ], + "score": 1.0, + "content": "reduces the number of required ensemble networks down to a tenth compared to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "score": 1.0, + "content": "the naive ensemble while achieving state-of-the-art performance on most of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 497, + 268, + 508 + ], + "spans": [ + { + "bbox": [ + 142, + 497, + 268, + 508 + ], + "score": 1.0, + "content": "D4RL benchmarks considered.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 17.5, + "bbox_fs": [ + 141, + 287, + 470, + 508 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 529, + 190, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 192, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 192, + 546 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 506, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "Over the recent years, deep reinforcement learning (deep RL) has achieved considerable success", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "in various domains such as robotics [20], recommendation systems [6], and strategy games [26].", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "However, a major drawback of RL algorithms is that they adopt an active learning procedure, where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "training steps require active interactions with the environment. This trial-and-error procedure can be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 507, + 612 + ], + "score": 1.0, + "content": "prohibitive when scaling RL to real-world applications such as autonomous driving and healthcare,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "as exploratory actions can cause critical damage to the agent or the environment [19]. 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Typically, if the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "coverage of the dataset is not sufficient, vanilla RL algorithms suffer severely from extrapolation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "error, overestimating the Q-values of out-of-distribution (OOD) state-action pairs [15]. To this end,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "most offline RL methods apply some constraints or penalty terms on top of the existing RL algorithms", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "to enforce the learning process to be more conservative. For example, some prior works explicitly", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "regularize the policy to be close to the behavior policy that was used to collect the data [11, 15]. A", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "more recent work instead penalizes the Q-values of OOD state-action pairs to enforce the Q-values to", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 210, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 210, + 140 + ], + "score": 1.0, + "content": "be more pessimistic [16].", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 38, + "bbox_fs": [ + 107, + 649, + 505, + 683 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "error, overestimating the Q-values of out-of-distribution (OOD) state-action pairs [15]. To this end,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "most offline RL methods apply some constraints or penalty terms on top of the existing RL algorithms", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "to enforce the learning process to be more conservative. For example, some prior works explicitly", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "regularize the policy to be close to the behavior policy that was used to collect the data [11, 15]. A", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "more recent work instead penalizes the Q-values of OOD state-action pairs to enforce the Q-values to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 210, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 210, + 140 + ], + "score": 1.0, + "content": "be more pessimistic [16].", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "While these methods achieve significant performance gains over vanilla RL methods, they either", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 104, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "require an estimation of the behavior policy or explicit sampling from OOD data points, which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "themselves can be non-trivial to solve. Furthermore, these methods do not utilize the generalization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "ability of the Q-function networks and prohibit the agent from approaching any OOD state-actions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "without any consideration on whether they are good or bad. However, if we can identify OOD data", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "points where we can predict their Q-values with high confidence, it is more effective not to restrain", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 278, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 278, + 222 + ], + "score": 1.0, + "content": "the agent from choosing those data points.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "From this intuition, we propose an uncertainty-based model-free offline RL method that effectively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "quantifies the uncertainty of the Q-value estimates by an ensemble of Q-function networks and does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "not require any estimation or sampling of the data distribution. To achieve this, we first show that a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "well-known technique from online RL, the clipped Q-learning [10], can be successfully leveraged as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "an uncertainty-based penalization term. Our experiments reveal that we can achieve state-of-the-art", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "performance on various offline RL tasks by solely using this technique with increased ensemble", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "size. To further improve the practical usability of the method, we develop an ensemble diversifying", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "objective that significantly reduces the number of required ensemble networks. We evaluate our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "proposed method on D4RL benchmarks [9] and verify that the proposed method outperforms the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 465, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 465, + 336 + ], + "score": 1.0, + "content": "previous state-of-the-art by a large margin on various types of environments and datasets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 195, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 348, + 196, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 196, + 366 + ], + "score": 1.0, + "content": "2 Preliminaries", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "We consider an environment formulated as a Markov Decision Process (MDP) defined by a tuple", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 177, + 398 + ], + "score": 0.91, + "content": "( S , A , T , r , d _ { 0 } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 386, + 209, + 399 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 210, + 387, + 218, + 396 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 386, + 295, + 399 + ], + "score": 1.0, + "content": "is the state space,", + "type": "text" + }, + { + "bbox": [ + 295, + 387, + 305, + 397 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 386, + 388, + 399 + ], + "score": 1.0, + "content": "is the action space,", + "type": "text" + }, + { + "bbox": [ + 388, + 387, + 437, + 398 + ], + "score": 0.91, + "content": "T ( \\mathbf { s } ^ { \\prime } \\mid \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "is the transition", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 206, + 410 + ], + "score": 1.0, + "content": "probability distribution,", + "type": "text" + }, + { + "bbox": [ + 206, + 398, + 273, + 408 + ], + "score": 0.91, + "content": "r : S \\times \\mathcal { A } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 397, + 369, + 410 + ], + "score": 1.0, + "content": "is the reward function,", + "type": "text" + }, + { + "bbox": [ + 369, + 398, + 380, + 408 + ], + "score": 0.87, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "is the initial state distribution,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 124, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 408, + 169, + 420 + ], + "score": 0.91, + "content": "\\gamma \\in \\mathsf { \\Gamma } ( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "is the discount factor. The goal of reinforcement learning is to find an optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 413, + 508, + 436 + ], + "spans": [ + { + "bbox": [ + 102, + 413, + 134, + 436 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 135, + 420, + 167, + 431 + ], + "score": 0.87, + "content": "\\pi ( \\mathbf { a } \\mid \\mathbf { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 413, + 372, + 436 + ], + "score": 1.0, + "content": "that maximizes the cumulative discounted reward", + "type": "text" + }, + { + "bbox": [ + 373, + 419, + 474, + 432 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 413, + 508, + 436 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 315, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 150, + 443 + ], + "score": 0.86, + "content": "\\mathbf { s } _ { 0 } \\sim d _ { 0 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 430, + 154, + 443 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 154, + 430, + 210, + 442 + ], + "score": 0.87, + "content": "\\mathbf { a } _ { t } \\sim \\pi ( \\cdot \\mid \\mathbf { s } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 430, + 231, + 443 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 231, + 430, + 311, + 443 + ], + "score": 0.92, + "content": "\\mathbf { s } _ { t + 1 } \\sim T ( \\cdot \\mid \\mathbf { s } _ { t } , \\mathbf { a } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 430, + 315, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "One of the major approaches for obtaining such a policy is Q-learning [12, 20] which learns a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 214, + 470 + ], + "score": 1.0, + "content": "state-action value function", + "type": "text" + }, + { + "bbox": [ + 214, + 457, + 250, + 469 + ], + "score": 0.93, + "content": "Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "parameterized by a neural network that represents the expected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "cumulative discounted reward when starting from state s and action a. Standard actor-critic approach", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 362, + 494 + ], + "score": 1.0, + "content": "[14] learns this Q-function by minimizing the Bellman residual", + "type": "text" + }, + { + "bbox": [ + 362, + 479, + 474, + 493 + ], + "score": 0.92, + "content": "\\big ( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) \\big ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 480, + 506, + 494 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 491, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 107, + 492, + 369, + 506 + ], + "score": 0.78, + "content": "B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) = \\mathbb { E } _ { \\mathbf { s } ^ { \\prime } \\sim T ( \\cdot | \\mathbf { s } , \\mathbf { a } ) } \\left[ r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } Q _ { \\phi } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 491, + 506, + 508 + ], + "score": 1.0, + "content": "is the Bellman operator. In the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 407, + 516 + ], + "score": 1.0, + "content": "context of offline RL, where transitions are sampled from a static dataset", + "type": "text" + }, + { + "bbox": [ + 407, + 505, + 416, + 514 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 503, + 505, + 516 + ], + "score": 1.0, + "content": ", the objective for the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 514, + 239, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 115, + 527 + ], + "score": 0.26, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 514, + 239, + 528 + ], + "score": 1.0, + "content": "-network becomes minimizing", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 531, + 470, + 559 + ], + "lines": [ + { + "bbox": [ + 139, + 531, + 470, + 559 + ], + "spans": [ + { + "bbox": [ + 139, + 531, + 470, + 559 + ], + "score": 0.92, + "content": "J _ { q } ( Q _ { \\phi } ) : = \\mathbb { E } _ { ( \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ Q _ { \\phi ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] ,", + "type": "interline_equation", + "image_path": "dd72bed5b712c6f643e4a71f2f1a93fde3332a7b416d74c86a8f6308169989dc.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 139, + 531, + 470, + 540.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 139, + 540.3333333333334, + 470, + 549.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 139, + 549.6666666666667, + 470, + 559.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 562, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 133, + 578 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 565, + 149, + 577 + ], + "score": 0.92, + "content": "Q _ { \\phi ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 562, + 506, + 578 + ], + "score": 1.0, + "content": "represents the target Q-network softly updated for algorithmic stability [20]. The policy,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "which is also parameterized by a neural network, is updated in an alternating fashion to maximize the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 586, + 338, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 144, + 601 + ], + "score": 1.0, + "content": "expected", + "type": "text" + }, + { + "bbox": [ + 144, + 587, + 153, + 597 + ], + "score": 0.33, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 586, + 182, + 601 + ], + "score": 1.0, + "content": "-value:", + "type": "text" + }, + { + "bbox": [ + 182, + 586, + 334, + 599 + ], + "score": 0.91, + "content": "J _ { p } ( \\pi _ { \\theta } ) : = \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi _ { \\theta } ( \\cdot | { \\mathbf s } ) } \\left[ Q _ { \\phi } ( { \\mathbf s } , { \\mathbf a } ) \\right] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 586, + 338, + 601 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 392, + 617 + ], + "score": 1.0, + "content": "However, as the policy is updated to maximize the Q-values, the actions", + "type": "text" + }, + { + "bbox": [ + 392, + 604, + 402, + 614 + ], + "score": 0.84, + "content": "\\mathbf { a } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "sampled from the current", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "policy in Equation (1) can be biased towards OOD actions with erroneously high Q-values. In the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "offline RL setting, such errors cannot be corrected by feedback from the environment as in online", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "RL. To handle the error propagation from these OOD actions, most offline RL algorithms regularize", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "either the policy [11, 15] or the Q-function [16] to be biased towards the given dataset. However, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "policy regularization methods typically require an accurate estimation of the behavior policy. 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Furthermore, these methods do not utilize the generalization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "ability of the Q-function networks and prohibit the agent from approaching any OOD state-actions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "without any consideration on whether they are good or bad. However, if we can identify OOD data", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "points where we can predict their Q-values with high confidence, it is more effective not to restrain", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 278, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 278, + 222 + ], + "score": 1.0, + "content": "the agent from choosing those data points.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 142, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "From this intuition, we propose an uncertainty-based model-free offline RL method that effectively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "quantifies the uncertainty of the Q-value estimates by an ensemble of Q-function networks and does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "not require any estimation or sampling of the data distribution. To achieve this, we first show that a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "well-known technique from online RL, the clipped Q-learning [10], can be successfully leveraged as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "an uncertainty-based penalization term. Our experiments reveal that we can achieve state-of-the-art", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "performance on various offline RL tasks by solely using this technique with increased ensemble", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "size. To further improve the practical usability of the method, we develop an ensemble diversifying", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "objective that significantly reduces the number of required ensemble networks. We evaluate our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "proposed method on D4RL benchmarks [9] and verify that the proposed method outperforms the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 465, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 465, + 336 + ], + "score": 1.0, + "content": "previous state-of-the-art by a large margin on various types of environments and datasets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 225, + 506, + 336 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 195, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 348, + 196, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 196, + 366 + ], + "score": 1.0, + "content": "2 Preliminaries", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "We consider an environment formulated as a Markov Decision Process (MDP) defined by a tuple", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 177, + 398 + ], + "score": 0.91, + "content": "( S , A , T , r , d _ { 0 } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 386, + 209, + 399 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 210, + 387, + 218, + 396 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 386, + 295, + 399 + ], + "score": 1.0, + "content": "is the state space,", + "type": "text" + }, + { + "bbox": [ + 295, + 387, + 305, + 397 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 386, + 388, + 399 + ], + "score": 1.0, + "content": "is the action space,", + "type": "text" + }, + { + "bbox": [ + 388, + 387, + 437, + 398 + ], + "score": 0.91, + "content": "T ( \\mathbf { s } ^ { \\prime } \\mid \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "is the transition", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 206, + 410 + ], + "score": 1.0, + "content": "probability distribution,", + "type": "text" + }, + { + "bbox": [ + 206, + 398, + 273, + 408 + ], + "score": 0.91, + "content": "r : S \\times \\mathcal { A } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 397, + 369, + 410 + ], + "score": 1.0, + "content": "is the reward function,", + "type": "text" + }, + { + "bbox": [ + 369, + 398, + 380, + 408 + ], + "score": 0.87, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "is the initial state distribution,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 124, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 408, + 169, + 420 + ], + "score": 0.91, + "content": "\\gamma \\in \\mathsf { \\Gamma } ( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "is the discount factor. The goal of reinforcement learning is to find an optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 413, + 508, + 436 + ], + "spans": [ + { + "bbox": [ + 102, + 413, + 134, + 436 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 135, + 420, + 167, + 431 + ], + "score": 0.87, + "content": "\\pi ( \\mathbf { a } \\mid \\mathbf { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 413, + 372, + 436 + ], + "score": 1.0, + "content": "that maximizes the cumulative discounted reward", + "type": "text" + }, + { + "bbox": [ + 373, + 419, + 474, + 432 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( { \\mathbf { s } } _ { t } , { \\mathbf { a } } _ { t } ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 413, + 508, + 436 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 315, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 150, + 443 + ], + "score": 0.86, + "content": "\\mathbf { s } _ { 0 } \\sim d _ { 0 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 430, + 154, + 443 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 154, + 430, + 210, + 442 + ], + "score": 0.87, + "content": "\\mathbf { a } _ { t } \\sim \\pi ( \\cdot \\mid \\mathbf { s } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 430, + 231, + 443 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 231, + 430, + 311, + 443 + ], + "score": 0.92, + "content": "\\mathbf { s } _ { t + 1 } \\sim T ( \\cdot \\mid \\mathbf { s } _ { t } , \\mathbf { a } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 430, + 315, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 102, + 374, + 508, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "One of the major approaches for obtaining such a policy is Q-learning [12, 20] which learns a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 214, + 470 + ], + "score": 1.0, + "content": "state-action value function", + "type": "text" + }, + { + "bbox": [ + 214, + 457, + 250, + 469 + ], + "score": 0.93, + "content": "Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "parameterized by a neural network that represents the expected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "cumulative discounted reward when starting from state s and action a. Standard actor-critic approach", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 362, + 494 + ], + "score": 1.0, + "content": "[14] learns this Q-function by minimizing the Bellman residual", + "type": "text" + }, + { + "bbox": [ + 362, + 479, + 474, + 493 + ], + "score": 0.92, + "content": "\\big ( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) \\big ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 480, + 506, + 494 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 491, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 107, + 492, + 369, + 506 + ], + "score": 0.78, + "content": "B ^ { \\pi _ { \\theta } } Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) = \\mathbb { E } _ { \\mathbf { s } ^ { \\prime } \\sim T ( \\cdot | \\mathbf { s } , \\mathbf { a } ) } \\left[ r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } Q _ { \\phi } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 491, + 506, + 508 + ], + "score": 1.0, + "content": "is the Bellman operator. In the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 407, + 516 + ], + "score": 1.0, + "content": "context of offline RL, where transitions are sampled from a static dataset", + "type": "text" + }, + { + "bbox": [ + 407, + 505, + 416, + 514 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 503, + 505, + 516 + ], + "score": 1.0, + "content": ", the objective for the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 514, + 239, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 115, + 527 + ], + "score": 0.26, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 514, + 239, + 528 + ], + "score": 1.0, + "content": "-network becomes minimizing", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 446, + 506, + 528 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 531, + 470, + 559 + ], + "lines": [ + { + "bbox": [ + 139, + 531, + 470, + 559 + ], + "spans": [ + { + "bbox": [ + 139, + 531, + 470, + 559 + ], + "score": 0.92, + "content": "J _ { q } ( Q _ { \\phi } ) : = \\mathbb { E } _ { ( \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ Q _ { \\phi ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] ,", + "type": "interline_equation", + "image_path": "dd72bed5b712c6f643e4a71f2f1a93fde3332a7b416d74c86a8f6308169989dc.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 139, + 531, + 470, + 540.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 139, + 540.3333333333334, + 470, + 549.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 139, + 549.6666666666667, + 470, + 559.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 562, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 133, + 578 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 565, + 149, + 577 + ], + "score": 0.92, + "content": "Q _ { \\phi ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 562, + 506, + 578 + ], + "score": 1.0, + "content": "represents the target Q-network softly updated for algorithmic stability [20]. The policy,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "which is also parameterized by a neural network, is updated in an alternating fashion to maximize the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 586, + 338, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 144, + 601 + ], + "score": 1.0, + "content": "expected", + "type": "text" + }, + { + "bbox": [ + 144, + 587, + 153, + 597 + ], + "score": 0.33, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 586, + 182, + 601 + ], + "score": 1.0, + "content": "-value:", + "type": "text" + }, + { + "bbox": [ + 182, + 586, + 334, + 599 + ], + "score": 0.91, + "content": "J _ { p } ( \\pi _ { \\theta } ) : = \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi _ { \\theta } ( \\cdot | { \\mathbf s } ) } \\left[ Q _ { \\phi } ( { \\mathbf s } , { \\mathbf a } ) \\right] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 586, + 338, + 601 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 562, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 392, + 617 + ], + "score": 1.0, + "content": "However, as the policy is updated to maximize the Q-values, the actions", + "type": "text" + }, + { + "bbox": [ + 392, + 604, + 402, + 614 + ], + "score": 0.84, + "content": "\\mathbf { a } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "sampled from the current", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "policy in Equation (1) can be biased towards OOD actions with erroneously high Q-values. In the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "offline RL setting, such errors cannot be corrected by feedback from the environment as in online", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "RL. To handle the error propagation from these OOD actions, most offline RL algorithms regularize", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "either the policy [11, 15] or the Q-function [16] to be biased towards the given dataset. However, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "policy regularization methods typically require an accurate estimation of the behavior policy. The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 683 + ], + "score": 1.0, + "content": "previous state-of-the-art method CQL [16] instead learns conservative Q-values without estimating", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 680, + 373, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 373, + 693 + ], + "score": 1.0, + "content": "the behavior policy by penalizing the Q-values of OOD actions by", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 603, + 506, + 693 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 697, + 445, + 720 + ], + "lines": [ + { + "bbox": [ + 163, + 697, + 445, + 720 + ], + "spans": [ + { + "bbox": [ + 163, + 697, + 445, + 720 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\phi } J _ { q } ( Q _ { \\phi } ) + \\alpha \\Big ( \\mathbb { E } _ { { \\mathbf { s } } \\sim \\mathcal { D } , { \\mathbf { a } } \\sim \\mu ( \\cdot \\vert \\mathbf { s } ) } \\left[ Q _ { \\phi } \\left( \\mathbf { s } , { \\mathbf { a } } \\right) \\right] - \\mathbb { E } _ { ( \\mathbf { s } , { \\mathbf { a } } ) \\sim \\mathcal { D } } \\left[ Q _ { \\phi } \\left( \\mathbf { s } , { \\mathbf { a } } \\right) \\right] \\Big ) ,", + "type": "interline_equation", + "image_path": "5200b1ab6df96c14f926c8a3dd66849ac370e617d49cbd25d3e463c66cab5da6.jpg" + } + ] + } + ], + "index": 51, + "virtual_lines": [ + { + "bbox": [ + 163, + 697, + 445, + 720 + ], + "spans": [], + "index": 51 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 134, + 86 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 75, + 142, + 84 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is an approximation of the policy that maximizes the current Q-function. While CQL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "does not need explicit behavior policy estimation, it requires sampling from an appropriate action", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 190, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 154, + 108 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 94, + 185, + 106 + ], + "score": 0.92, + "content": "\\mu ( \\cdot | \\mathbf { \\bar { s } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 93, + 190, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 106, + 121, + 342, + 135 + ], + "lines": [ + { + "bbox": [ + 103, + 118, + 343, + 138 + ], + "spans": [ + { + "bbox": [ + 103, + 118, + 343, + 138 + ], + "score": 1.0, + "content": "3 Uncertainty penalization with Q-ensemble", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 122, + 152, + 486, + 293 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 152, + 486, + 293 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 152, + 486, + 293 + ], + "spans": [ + { + "bbox": [ + 122, + 152, + 486, + 293 + ], + "score": 0.971, + "type": "image", + "image_path": "5ef0099c6b2f91502d99a5bd4e4ae37be48755a4861064068d4c78cfb8b708b5.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 122, + 152, + 486, + 199.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 122, + 199.0, + 486, + 246.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 122, + 246.0, + 486, + 293.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 301, + 505, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 229, + 316 + ], + "score": 1.0, + "content": "Figure 1: Performance of SAC-", + "type": "text" + }, + { + "bbox": [ + 230, + 302, + 240, + 312 + ], + "score": 0.71, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 299, + 505, + 316 + ], + "score": 1.0, + "content": "on halfcheetah-medium and hopper-medium datasets while varying", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 313, + 117, + 322 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ", compared to CQL. ‘Average Return’ denotes the undiscounted return of each policies on evaluation.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 230, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 230, + 336 + ], + "score": 1.0, + "content": "Results averaged over 4 seeds.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 506, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "In this section, we turn our attention to a conventional technique from online RL, Clipped Double Q-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 486, + 371 + ], + "score": 1.0, + "content": "Learning [10], which uses the minimum value of two parallel Q-networks as the Bellman target:", + "type": "text" + }, + { + "bbox": [ + 487, + 360, + 506, + 370 + ], + "score": 0.85, + "content": "y =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 368, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 298, + 388 + ], + "score": 0.89, + "content": "\\begin{array} { r } { r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ \\operatorname* { m i n } _ { j = 1 , 2 } Q _ { \\phi _ { j } ^ { \\prime } } ( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 371, + 506, + 387 + ], + "score": 1.0, + "content": ". Although this technique was originally proposed in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "online RL to mitigate the overestimation from general prediction errors, some offline RL algorithms", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 332, + 411 + ], + "score": 1.0, + "content": "[11, 15, 28] also utilize this technique to enforce their", + "type": "text" + }, + { + "bbox": [ + 332, + 398, + 342, + 409 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "-value estimates to be more pessimistic.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "However, the isolated effect of the clipped Q-learning in offline RL was not fully analyzed in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "score": 1.0, + "content": "previous works, as they use the technique only as an auxiliary term that adds up to their core methods.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 504, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "To examine the ability of clipped Q-learning to prevent the overestimation in offline RL on its own,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 444, + 415, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 306, + 457 + ], + "score": 1.0, + "content": "we modify SAC [12] by increasing the number of", + "type": "text" + }, + { + "bbox": [ + 306, + 446, + 315, + 456 + ], + "score": 0.27, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 444, + 401, + 457 + ], + "score": 1.0, + "content": "-ensembles from 2 to", + "type": "text" + }, + { + "bbox": [ + 401, + 445, + 411, + 455 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 444, + 415, + 457 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 460, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 111, + 460, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 111, + 460, + 506, + 517 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\underset { \\phi _ { i } } { \\mathrm { n i n } } \\ : \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi \\theta \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } ^ { \\prime } \\right) } \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } ^ { \\prime } \\mid \\mathbf { s } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] } \\\\ & { \\underset { \\theta } { \\mathrm { n a x } } \\ : \\mathbb { E } _ { \\mathbf { s } \\sim \\mathcal { D } , \\mathbf { a } \\sim \\pi _ { \\theta } \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } \\right) } \\ : \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } \\mid \\mathbf { s } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "2774320fff9c4e1fa2ee39692e9d78ca73bb14e4e792e8943040295899bad1a6.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 111, + 460, + 506, + 479.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 111, + 479.0, + 506, + 498.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 111, + 498.0, + 506, + 517.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 366, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 367, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 120, + 532 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 520, + 174, + 531 + ], + "score": 0.93, + "content": "i = 1 , \\ldots , N", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 519, + 353, + 532 + ], + "score": 1.0, + "content": ". We denote this modified algorithm as SAC-", + "type": "text" + }, + { + "bbox": [ + 353, + 520, + 363, + 529 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 519, + 367, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "Figure 1 shows the preliminary experiments on D4RL halfcheetah-medium and hopper-medium", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 215, + 559 + ], + "score": 1.0, + "content": "datasets [9] while varying", + "type": "text" + }, + { + "bbox": [ + 216, + 547, + 226, + 557 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 547, + 504, + 559 + ], + "score": 1.0, + "content": ". Note that these datasets are constructed from suboptimal behavior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 295, + 570 + ], + "score": 1.0, + "content": "policies. Surprisingly, as we gradually increase", + "type": "text" + }, + { + "bbox": [ + 295, + 559, + 305, + 568 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ", we can successfully find policies that outperform", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "score": 1.0, + "content": "the previous state-of-the-art method (CQL) by a large margin. In fact, as we will present in Section 5,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 578, + 464, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 129, + 594 + ], + "score": 1.0, + "content": "SAC-", + "type": "text" + }, + { + "bbox": [ + 129, + 580, + 139, + 590 + ], + "score": 0.66, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 578, + 464, + 594 + ], + "score": 1.0, + "content": "outperforms CQL on various types of environments and data-collection policies.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 610 + ], + "score": 1.0, + "content": "To understand why this simple technique works so well, we can first interpret the clipping procedure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "(choosing the minimum value from the ensemble) as penalizing state-action pairs with high-variance", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "Q-value estimates, which encourages the policy to favor actions that appeared in the dataset [11]. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "dataset samples will naturally have lower variance compared to the OOD samples as the Bellman", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "residual term in Equation (2) explicitly aligns the Q-value predictions for the dataset samples. More", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "formally, we can regard this difference in variance as accounting for epistemic uncertainty [8] which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 382, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 382, + 675 + ], + "score": 1.0, + "content": "refers to the uncertainty stemming from limited data and knowledge.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Utilization of the clipped Q-value relates to methods that consider the confidence bound of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "Q-value estimates [24]. Online RL methods typically utilize the Q-ensemble to form an optimistic", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "estimate of the Q-value, by adding the standard deviation to the mean of the Q-ensembles [18]. This", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "optimistic Q-value, also known as the upper-confidence bound (UCB), can encourage the exploration", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 134, + 86 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 75, + 142, + 84 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is an approximation of the policy that maximizes the current Q-function. While CQL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "does not need explicit behavior policy estimation, it requires sampling from an appropriate action", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 190, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 154, + 108 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 94, + 185, + 106 + ], + "score": 0.92, + "content": "\\mu ( \\cdot | \\mathbf { \\bar { s } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 93, + 190, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 72, + 505, + 108 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 121, + 342, + 135 + ], + "lines": [ + { + "bbox": [ + 103, + 118, + 343, + 138 + ], + "spans": [ + { + "bbox": [ + 103, + 118, + 343, + 138 + ], + "score": 1.0, + "content": "3 Uncertainty penalization with Q-ensemble", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 122, + 152, + 486, + 293 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 152, + 486, + 293 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 152, + 486, + 293 + ], + "spans": [ + { + "bbox": [ + 122, + 152, + 486, + 293 + ], + "score": 0.971, + "type": "image", + "image_path": "5ef0099c6b2f91502d99a5bd4e4ae37be48755a4861064068d4c78cfb8b708b5.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 122, + 152, + 486, + 199.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 122, + 199.0, + 486, + 246.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 122, + 246.0, + 486, + 293.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 301, + 505, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 229, + 316 + ], + "score": 1.0, + "content": "Figure 1: Performance of SAC-", + "type": "text" + }, + { + "bbox": [ + 230, + 302, + 240, + 312 + ], + "score": 0.71, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 299, + 505, + 316 + ], + "score": 1.0, + "content": "on halfcheetah-medium and hopper-medium datasets while varying", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 313, + 117, + 322 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ", compared to CQL. ‘Average Return’ denotes the undiscounted return of each policies on evaluation.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 230, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 230, + 336 + ], + "score": 1.0, + "content": "Results averaged over 4 seeds.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 506, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "In this section, we turn our attention to a conventional technique from online RL, Clipped Double Q-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 486, + 371 + ], + "score": 1.0, + "content": "Learning [10], which uses the minimum value of two parallel Q-networks as the Bellman target:", + "type": "text" + }, + { + "bbox": [ + 487, + 360, + 506, + 370 + ], + "score": 0.85, + "content": "y =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 368, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 298, + 388 + ], + "score": 0.89, + "content": "\\begin{array} { r } { r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | \\mathbf { s } ^ { \\prime } ) } \\left[ \\operatorname* { m i n } _ { j = 1 , 2 } Q _ { \\phi _ { j } ^ { \\prime } } ( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 371, + 506, + 387 + ], + "score": 1.0, + "content": ". Although this technique was originally proposed in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "online RL to mitigate the overestimation from general prediction errors, some offline RL algorithms", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 332, + 411 + ], + "score": 1.0, + "content": "[11, 15, 28] also utilize this technique to enforce their", + "type": "text" + }, + { + "bbox": [ + 332, + 398, + 342, + 409 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "-value estimates to be more pessimistic.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "However, the isolated effect of the clipped Q-learning in offline RL was not fully analyzed in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "score": 1.0, + "content": "previous works, as they use the technique only as an auxiliary term that adds up to their core methods.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 345, + 507, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 504, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "To examine the ability of clipped Q-learning to prevent the overestimation in offline RL on its own,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 444, + 415, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 306, + 457 + ], + "score": 1.0, + "content": "we modify SAC [12] by increasing the number of", + "type": "text" + }, + { + "bbox": [ + 306, + 446, + 315, + 456 + ], + "score": 0.27, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 444, + 401, + 457 + ], + "score": 1.0, + "content": "-ensembles from 2 to", + "type": "text" + }, + { + "bbox": [ + 401, + 445, + 411, + 455 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 444, + 415, + 457 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 434, + 506, + 457 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 460, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 111, + 460, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 111, + 460, + 506, + 517 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\underset { \\phi _ { i } } { \\mathrm { n i n } } \\ : \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } \\sim \\mathcal { D } } \\left[ \\left( Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) - \\left( r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a } ^ { \\prime } \\sim \\pi \\theta \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } ^ { \\prime } \\right) } \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } ^ { \\prime } } \\left( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } \\right) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } ^ { \\prime } \\mid \\mathbf { s } ^ { \\prime } \\right) \\right] \\right) \\right) ^ { 2 } \\right] } \\\\ & { \\underset { \\theta } { \\mathrm { n a x } } \\ : \\mathbb { E } _ { \\mathbf { s } \\sim \\mathcal { D } , \\mathbf { a } \\sim \\pi _ { \\theta } \\cdot \\left( \\mathbf { \\cdot } \\mathbf { s } \\right) } \\ : \\left[ \\underset { j = 1 , \\ldots , N } { \\operatorname* { m i n } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) - \\beta \\log \\pi _ { \\theta } \\left( \\mathbf { a } \\mid \\mathbf { s } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "2774320fff9c4e1fa2ee39692e9d78ca73bb14e4e792e8943040295899bad1a6.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 111, + 460, + 506, + 479.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 111, + 479.0, + 506, + 498.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 111, + 498.0, + 506, + 517.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 366, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 367, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 120, + 532 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 520, + 174, + 531 + ], + "score": 0.93, + "content": "i = 1 , \\ldots , N", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 519, + 353, + 532 + ], + "score": 1.0, + "content": ". We denote this modified algorithm as SAC-", + "type": "text" + }, + { + "bbox": [ + 353, + 520, + 363, + 529 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 519, + 367, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 519, + 367, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "Figure 1 shows the preliminary experiments on D4RL halfcheetah-medium and hopper-medium", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 215, + 559 + ], + "score": 1.0, + "content": "datasets [9] while varying", + "type": "text" + }, + { + "bbox": [ + 216, + 547, + 226, + 557 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 547, + 504, + 559 + ], + "score": 1.0, + "content": ". Note that these datasets are constructed from suboptimal behavior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 295, + 570 + ], + "score": 1.0, + "content": "policies. Surprisingly, as we gradually increase", + "type": "text" + }, + { + "bbox": [ + 295, + 559, + 305, + 568 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ", we can successfully find policies that outperform", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "score": 1.0, + "content": "the previous state-of-the-art method (CQL) by a large margin. In fact, as we will present in Section 5,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 578, + 464, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 129, + 594 + ], + "score": 1.0, + "content": "SAC-", + "type": "text" + }, + { + "bbox": [ + 129, + 580, + 139, + 590 + ], + "score": 0.66, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 578, + 464, + 594 + ], + "score": 1.0, + "content": "outperforms CQL on various types of environments and data-collection policies.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 536, + 507, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 610 + ], + "score": 1.0, + "content": "To understand why this simple technique works so well, we can first interpret the clipping procedure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "(choosing the minimum value from the ensemble) as penalizing state-action pairs with high-variance", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "Q-value estimates, which encourages the policy to favor actions that appeared in the dataset [11]. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "dataset samples will naturally have lower variance compared to the OOD samples as the Bellman", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "residual term in Equation (2) explicitly aligns the Q-value predictions for the dataset samples. More", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "formally, we can regard this difference in variance as accounting for epistemic uncertainty [8] which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 382, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 382, + 675 + ], + "score": 1.0, + "content": "refers to the uncertainty stemming from limited data and knowledge.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 595, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Utilization of the clipped Q-value relates to methods that consider the confidence bound of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "Q-value estimates [24]. Online RL methods typically utilize the Q-ensemble to form an optimistic", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "estimate of the Q-value, by adding the standard deviation to the mean of the Q-ensembles [18]. This", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "optimistic Q-value, also known as the upper-confidence bound (UCB), can encourage the exploration", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "of unseen actions with high uncertainty. However, in offline RL, the dataset available during training", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "is fixed, and we have to focus on exploiting the given data. For this purpose, it is natural to utilize the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "lower-confidence bound (LCB) of the Q-value estimates, for example by subtracting the standard", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 387, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 387, + 118 + ], + "score": 1.0, + "content": "deviation from the mean, which allows us to avoid risky state-actions.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 677, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "of unseen actions with high uncertainty. However, in offline RL, the dataset available during training", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "is fixed, and we have to focus on exploiting the given data. For this purpose, it is natural to utilize the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "lower-confidence bound (LCB) of the Q-value estimates, for example by subtracting the standard", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 387, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 387, + 118 + ], + "score": 1.0, + "content": "deviation from the mean, which allows us to avoid risky state-actions.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "The clipped Q-learning algorithm, which chooses the worst-case Q-value instead to compute the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 386, + 146 + ], + "score": 1.0, + "content": "pessimistic estimate, can also be interpreted as utilizing the LCB of the", + "type": "text" + }, + { + "bbox": [ + 387, + 133, + 396, + 144 + ], + "score": 0.3, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "-value predictions. Suppose", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 107, + 144, + 137, + 155 + ], + "score": 0.92, + "content": "Q ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 143, + 309, + 156 + ], + "score": 1.0, + "content": "follows a Gaussian distribution with mean", + "type": "text" + }, + { + "bbox": [ + 309, + 144, + 341, + 156 + ], + "score": 0.93, + "content": "m ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 143, + 435, + 156 + ], + "score": 1.0, + "content": "and standard deviation", + "type": "text" + }, + { + "bbox": [ + 435, + 144, + 464, + 156 + ], + "score": 0.92, + "content": "\\sigma ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 143, + 505, + 156 + ], + "score": 1.0, + "content": ". Also, let", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 149, + 509, + 170 + ], + "spans": [ + { + "bbox": [ + 107, + 154, + 166, + 168 + ], + "score": 0.91, + "content": "\\{ Q _ { j } ( \\mathbf { s } , \\mathbf { \\bar { a } } ) \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 149, + 240, + 170 + ], + "score": 1.0, + "content": "be realizations of", + "type": "text" + }, + { + "bbox": [ + 240, + 155, + 271, + 166 + ], + "score": 0.92, + "content": "Q ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 149, + 509, + 170 + ], + "score": 1.0, + "content": ". Then, we can approximate the expected minimum of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 311, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 311, + 178 + ], + "score": 1.0, + "content": "realizations following the work of Royston [23] as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 181, + 436, + 209 + ], + "lines": [ + { + "bbox": [ + 174, + 181, + 436, + 209 + ], + "spans": [ + { + "bbox": [ + 174, + 181, + 436, + 209 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { j } ( \\mathbf { s } , \\mathbf { a } ) \\right] \\approx m ( \\mathbf { s } , \\mathbf { a } ) - \\Phi ^ { - 1 } \\left( \\frac { N - \\frac { \\pi } { 8 } } { N - \\frac { \\pi } { 4 } + 1 } \\right) \\sigma ( \\mathbf { s } , \\mathbf { a } ) ,", + "type": "interline_equation", + "image_path": "ca4f68505ad91174a21804ab1d2728e4d21634a54371811adf093c2e43133fa2.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 174, + 181, + 436, + 190.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 174, + 190.33333333333334, + 436, + 199.66666666666669 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 199.66666666666669, + 436, + 209.00000000000003 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 134, + 226 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 215, + 142, + 224 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "is the CDF of the standard Gaussian distribution. This relation indicates that using the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "clipped Q-value is similar to penalizing the ensemble mean of the Q-values with the standard deviation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 268, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 253, + 248 + ], + "score": 1.0, + "content": "scaled by a coefficient dependent on", + "type": "text" + }, + { + "bbox": [ + 254, + 237, + 264, + 246 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 236, + 268, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "image", + "bbox": [ + 114, + 259, + 496, + 369 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 259, + 496, + 369 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 259, + 496, + 369 + ], + "spans": [ + { + "bbox": [ + 114, + 259, + 496, + 369 + ], + "score": 0.97, + "type": "image", + "image_path": "ce40e054b643c8f22be0cd242e5ba33fb71eb6ea668533184970fad707c33dc5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 114, + 259, + 496, + 295.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 114, + 295.6666666666667, + 496, + 332.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 114, + 332.33333333333337, + 496, + 369.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 375, + 505, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "Figure 2: (a) and (b) each plots the size of the clip penalty and the standard deviation of the Q-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "value estimates for in-distribution (behavior) and OOD (random) actions while training SAC-10 on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "halfcheetah-medium dataset. (c) plots the gap of the clip penalty between the in-distribution and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 408, + 357, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 218, + 420 + ], + "score": 1.0, + "content": "OOD actions while varying", + "type": "text" + }, + { + "bbox": [ + 219, + 408, + 229, + 418 + ], + "score": 0.69, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 408, + 357, + 420 + ], + "score": 1.0, + "content": ". Results averaged over 4 seeds.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + } + ], + "index": 17.75 + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "We now move on to the empirical analysis of the clipped Q-learning. Figure 2a compares the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "strength of the uncertainty penalty on in-distribution and OOD actions. Specifically, we compare", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "actions sampled from two types of policies: (1) the behavior policy which was used to collect the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "dataset, and (2) the random policy which samples actions uniformly from the action space. For each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 475, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 357, + 494 + ], + "score": 1.0, + "content": "policy, we measure the size of the penalty from the clipping as", + "type": "text" + }, + { + "bbox": [ + 358, + 475, + 505, + 491 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi ( \\cdot | { \\mathbf s } ) } [ \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } Q _ { \\phi _ { j } } ( { \\mathbf s } , { \\mathbf a } ) - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 107, + 490, + 201, + 503 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 490, + 506, + 503 + ], + "score": 1.0, + "content": ". Figure 2a shows that the clipping term penalizes the random state-action", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "pairs much stronger than the in-distribution pairs throughout the training. For comparison, we also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 255, + 524 + ], + "score": 1.0, + "content": "measure the standard deviation of the", + "type": "text" + }, + { + "bbox": [ + 256, + 513, + 265, + 523 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "-values for each policy. The results in Figure 2b show that as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "we conjectured, the Q-value predictions for the OOD actions have a higher variance. We also find that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "the size of the penalty and the standard deviation are highly correlated, as we noted in Equation (3).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "As we observe that OOD actions have higher variance on Q-value estimates, the effect of increasing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 117, + 571 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "becomes obvious: it strengthens the penalty applied to the OOD samples compared to the dataset", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "samples. To verify this, we measured the relative penalty applied to the OOD samples in Figure 2c", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 457, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 404, + 596 + ], + "score": 1.0, + "content": "and found that indeed the OOD samples are penalized relatively further as", + "type": "text" + }, + { + "bbox": [ + 405, + 583, + 415, + 593 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 583, + 457, + 596 + ], + "score": 1.0, + "content": "increases.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 609, + 297, + 623 + ], + "lines": [ + { + "bbox": [ + 104, + 609, + 298, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 609, + 298, + 626 + ], + "score": 1.0, + "content": "4 Ensemble gradient diversification", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 186, + 647 + ], + "score": 1.0, + "content": "Even though SAC-", + "type": "text" + }, + { + "bbox": [ + 186, + 635, + 196, + 645 + ], + "score": 0.69, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 633, + 505, + 647 + ], + "score": 1.0, + "content": "outperforms existing methods on various tasks, it sometimes requires an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 352, + 658 + ], + "score": 1.0, + "content": "excessively large number of ensembles to learn stably (e.g.,", + "type": "text" + }, + { + "bbox": [ + 352, + 645, + 388, + 655 + ], + "score": 0.89, + "content": "N = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "for hopper-medium). While", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 363, + 668 + ], + "score": 1.0, + "content": "investigating its reason, we found that the performance of SAC-", + "type": "text" + }, + { + "bbox": [ + 363, + 657, + 373, + 666 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "is negatively correlated with the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 312, + 680 + ], + "score": 1.0, + "content": "degree to which the input gradients of Q-functions", + "type": "text" + }, + { + "bbox": [ + 313, + 667, + 366, + 680 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "are aligned, which decreases with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 117, + 688 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 678, + 505, + 690 + ], + "score": 1.0, + "content": ". Figure 4 measures the minimum cosine similarity between the gradients of the Q-functions", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 253, + 702 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\operatorname* { m i n } _ { i \\neq j } \\langle \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 687, + 481, + 702 + ], + "score": 1.0, + "content": "to examine the alignment of the gradients while varying", + "type": "text" + }, + { + "bbox": [ + 481, + 689, + 491, + 699 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "the D4RL hopper-medium dataset. The results imply that the performance of the learned policy", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 407, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 407, + 723 + ], + "score": 1.0, + "content": "degrades significantly when the Q-functions share a similar local structure.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 506, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "The clipped Q-learning algorithm, which chooses the worst-case Q-value instead to compute the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 386, + 146 + ], + "score": 1.0, + "content": "pessimistic estimate, can also be interpreted as utilizing the LCB of the", + "type": "text" + }, + { + "bbox": [ + 387, + 133, + 396, + 144 + ], + "score": 0.3, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "-value predictions. Suppose", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 107, + 144, + 137, + 155 + ], + "score": 0.92, + "content": "Q ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 143, + 309, + 156 + ], + "score": 1.0, + "content": "follows a Gaussian distribution with mean", + "type": "text" + }, + { + "bbox": [ + 309, + 144, + 341, + 156 + ], + "score": 0.93, + "content": "m ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 143, + 435, + 156 + ], + "score": 1.0, + "content": "and standard deviation", + "type": "text" + }, + { + "bbox": [ + 435, + 144, + 464, + 156 + ], + "score": 0.92, + "content": "\\sigma ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 143, + 505, + 156 + ], + "score": 1.0, + "content": ". Also, let", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 149, + 509, + 170 + ], + "spans": [ + { + "bbox": [ + 107, + 154, + 166, + 168 + ], + "score": 0.91, + "content": "\\{ Q _ { j } ( \\mathbf { s } , \\mathbf { \\bar { a } } ) \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 149, + 240, + 170 + ], + "score": 1.0, + "content": "be realizations of", + "type": "text" + }, + { + "bbox": [ + 240, + 155, + 271, + 166 + ], + "score": 0.92, + "content": "Q ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 149, + 509, + 170 + ], + "score": 1.0, + "content": ". Then, we can approximate the expected minimum of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 311, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 311, + 178 + ], + "score": 1.0, + "content": "realizations following the work of Royston [23] as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 122, + 509, + 178 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 181, + 436, + 209 + ], + "lines": [ + { + "bbox": [ + 174, + 181, + 436, + 209 + ], + "spans": [ + { + "bbox": [ + 174, + 181, + 436, + 209 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { j } ( \\mathbf { s } , \\mathbf { a } ) \\right] \\approx m ( \\mathbf { s } , \\mathbf { a } ) - \\Phi ^ { - 1 } \\left( \\frac { N - \\frac { \\pi } { 8 } } { N - \\frac { \\pi } { 4 } + 1 } \\right) \\sigma ( \\mathbf { s } , \\mathbf { a } ) ,", + "type": "interline_equation", + "image_path": "ca4f68505ad91174a21804ab1d2728e4d21634a54371811adf093c2e43133fa2.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 174, + 181, + 436, + 190.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 174, + 190.33333333333334, + 436, + 199.66666666666669 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 199.66666666666669, + 436, + 209.00000000000003 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 134, + 226 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 215, + 142, + 224 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "is the CDF of the standard Gaussian distribution. This relation indicates that using the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "clipped Q-value is similar to penalizing the ensemble mean of the Q-values with the standard deviation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 268, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 253, + 248 + ], + "score": 1.0, + "content": "scaled by a coefficient dependent on", + "type": "text" + }, + { + "bbox": [ + 254, + 237, + 264, + 246 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 236, + 268, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 213, + 505, + 248 + ] + }, + { + "type": "image", + "bbox": [ + 114, + 259, + 496, + 369 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 259, + 496, + 369 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 259, + 496, + 369 + ], + "spans": [ + { + "bbox": [ + 114, + 259, + 496, + 369 + ], + "score": 0.97, + "type": "image", + "image_path": "ce40e054b643c8f22be0cd242e5ba33fb71eb6ea668533184970fad707c33dc5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 114, + 259, + 496, + 295.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 114, + 295.6666666666667, + 496, + 332.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 114, + 332.33333333333337, + 496, + 369.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 375, + 505, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "Figure 2: (a) and (b) each plots the size of the clip penalty and the standard deviation of the Q-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "value estimates for in-distribution (behavior) and OOD (random) actions while training SAC-10 on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "halfcheetah-medium dataset. (c) plots the gap of the clip penalty between the in-distribution and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 408, + 357, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 218, + 420 + ], + "score": 1.0, + "content": "OOD actions while varying", + "type": "text" + }, + { + "bbox": [ + 219, + 408, + 229, + 418 + ], + "score": 0.69, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 408, + 357, + 420 + ], + "score": 1.0, + "content": ". Results averaged over 4 seeds.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + } + ], + "index": 17.75 + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "We now move on to the empirical analysis of the clipped Q-learning. Figure 2a compares the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "strength of the uncertainty penalty on in-distribution and OOD actions. Specifically, we compare", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "actions sampled from two types of policies: (1) the behavior policy which was used to collect the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "dataset, and (2) the random policy which samples actions uniformly from the action space. For each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 475, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 357, + 494 + ], + "score": 1.0, + "content": "policy, we measure the size of the penalty from the clipping as", + "type": "text" + }, + { + "bbox": [ + 358, + 475, + 505, + 491 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } _ { { \\mathbf s } \\sim \\mathcal { D } , { \\mathbf a } \\sim \\pi ( \\cdot | { \\mathbf s } ) } [ \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } Q _ { \\phi _ { j } } ( { \\mathbf s } , { \\mathbf a } ) - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 107, + 490, + 201, + 503 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { j = 1 , \\dots , N } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 490, + 506, + 503 + ], + "score": 1.0, + "content": ". Figure 2a shows that the clipping term penalizes the random state-action", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "pairs much stronger than the in-distribution pairs throughout the training. For comparison, we also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 255, + 524 + ], + "score": 1.0, + "content": "measure the standard deviation of the", + "type": "text" + }, + { + "bbox": [ + 256, + 513, + 265, + 523 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "-values for each policy. The results in Figure 2b show that as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "we conjectured, the Q-value predictions for the OOD actions have a higher variance. We also find that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "the size of the penalty and the standard deviation are highly correlated, as we noted in Equation (3).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 432, + 506, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "As we observe that OOD actions have higher variance on Q-value estimates, the effect of increasing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 117, + 571 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "becomes obvious: it strengthens the penalty applied to the OOD samples compared to the dataset", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "samples. To verify this, we measured the relative penalty applied to the OOD samples in Figure 2c", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 457, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 404, + 596 + ], + "score": 1.0, + "content": "and found that indeed the OOD samples are penalized relatively further as", + "type": "text" + }, + { + "bbox": [ + 405, + 583, + 415, + 593 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 583, + 457, + 596 + ], + "score": 1.0, + "content": "increases.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 549, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 609, + 297, + 623 + ], + "lines": [ + { + "bbox": [ + 104, + 609, + 298, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 609, + 298, + 626 + ], + "score": 1.0, + "content": "4 Ensemble gradient diversification", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 186, + 647 + ], + "score": 1.0, + "content": "Even though SAC-", + "type": "text" + }, + { + "bbox": [ + 186, + 635, + 196, + 645 + ], + "score": 0.69, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 633, + 505, + 647 + ], + "score": 1.0, + "content": "outperforms existing methods on various tasks, it sometimes requires an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 352, + 658 + ], + "score": 1.0, + "content": "excessively large number of ensembles to learn stably (e.g.,", + "type": "text" + }, + { + "bbox": [ + 352, + 645, + 388, + 655 + ], + "score": 0.89, + "content": "N = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "for hopper-medium). While", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 363, + 668 + ], + "score": 1.0, + "content": "investigating its reason, we found that the performance of SAC-", + "type": "text" + }, + { + "bbox": [ + 363, + 657, + 373, + 666 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "is negatively correlated with the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 312, + 680 + ], + "score": 1.0, + "content": "degree to which the input gradients of Q-functions", + "type": "text" + }, + { + "bbox": [ + 313, + 667, + 366, + 680 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "are aligned, which decreases with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 117, + 688 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 678, + 505, + 690 + ], + "score": 1.0, + "content": ". Figure 4 measures the minimum cosine similarity between the gradients of the Q-functions", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 253, + 702 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\operatorname* { m i n } _ { i \\neq j } \\langle \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 687, + 481, + 702 + ], + "score": 1.0, + "content": "to examine the alignment of the gradients while varying", + "type": "text" + }, + { + "bbox": [ + 481, + 689, + 491, + 699 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "the D4RL hopper-medium dataset. The results imply that the performance of the learned policy", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 407, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 407, + 723 + ], + "score": 1.0, + "content": "degrades significantly when the Q-functions share a similar local structure.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 633, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 71, + 494, + 238 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 71, + 494, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 71, + 494, + 238 + ], + "spans": [ + { + "bbox": [ + 124, + 71, + 494, + 238 + ], + "score": 0.811, + "type": "image", + "image_path": "968b107276cad894e397102039b5ec06b011948bd1a659ae32e5bb3c98ac6b5a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 71, + 494, + 126.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 126.66666666666666, + 494, + 182.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 182.33333333333331, + 494, + 237.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 504, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 421, + 256 + ], + "score": 1.0, + "content": "Figure 3: Illustration of the ensemble gradient diversification. The vector", + "type": "text" + }, + { + "bbox": [ + 421, + 244, + 443, + 254 + ], + "score": 0.92, + "content": "\\lambda _ { i } \\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 253, + 432, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 202, + 268 + ], + "score": 1.0, + "content": "normalized eigenvector", + "type": "text" + }, + { + "bbox": [ + 202, + 256, + 215, + 265 + ], + "score": 0.85, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 253, + 227, + 268 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 227, + 254, + 303, + 267 + ], + "score": 0.92, + "content": "\\mathrm { V a r } ( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 253, + 418, + 268 + ], + "score": 1.0, + "content": "multiplied by its eigenvalue", + "type": "text" + }, + { + "bbox": [ + 418, + 255, + 428, + 265 + ], + "score": 0.87, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 253, + 432, + 268 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 340, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 341, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 341, + 298 + ], + "score": 1.0, + "content": "We now show that the alignment of the input gradients", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 297, + 340, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 340, + 308 + ], + "score": 1.0, + "content": "can induce insufficient penalization of near-distribution", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 307, + 342, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 342, + 319 + ], + "score": 1.0, + "content": "data points, which leads to requiring a large number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 318, + 342, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 209, + 331 + ], + "score": 1.0, + "content": "ensemble networks. Let", + "type": "text" + }, + { + "bbox": [ + 210, + 318, + 263, + 331 + ], + "score": 0.92, + "content": "\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 318, + 342, + 331 + ], + "score": 1.0, + "content": "be the gradient of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 329, + 341, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 121, + 342 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 330, + 128, + 341 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 329, + 341, + 342 + ], + "score": 1.0, + "content": "-th Q-function with respect to the behavior action a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 340, + 341, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 341, + 352 + ], + "score": 1.0, + "content": "and assume the gradient is normalized for simplicity. If the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 351, + 341, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 341, + 363 + ], + "score": 1.0, + "content": "gradients of the Q-functions are well-aligned as illustrated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 341, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 341, + 373 + ], + "score": 1.0, + "content": "in Figure 3a, then there exists a unit vector w such that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 373, + 341, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 341, + 385 + ], + "score": 1.0, + "content": "the Q-values for the OOD actions along the direction of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 385, + 340, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 340, + 395 + ], + "score": 1.0, + "content": "w have a low variance. 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} } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } , } \\end{array}", + "type": "interline_equation", + "image_path": "c516ccb12b1c7cec2cadc3c22c4e3c9c00941050f096e5a6804326e4b666a370.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 171, + 480, + 439, + 502.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 171, + 502.0, + 439, + 524.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 171, + 524.0, + 439, + 546.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 506, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 133, + 565 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 551, + 152, + 563 + ], + "score": 0.91, + "content": "\\langle \\cdot , \\cdot \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 550, + 257, + 565 + ], + "score": 1.0, + "content": "denotes an inner-product,", + "type": "text" + }, + { + "bbox": [ + 258, + 551, + 284, + 562 + ], + "score": 0.9, + "content": "k \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 550, + 322, + 565 + ], + "score": 1.0, + "content": ", and Var", + "type": "text" + }, + { + "bbox": [ + 323, + 550, + 384, + 564 + ], + "score": 0.89, + "content": "\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "is the sample variance matrix", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 198, + 577 + ], + "score": 1.0, + "content": "for the input gradients", + "type": "text" + }, + { + "bbox": [ + 198, + 563, + 251, + 576 + ], + "score": 0.93, + "content": "\\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 562, + 506, + 577 + ], + "score": 1.0, + "content": ". One interesting property of the variance matrix is that its total", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "variance, which is equivalent to the sum of its eigenvalues, can be represented as a function of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 283, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 283, + 597 + ], + "score": 1.0, + "content": "norm of the average gradients by Lemma 1.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 105, + 599, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 289, + 614 + ], + "score": 1.0, + "content": "Lemma 1. The total variance of the matrix", + "type": "text" + }, + { + "bbox": [ + 289, + 599, + 368, + 613 + ], + "score": 0.61, + "content": "\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 597, + 416, + 614 + ], + "score": 1.0, + "content": "is equal to", + "type": "text" + }, + { + "bbox": [ + 416, + 599, + 454, + 612 + ], + "score": 0.91, + "content": "1 - \\| \\bar { q } \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 597, + 486, + 614 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 486, + 600, + 506, + 612 + ], + "score": 0.83, + "content": "\\bar { q } =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 608, + 203, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 198, + 628 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 608, + 203, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 121, + 650 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 637, + 142, + 649 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 637, + 255, + 650 + ], + "score": 1.0, + "content": "be the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 256, + 636, + 334, + 650 + ], + "score": 0.91, + "content": "\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 637, + 352, + 650 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 639, + 375, + 649 + ], + "score": 0.86, + "content": "\\mathbf { w } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "be the corresponding normalized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 197, + 664 + ], + "score": 1.0, + "content": "eigenvector. Also, let", + "type": "text" + }, + { + "bbox": [ + 197, + 650, + 222, + 660 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 649, + 316, + 664 + ], + "score": 1.0, + "content": "be the value such that", + "type": "text" + }, + { + "bbox": [ + 316, + 649, + 504, + 663 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { i \\neq j } \\left. \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. = 1 - \\epsilon . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 659, + 504, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 343, + 676 + ], + "score": 1.0, + "content": "Then, using Lemma 1, we can prove that the variance of the", + "type": "text" + }, + { + "bbox": [ + 343, + 663, + 352, + 672 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 659, + 480, + 676 + ], + "score": 1.0, + "content": "-values for an OOD action along", + "type": "text" + }, + { + "bbox": [ + 480, + 663, + 504, + 672 + ], + "score": 0.78, + "content": "\\mathbf { w } _ { \\mathrm { m i n } }", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 437, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 297, + 685 + ], + "score": 1.0, + "content": "is upper-bounded by some constant multiple of", + "type": "text" + }, + { + "bbox": [ + 297, + 675, + 302, + 682 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 672, + 437, + 685 + ], + "score": 1.0, + "content": ", which is given by Proposition 1.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 505, + 724 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 212, + 701 + ], + "score": 1.0, + "content": "Proposition 1. Suppose", + "type": "text" + }, + { + "bbox": [ + 212, + 686, + 301, + 699 + ], + "score": 0.91, + "content": "Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\ = \\ Q ( { \\bf s } , { \\bf a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 685, + 324, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 687, + 361, + 700 + ], + "score": 0.91, + "content": "Q _ { \\phi _ { j } } ( \\mathbf { s } , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 685, + 507, + 701 + ], + "score": 1.0, + "content": "is locally linear in the neighbor-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 180, + 712 + ], + "score": 1.0, + "content": "hood of a for all", + "type": "text" + }, + { + "bbox": [ + 180, + 699, + 216, + 711 + ], + "score": 0.91, + "content": "j \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 697, + 240, + 712 + ], + "score": 1.0, + "content": ". 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One interesting property of the variance matrix is that its total", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "variance, which is equivalent to the sum of its eigenvalues, can be represented as a function of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 283, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 283, + 597 + ], + "score": 1.0, + "content": "norm of the average gradients by Lemma 1.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 550, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 599, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 289, + 614 + ], + "score": 1.0, + "content": "Lemma 1. The total variance of the matrix", + "type": "text" + }, + { + "bbox": [ + 289, + 599, + 368, + 613 + ], + "score": 0.61, + "content": "\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 597, + 416, + 614 + ], + "score": 1.0, + "content": "is equal to", + "type": "text" + }, + { + "bbox": [ + 416, + 599, + 454, + 612 + ], + "score": 0.91, + "content": "1 - \\| \\bar { q } \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 597, + 486, + 614 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 486, + 600, + 506, + 612 + ], + "score": 0.83, + "content": "\\bar { q } =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 608, + 203, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 198, + 628 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 608, + 203, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 597, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 121, + 650 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 637, + 142, + 649 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 637, + 255, + 650 + ], + "score": 1.0, + "content": "be the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 256, + 636, + 334, + 650 + ], + "score": 0.91, + "content": "\\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 637, + 352, + 650 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 639, + 375, + 649 + ], + "score": 0.86, + "content": "\\mathbf { w } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "be the corresponding normalized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 197, + 664 + ], + "score": 1.0, + "content": "eigenvector. Also, let", + "type": "text" + }, + { + "bbox": [ + 197, + 650, + 222, + 660 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 649, + 316, + 664 + ], + "score": 1.0, + "content": "be the value such that", + "type": "text" + }, + { + "bbox": [ + 316, + 649, + 504, + 663 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { i \\neq j } \\left. \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. = 1 - \\epsilon . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 659, + 504, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 343, + 676 + ], + "score": 1.0, + "content": "Then, using Lemma 1, we can prove that the variance of the", + "type": "text" + }, + { + "bbox": [ + 343, + 663, + 352, + 672 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 659, + 480, + 676 + ], + "score": 1.0, + "content": "-values for an OOD action along", + "type": "text" + }, + { + "bbox": [ + 480, + 663, + 504, + 672 + ], + "score": 0.78, + "content": "\\mathbf { w } _ { \\mathrm { m i n } }", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 437, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 297, + 685 + ], + "score": 1.0, + "content": "is upper-bounded by some constant multiple of", + "type": "text" + }, + { + "bbox": [ + 297, + 675, + 302, + 682 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 672, + 437, + 685 + ], + "score": 1.0, + "content": ", which is given by Proposition 1.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 636, + 505, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 505, + 724 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 212, + 701 + ], + "score": 1.0, + "content": "Proposition 1. Suppose", + "type": "text" + }, + { + "bbox": [ + 212, + 686, + 301, + 699 + ], + "score": 0.91, + "content": "Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\ = \\ Q ( { \\bf s } , { \\bf a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 685, + 324, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 687, + 361, + 700 + ], + "score": 0.91, + "content": "Q _ { \\phi _ { j } } ( \\mathbf { s } , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 685, + 507, + 701 + ], + "score": 1.0, + "content": "is locally linear in the neighbor-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 180, + 712 + ], + "score": 1.0, + "content": "hood of a for all", + "type": "text" + }, + { + "bbox": [ + 180, + 699, + 216, + 711 + ], + "score": 0.91, + "content": "j \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 697, + 240, + 712 + ], + "score": 1.0, + "content": ". 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Proposition 1 implies that if there exists such", + "type": "text" + }, + { + "bbox": [ + 453, + 150, + 477, + 160 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "that is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "small, which means the gradients of Q-function are well-aligned, then the variance of the Q-values", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "for an OOD action along a specific direction will also be small. This in turn degrades the ability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "of the ensembles to penalize OOD actions, which ultimately leads to requiring a large number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 187, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 187, + 205 + ], + "score": 1.0, + "content": "ensemble networks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 507, + 223 + ], + "score": 1.0, + "content": "To address this problem, we propose a regularizer that effectively increases the variance of the Q-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "values for near-distribution OOD actions. Note that the variance is lower-bounded by some constant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 273, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 248, + 245 + ], + "score": 1.0, + "content": "multiple of the smallest eigenvalue", + "type": "text" + }, + { + "bbox": [ + 248, + 232, + 268, + 243 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 230, + 273, + 245 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 245, + 426, + 294 + ], + "lines": [ + { + "bbox": [ + 183, + 245, + 426, + 294 + ], + "spans": [ + { + "bbox": [ + 183, + 245, + 426, + 294 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } ) \\right) \\approx k ^ { 2 } \\mathbf { w } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } } \\\\ & { \\quad \\quad \\quad \\quad \\geq k ^ { 2 } \\mathbf { w } _ { \\operatorname* { m i n } } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } _ { \\operatorname* { m i n } } } \\\\ & { \\quad \\quad \\quad = k ^ { 2 } \\lambda _ { \\operatorname* { m i n } } . } \\end{array}", + "type": "interline_equation", + "image_path": "811413a540f1df9255ac01cd613a36f965755918dbbfb18243bb3a4f2c812718.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 183, + 245, + 426, + 261.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 183, + 261.3333333333333, + 426, + 277.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 183, + 277.66666666666663, + 426, + 293.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "Therefore, an obvious way to increase this variance is to maximize the smallest eigenvalue of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 303, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 124, + 321 + ], + "score": 1.0, + "content": "Var", + "type": "text" + }, + { + "bbox": [ + 125, + 308, + 185, + 321 + ], + "score": 0.9, + "content": "\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 307, + 303, + 321 + ], + "score": 1.0, + "content": ", which can be formulated as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 324, + 406, + 346 + ], + "lines": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "spans": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { \\phi } { \\mathrm { m a x i m i z e } } ~ \\mathbb { E } _ { { \\mathbf s } , { \\mathbf a } \\sim \\mathcal { D } } \\left[ \\lambda _ { \\mathrm { m i n } } \\left( { \\mathrm { V a r } \\left( { \\nabla _ { { \\mathbf a } } { Q _ { \\phi _ { j } } } \\left( { \\mathbf s } , { \\mathbf a } \\right) } \\right) } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "d430e1e503420db590a293421a6ff423942fda889c72749104eed69bbfbf1637.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 132, + 367 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 352, + 140, + 362 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 347, + 299, + 367 + ], + "score": 1.0, + "content": "denotes the collection of the parameters", + "type": "text" + }, + { + "bbox": [ + 300, + 350, + 335, + 364 + ], + "score": 0.92, + "content": "\\{ \\phi _ { j } \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 347, + 507, + 367 + ], + "score": 1.0, + "content": ". There are several methods to compute the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "smallest eigenvalue, such as the power method or the QR algorithm [27]. However, these iterative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "methods require constructing huge computation graphs, which makes optimizing the eigenvalue using", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "back-propagation inefficient. Instead, we aim to maximize the sum of all eigenvalues, which is equal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "to the total variance. By Lemma 1, it is equivalent to minimizing the norm of the average gradients:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 410, + 444, + 450 + ], + "lines": [ + { + "bbox": [ + 165, + 410, + 444, + 450 + ], + "spans": [ + { + "bbox": [ + 165, + 410, + 444, + 450 + ], + "score": 0.94, + "content": "\\underset { \\phi } { \\mathrm { m i n i m i z e } } \\ \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } \\left[ \\left. \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right] .", + "type": "interline_equation", + "image_path": "a2ca5fc1796f265830d4bb41d4f48e96f8e49c295e79a3291b4b822c1b02d1a5.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 165, + 410, + 444, + 423.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 165, + 423.3333333333333, + 444, + 436.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 436.66666666666663, + 444, + 449.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 504, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "With simple modification, we can reformulate Equation (4) as diversifying the gradients of each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 465, + 294, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 294, + 476 + ], + "score": 1.0, + "content": "Q-function network for in-distribution actions:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 478, + 473, + 531 + ], + "lines": [ + { + "bbox": [ + 137, + 478, + 473, + 531 + ], + "spans": [ + { + "bbox": [ + 137, + 478, + 473, + 531 + ], + "score": 0.95, + "content": "\\operatorname* { m i n i m i z e } J _ { \\mathrm { E S } } ( Q _ { \\phi } ) : = \\mathbb { E } _ { { \\bf s } , { \\bf a } \\sim \\mathcal { D } } \\left[ \\frac { 1 } { N - 1 } \\sum _ { 1 \\leq i \\neq j \\leq N } \\underbrace { \\left. \\nabla _ { { \\bf a } } Q _ { \\phi _ { i } } ( { \\bf s } , { \\bf a } ) , \\nabla _ { { \\bf a } } Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\right. } _ { \\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( { \\bf s } , { \\bf a } ) } \\right] .", + "type": "interline_equation", + "image_path": "0bb016ef1444d7ea26e6acd18f0bbb0053254181f6d9f0db8777ddf9750ab08f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 137, + 478, + 473, + 495.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 137, + 495.6666666666667, + 473, + 513.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 137, + 513.3333333333334, + 473, + 531.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Concretely, our final objective can be interpreted as measuring the pairwise alignment of the gradients", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 429, + 559 + ], + "score": 1.0, + "content": "using cosine similarity, which we denote as the Ensemble Similarity (ES) metric", + "type": "text" + }, + { + "bbox": [ + 430, + 545, + 484, + 558 + ], + "score": 0.93, + "content": "\\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 544, + 506, + 559 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "minimizing the ES values for every pair in the Q-ensemble with regard to the dataset state-actions.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "The illustration of the ensemble gradient diversification is shown in Figure 3b. Note that we instead", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "maximize the total variance to reduce the computational burden. Nevertheless, the modified objective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "is closely related to maximizing the smallest eigenvalue. The detailed explanation can be found in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 168, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 168, + 612 + ], + "score": 1.0, + "content": "Appendix A.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "We name the resulting actor-critic algorithm as Ensemble-Diversified Actor Critic (EDAC) and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "present the detailed procedure in Algorithm 1 (differences with the original SAC algorithm marked", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 302, + 650 + ], + "score": 1.0, + "content": "in blue). Note that Algorithm 1 reduces to SAC-", + "type": "text" + }, + { + "bbox": [ + 302, + 638, + 312, + 648 + ], + "score": 0.7, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 637, + 337, + 650 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 338, + 638, + 359, + 649 + ], + "score": 0.9, + "content": "\\eta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 637, + 506, + 650 + ], + "score": 1.0, + "content": ", and further reduces to vanilla SAC", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 178, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 149, + 660 + ], + "score": 1.0, + "content": "when also", + "type": "text" + }, + { + "bbox": [ + 149, + 649, + 174, + 659 + ], + "score": 0.89, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 649, + 178, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 107, + 675, + 191, + 689 + ], + "lines": [ + { + "bbox": [ + 104, + 673, + 193, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 193, + 692 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "We evaluate our proposed methods against the previous offline RL algorithms on the standard D4RL", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "benchmark [9] . Concretely, we perform our evaluation on MuJoCo Gym (Section 5.1) and Adroit", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 504, + 96 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 71, + 506, + 96 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 99, + 396, + 125 + ], + "lines": [ + { + "bbox": [ + 214, + 99, + 396, + 125 + ], + "spans": [ + { + "bbox": [ + 214, + 99, + 396, + 125 + ], + "score": 0.93, + "content": "\\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } _ { \\mathrm { m i n } } ) \\right) \\leq \\frac { 1 } { | \\mathcal { A } | } \\frac { N - 1 } { N } k ^ { 2 } \\epsilon ,", + "type": "interline_equation", + "image_path": "ee43964a993eee20e022a5c8c485989210b8ae0300a2d8436215b0db442b3d11.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 214, + 99, + 396, + 125 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 129, + 269, + 142 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 271, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 133, + 142 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 130, + 147, + 142 + ], + "score": 0.89, + "content": "| { \\cal A } |", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 129, + 271, + 142 + ], + "score": 1.0, + "content": "is the action space dimension.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 129, + 271, + 142 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 453, + 161 + ], + "score": 1.0, + "content": "We provide the proofs in Appendix A.1. Proposition 1 implies that if there exists such", + "type": "text" + }, + { + "bbox": [ + 453, + 150, + 477, + 160 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "that is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "small, which means the gradients of Q-function are well-aligned, then the variance of the Q-values", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "for an OOD action along a specific direction will also be small. This in turn degrades the ability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "of the ensembles to penalize OOD actions, which ultimately leads to requiring a large number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 187, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 187, + 205 + ], + "score": 1.0, + "content": "ensemble networks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 149, + 506, + 205 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 507, + 223 + ], + "score": 1.0, + "content": "To address this problem, we propose a regularizer that effectively increases the variance of the Q-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "values for near-distribution OOD actions. Note that the variance is lower-bounded by some constant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 273, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 248, + 245 + ], + "score": 1.0, + "content": "multiple of the smallest eigenvalue", + "type": "text" + }, + { + "bbox": [ + 248, + 232, + 268, + 243 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 230, + 273, + 245 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 208, + 507, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 245, + 426, + 294 + ], + "lines": [ + { + "bbox": [ + 183, + 245, + 426, + 294 + ], + "spans": [ + { + "bbox": [ + 183, + 245, + 426, + 294 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathrm { V a r } \\left( Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } + k \\mathbf { w } ) \\right) \\approx k ^ { 2 } \\mathbf { w } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } } \\\\ & { \\quad \\quad \\quad \\quad \\geq k ^ { 2 } \\mathbf { w } _ { \\operatorname* { m i n } } ^ { \\top } \\mathrm { V a r } \\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right) \\mathbf { w } _ { \\operatorname* { m i n } } } \\\\ & { \\quad \\quad \\quad = k ^ { 2 } \\lambda _ { \\operatorname* { m i n } } . } \\end{array}", + "type": "interline_equation", + "image_path": "811413a540f1df9255ac01cd613a36f965755918dbbfb18243bb3a4f2c812718.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 183, + 245, + 426, + 261.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 183, + 261.3333333333333, + 426, + 277.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 183, + 277.66666666666663, + 426, + 293.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "Therefore, an obvious way to increase this variance is to maximize the smallest eigenvalue of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 303, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 124, + 321 + ], + "score": 1.0, + "content": "Var", + "type": "text" + }, + { + "bbox": [ + 125, + 308, + 185, + 321 + ], + "score": 0.9, + "content": "\\left( \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 307, + 303, + 321 + ], + "score": 1.0, + "content": ", which can be formulated as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 295, + 506, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 324, + 406, + 346 + ], + "lines": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "spans": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { \\phi } { \\mathrm { m a x i m i z e } } ~ \\mathbb { E } _ { { \\mathbf s } , { \\mathbf a } \\sim \\mathcal { D } } \\left[ \\lambda _ { \\mathrm { m i n } } \\left( { \\mathrm { V a r } \\left( { \\nabla _ { { \\mathbf a } } { Q _ { \\phi _ { j } } } \\left( { \\mathbf s } , { \\mathbf a } \\right) } \\right) } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "d430e1e503420db590a293421a6ff423942fda889c72749104eed69bbfbf1637.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 324, + 406, + 346 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 132, + 367 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 352, + 140, + 362 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 347, + 299, + 367 + ], + "score": 1.0, + "content": "denotes the collection of the parameters", + "type": "text" + }, + { + "bbox": [ + 300, + 350, + 335, + 364 + ], + "score": 0.92, + "content": "\\{ \\phi _ { j } \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 347, + 507, + 367 + ], + "score": 1.0, + "content": ". There are several methods to compute the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "smallest eigenvalue, such as the power method or the QR algorithm [27]. However, these iterative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "methods require constructing huge computation graphs, which makes optimizing the eigenvalue using", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "back-propagation inefficient. Instead, we aim to maximize the sum of all eigenvalues, which is equal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "to the total variance. By Lemma 1, it is equivalent to minimizing the norm of the average gradients:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 347, + 507, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 410, + 444, + 450 + ], + "lines": [ + { + "bbox": [ + 165, + 410, + 444, + 450 + ], + "spans": [ + { + "bbox": [ + 165, + 410, + 444, + 450 + ], + "score": 0.94, + "content": "\\underset { \\phi } { \\mathrm { m i n i m i z e } } \\ \\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } \\left[ \\left. \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { i } } ( \\mathbf { s } , \\mathbf { a } ) , \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\nabla _ { \\mathbf { a } } Q _ { \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } ) \\right. \\right] .", + "type": "interline_equation", + "image_path": "a2ca5fc1796f265830d4bb41d4f48e96f8e49c295e79a3291b4b822c1b02d1a5.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 165, + 410, + 444, + 423.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 165, + 423.3333333333333, + 444, + 436.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 436.66666666666663, + 444, + 449.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 504, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "With simple modification, we can reformulate Equation (4) as diversifying the gradients of each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 465, + 294, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 294, + 476 + ], + "score": 1.0, + "content": "Q-function network for in-distribution actions:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 452, + 506, + 476 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 478, + 473, + 531 + ], + "lines": [ + { + "bbox": [ + 137, + 478, + 473, + 531 + ], + "spans": [ + { + "bbox": [ + 137, + 478, + 473, + 531 + ], + "score": 0.95, + "content": "\\operatorname* { m i n i m i z e } J _ { \\mathrm { E S } } ( Q _ { \\phi } ) : = \\mathbb { E } _ { { \\bf s } , { \\bf a } \\sim \\mathcal { D } } \\left[ \\frac { 1 } { N - 1 } \\sum _ { 1 \\leq i \\neq j \\leq N } \\underbrace { \\left. \\nabla _ { { \\bf a } } Q _ { \\phi _ { i } } ( { \\bf s } , { \\bf a } ) , \\nabla _ { { \\bf a } } Q _ { \\phi _ { j } } ( { \\bf s } , { \\bf a } ) \\right. } _ { \\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( { \\bf s } , { \\bf a } ) } \\right] .", + "type": "interline_equation", + "image_path": "0bb016ef1444d7ea26e6acd18f0bbb0053254181f6d9f0db8777ddf9750ab08f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 137, + 478, + 473, + 495.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 137, + 495.6666666666667, + 473, + 513.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 137, + 513.3333333333334, + 473, + 531.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Concretely, our final objective can be interpreted as measuring the pairwise alignment of the gradients", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 429, + 559 + ], + "score": 1.0, + "content": "using cosine similarity, which we denote as the Ensemble Similarity (ES) metric", + "type": "text" + }, + { + "bbox": [ + 430, + 545, + 484, + 558 + ], + "score": 0.93, + "content": "\\mathrm { E S } _ { \\phi _ { i } , \\phi _ { j } } ( \\mathbf { s } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 544, + 506, + 559 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "minimizing the ES values for every pair in the Q-ensemble with regard to the dataset state-actions.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "The illustration of the ensemble gradient diversification is shown in Figure 3b. Note that we instead", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "maximize the total variance to reduce the computational burden. Nevertheless, the modified objective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "is closely related to maximizing the smallest eigenvalue. The detailed explanation can be found in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 168, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 168, + 612 + ], + "score": 1.0, + "content": "Appendix A.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 534, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "We name the resulting actor-critic algorithm as Ensemble-Diversified Actor Critic (EDAC) and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "present the detailed procedure in Algorithm 1 (differences with the original SAC algorithm marked", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 302, + 650 + ], + "score": 1.0, + "content": "in blue). 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For the implementation details of our algorithm and the baselines, please", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "refer to Appendix B and C. Also, we provide more experiments such as comparison with more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 189, + 461 + ], + "score": 1.0, + "content": "baselines, CQL with", + "type": "text" + }, + { + "bbox": [ + 189, + 448, + 199, + 457 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 446, + 505, + 461 + ], + "score": 1.0, + "content": "Q-networks, and hyperparameter sensitivity from Appendix E to H. The code", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 458, + 188, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 188, + 470 + ], + "score": 1.0, + "content": "is available online3.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 106, + 483, + 306, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 306, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 306, + 497 + ], + "score": 1.0, + "content": "5.1 Evaluation on D4RL MuJoCo Gym tasks", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "We first evaluate each method on D4RL MuJoCo Gym tasks which consist of three environments,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "halfcheetah, hopper, and walker2d, each with six datasets from different data-collecting policies.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "In detail, the considered policies are random: a uniform random policy, expert: a fully trained", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "online expert, medium: a suboptimal policy with approximately 1/3 the performance of the expert,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "medium-expert: a mixture of medium and expert policies, medium-replay: the replay buffer of a policy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "trained up to the performance of the medium agent, and full-replay: the final replay buffer of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 569, + 507, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 507, + 583 + ], + "score": 1.0, + "content": "expert policy. 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+ ], + "score": 0.89, + "content": "\\phi _ { i } ^ { \\prime } \\rho \\phi _ { i } ^ { \\prime } + ( 1 - \\rho ) \\phi _ { i }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "(Section 5.2) domains. We consider the following baselines: SAC, the backbone algorithm of our", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "method, CQL, the previous state-of-the-art on the D4RL benchmark, REM [2], an offline RL method", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "score": 1.0, + "content": "which utilized Q-network ensemble on discrete control environments, and BC, the behavior cloning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "method. We evaluate each method under the normalized average return metric where the average", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "return is scaled such that 0 and 100 each equals the performance of a random policy and an online", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "score": 1.0, + "content": "expert policy. In addition to the performance evaluation, we compare the computational cost of each", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "method (Section 5.3). For the implementation details of our algorithm and the baselines, please", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "refer to Appendix B and C. Also, we provide more experiments such as comparison with more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 189, + 461 + ], + "score": 1.0, + "content": "baselines, CQL with", + "type": "text" + }, + { + "bbox": [ + 189, + 448, + 199, + 457 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 446, + 505, + 461 + ], + "score": 1.0, + "content": "Q-networks, and hyperparameter sensitivity from Appendix E to H. The code", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 458, + 188, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 188, + 470 + ], + "score": 1.0, + "content": "is available online3.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18, + "bbox_fs": [ + 111, + 321, + 345, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 505, + 469 + ], + "lines": [], + "index": 23.5, + "bbox_fs": [ + 105, + 360, + 506, + 470 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 106, + 483, + 306, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 306, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 306, + 497 + ], + "score": 1.0, + "content": "5.1 Evaluation on D4RL MuJoCo Gym tasks", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "We first evaluate each method on D4RL MuJoCo Gym tasks which consist of three environments,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "halfcheetah, hopper, and walker2d, each with six datasets from different data-collecting policies.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "In detail, the considered policies are random: a uniform random policy, expert: a fully trained", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "online expert, medium: a suboptimal policy with approximately 1/3 the performance of the expert,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "medium-expert: a mixture of medium and expert policies, medium-replay: the replay buffer of a policy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "trained up to the performance of the medium agent, and full-replay: the final replay buffer of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 569, + 507, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 507, + 583 + ], + "score": 1.0, + "content": "expert policy. Each dataset consists of 1M transitions except for medium-expert and medium-replay.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 502, + 507, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 506, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 332, + 597 + ], + "score": 1.0, + "content": "The experiment results in Table 1 show EDAC and SAC-", + "type": "text" + }, + { + "bbox": [ + 333, + 586, + 342, + 596 + ], + "score": 0.7, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "both outperform or are competitive with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "the previous state-of-the-art on all of the tasks considered. Notably, the performance gap is especially", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "high for random, medium, and medium-replay datasets, where the performances of the previous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "works are relatively low. Both the proposed methods achieve average normalized scores over 80,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 275, + 641 + ], + "score": 1.0, + "content": "reducing the gap with the online expert by", + "type": "text" + }, + { + "bbox": [ + 276, + 630, + 295, + 640 + ], + "score": 0.85, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 630, + 506, + 641 + ], + "score": 1.0, + "content": "compared to CQL. While the performance of EDAC", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 640, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 640, + 304, + 652 + ], + "score": 1.0, + "content": "is marginally better than the performance of SAC-", + "type": "text" + }, + { + "bbox": [ + 304, + 641, + 313, + 650 + ], + "score": 0.69, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 640, + 506, + 652 + ], + "score": 1.0, + "content": ", EDAC achieves this result with a much smaller", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 651, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 360, + 663 + ], + "score": 1.0, + "content": "Q-ensemble size. As noted in Figure 5, on hopper tasks, SAC-", + "type": "text" + }, + { + "bbox": [ + 361, + 651, + 371, + 661 + ], + "score": 0.67, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 651, + 435, + 663 + ], + "score": 1.0, + "content": "requires 200 to", + "type": "text" + }, + { + "bbox": [ + 435, + 651, + 462, + 662 + ], + "score": 0.27, + "content": "5 0 0 \\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 651, + 506, + 663 + ], + "score": 1.0, + "content": "-networks,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 662, + 246, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 246, + 674 + ], + "score": 1.0, + "content": "while EDAC requires less than 50.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 586, + 506, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 504, + 690 + ], + "score": 1.0, + "content": "Figure 6 compares the distance between the actions chosen by each method and the dataset actions.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 201, + 704 + ], + "score": 1.0, + "content": "Concretely, we measure", + "type": "text" + }, + { + "bbox": [ + 201, + 689, + 315, + 702 + ], + "score": 0.89, + "content": "\\mathbb { E } _ { ( \\mathbf { s } , \\mathbf { a } ) \\sim \\mathcal { D } , \\hat { \\mathbf { a } } \\sim \\pi _ { \\boldsymbol { \\theta } } ( \\cdot | \\mathbf { s } ) } [ | \\hat { \\mathbf { a } } - \\mathbf { a } | | _ { 2 } ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 688, + 381, + 704 + ], + "score": 1.0, + "content": "for EDAC, SAC-", + "type": "text" + }, + { + "bbox": [ + 381, + 690, + 391, + 699 + ], + "score": 0.72, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 688, + 506, + 704 + ], + "score": 1.0, + "content": ", CQL, SAC-2, and a random", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 72, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 146, + 87 + ], + "score": 1.0, + "content": "policy on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 146, + 75, + 152, + 83 + ], + "score": 0.7, + "content": "^ *", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 153, + 72, + 506, + 87 + ], + "score": 1.0, + "content": "-medium datasets. 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This shows the advantage of the uncertainty-based penalization which", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 406, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 406, + 107 + ], + "score": 1.0, + "content": "considers the prediction confidence other than penalizing all OOD actions.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table", + "bbox": [ + 106, + 147, + 506, + 349 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 124, + 504, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "score": 1.0, + "content": "Table 1: Normalized average returns on D4RL Gym tasks, averaged over 4 random seeds. 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Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
halfcheetah-random2.2±0.029.7±1.4-0.8±1.135.431.3±3.528.0±0.928.4±1.0
halfcheetah-medium43.2±0.655.2±27.8-0.8±1.344.446.9±0.467.5±1.265.9±0.6
halfcheetah-expert91.8±1.5-0.8±1.84.1±5.7104.897.3±1.1105.2±2.6106.8±3.4
halfcheetah-medium-expert44.0±1.628.4±19.40.7±3.762.495.0±1.4107.1±2.0106.3±1.9
halfcheetah-medium-replay37.6±2.10.8±1.06.6±11.046.245.3±0.363.9±0.861.3±1.9
halfcheetah-full-replay62.9±0.886.8±1.027.8±35.4-76.9±0.984.5±1.284.6±0.9
hopper-random3.7±0.69.9±1.53.4±2.210.85.3±0.631.3±0.025.3±10.4
hopper-medium54.1±3.80.8±0.00.7±0.086.661.9±6.4100.3±0.3101.6±0.6
hopper-expert107.7±9.70.7±0.00.8±0.0109.9106.5±9.1110.3±0.3110.1±0.1
hopper-medium-expert53.9±4.70.7±0.00.8±0.0111.096.9±15.1110.1±0.3110.7±0.1
hopper-medium-replay16.6±4.87.4±0.527.5±15.248.686.3±7.3101.8±0.5101.0±0.5
hopper-full-replay19.9±12.941.1±17.919.7±24.6-101.9±0.6102.9±0.3105.4±0.7
walker2d-random1.3±0.10.9±0.86.9±8.37.05.4±1.721.7±0.016.6±7.0
walker2d-medium70.9±11.0-0.3±0.20.2±0.774.579.5±3.287.9±0.292.5±0.8
walker2d-expert108.7±0.20.7±0.31.0±2.3121.6109.3±0.1107.4±2.4115.1±1.9
walker2d-medium-expert90.1±13.21.9±3.9-0.1±0.098.7109.1±0.2116.7±0.4114.7±0.9
walker2d-medium-replay20.3±9.8-0.4±0.312.5±6.232.676.8±10.078.7±0.787.1±2.3
walker2d-full-replay68.8±17.727.9±47.3-0.2±0.3-94.2±1.994.6±0.599.8±0.7
Average49.916.26.2-73.784.585.2
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For more details of the experiment,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 701, + 219, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 219, + 712 + ], + "score": 1.0, + "content": "please refer to Appendix C.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 17.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 504, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 72, + 506, + 107 + ], + "lines_deleted": true + }, + { + "type": "table", + "bbox": [ + 106, + 147, + 506, + 349 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 124, + 504, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "score": 1.0, + "content": "Table 1: Normalized average returns on D4RL Gym tasks, averaged over 4 random seeds. CQL", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 334, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 334, + 149 + ], + "score": 1.0, + "content": "(Paper) denotes the results reported in the original paper.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 147, + 506, + 349 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 147, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 349 + ], + "score": 0.984, + "html": "
Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
halfcheetah-random2.2±0.029.7±1.4-0.8±1.135.431.3±3.528.0±0.928.4±1.0
halfcheetah-medium43.2±0.655.2±27.8-0.8±1.344.446.9±0.467.5±1.265.9±0.6
halfcheetah-expert91.8±1.5-0.8±1.84.1±5.7104.897.3±1.1105.2±2.6106.8±3.4
halfcheetah-medium-expert44.0±1.628.4±19.40.7±3.762.495.0±1.4107.1±2.0106.3±1.9
halfcheetah-medium-replay37.6±2.10.8±1.06.6±11.046.245.3±0.363.9±0.861.3±1.9
halfcheetah-full-replay62.9±0.886.8±1.027.8±35.4-76.9±0.984.5±1.284.6±0.9
hopper-random3.7±0.69.9±1.53.4±2.210.85.3±0.631.3±0.025.3±10.4
hopper-medium54.1±3.80.8±0.00.7±0.086.661.9±6.4100.3±0.3101.6±0.6
hopper-expert107.7±9.70.7±0.00.8±0.0109.9106.5±9.1110.3±0.3110.1±0.1
hopper-medium-expert53.9±4.70.7±0.00.8±0.0111.096.9±15.1110.1±0.3110.7±0.1
hopper-medium-replay16.6±4.87.4±0.527.5±15.248.686.3±7.3101.8±0.5101.0±0.5
hopper-full-replay19.9±12.941.1±17.919.7±24.6-101.9±0.6102.9±0.3105.4±0.7
walker2d-random1.3±0.10.9±0.86.9±8.37.05.4±1.721.7±0.016.6±7.0
walker2d-medium70.9±11.0-0.3±0.20.2±0.774.579.5±3.287.9±0.292.5±0.8
walker2d-expert108.7±0.20.7±0.31.0±2.3121.6109.3±0.1107.4±2.4115.1±1.9
walker2d-medium-expert90.1±13.21.9±3.9-0.1±0.098.7109.1±0.2116.7±0.4114.7±0.9
walker2d-medium-replay20.3±9.8-0.4±0.312.5±6.232.676.8±10.078.7±0.787.1±2.3
walker2d-full-replay68.8±17.727.9±47.3-0.2±0.3-94.2±1.994.6±0.599.8±0.7
Average49.916.26.2-73.784.585.2
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M-E denotes medium-expert. We omit the results of medium-replay and full-replay as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 129, + 537 + ], + "score": 1.0, + "content": "SAC-", + "type": "text" + }, + { + "bbox": [ + 129, + 525, + 140, + 535 + ], + "score": 0.61, + "content": ". N", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "already works well with a small number of ensembles (less than or equal to 5). 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For more details of the experiment,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 701, + 219, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 219, + 712 + ], + "score": 1.0, + "content": "please refer to Appendix C.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 17.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 72, + 273, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 274, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 274, + 86 + ], + "score": 1.0, + "content": "5.2 Evaluation on D4RL Adroit tasks", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 92, + 506, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 91, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 506, + 106 + ], + "score": 1.0, + "content": "We also experiment on the more complex D4RL Adroit tasks that require controlling a 24-DoF robotic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "hand to perform tasks such as aligning a pen, hammering a nail, opening a door, or relocating a ball.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "We use two types of datasets for each environment: human, containing 25 trajectories of human", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "demonstrations, and cloned, a 50-50 mixture between the demonstration data and the behavioral", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "cloned policy on the demonstrations. 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Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
pen-human25.8±8.84.3±3.85.4±4.355.835.2±6.69.5±1.152.1±8.6
hammer-human3.1±3.20.2±0.00.3±0.02.10.6±0.50.3±0.00.8±0.4
door-human2.8±0.7-0.3±0.0-0.3±0.09.11.2±1.8-0.3±0.010.7±6.8
relocate-human0.0±0.0-0.3±0.0-0.3±0.00.350.0±0.0-0.1±0.10.1±0.1
pen-cloned38.3±11.9-0.8±3.2-1.0±0.140.327.2±11.364.1±8.768.2±7.3
hammer-cloned0.7±0.30.1±0.1-0.3±0.05.71.4±2.10.2±0.20.3±0.0
door-cloned0.0±0.0-0.3±0.1-0.3±0.03.52.4±2.4-0.3±0.09.6±8.3
relocate-cloned0.1±0.0-0.1±0.1-0.2±0.2-0.10.0±0.00.0±0.00.0±0.0
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For pen-", + "type": "text" + }, + { + "bbox": [ + 341, + 339, + 347, + 347 + ], + "score": 0.76, + "content": "^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "tasks, where the considered algorithms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "achieve meaningful performance, EDAC outperforms or matches with the previous state-of-the-art.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 309, + 371 + ], + "score": 1.0, + "content": "Especially, for pen-cloned, both EDAC and SAC-", + "type": "text" + }, + { + "bbox": [ + 309, + 359, + 319, + 369 + ], + "score": 0.64, + "content": ". 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However, our method with ensemble diversification successfully overcomes this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 147, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 147, + 416 + ], + "score": 1.0, + "content": "difficulty.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 265, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 267, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 267, + 441 + ], + "score": 1.0, + "content": "5.3 Computational cost comparison", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 336, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 336, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 336, + 458 + ], + "score": 1.0, + "content": "We compared the computational cost of our methods with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 337, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 337, + 470 + ], + "score": 1.0, + "content": "vanilla SAC and CQL on hopper-medium-v2, where our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 337, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 337, + 480 + ], + "score": 1.0, + "content": "methods require the largest number of Q-networks. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 337, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 337, + 492 + ], + "score": 1.0, + "content": "each method, we measure the runtime per training epoch", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 338, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 338, + 503 + ], + "score": 1.0, + "content": "(1000 gradient steps) along with GPU memory consump-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 337, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 337, + 513 + ], + "score": 1.0, + "content": "tion. We run our experiments on a single machine with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 512, + 328, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 328, + 524 + ], + "score": 1.0, + "content": "one RTX 3090 GPU and provide the results in Table 3.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "table", + "bbox": [ + 348, + 476, + 498, + 554 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 343, + 447, + 504, + 469 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 342, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 342, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "Table 3: Computational costs of each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 343, + 458, + 379, + 469 + ], + "spans": [ + { + "bbox": [ + 343, + 458, + 379, + 469 + ], + "score": 1.0, + "content": "method.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 348, + 476, + 498, + 554 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 348, + 476, + 498, + 554 + ], + "spans": [ + { + "bbox": [ + 348, + 476, + 498, + 554 + ], + "score": 0.975, + "html": "
Runtime (s/epoch)GPU Mem. (GB)
SAC21.41.3
CQL38.21.4
SAC-50044.15.1
EDAC30.81.8
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Note that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 336, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 336, + 562 + ], + "score": 1.0, + "content": "CQL is about twice as slower than vanilla SAC due to the", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "additional computations for Q-value regularization (e.g., dual update and approximate logsumexp", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 434, + 584 + ], + "score": 1.0, + "content": "via sampling). Meanwhile, the inference to the Q-network ensemble in SAC-", + "type": "text" + }, + { + "bbox": [ + 434, + 572, + 444, + 582 + ], + "score": 0.62, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "and EDAC is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "embarrassingly parallelizable, minimizing the runtime increase with the number of Q-networks.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Also, we emphasize that our gradient diversification term in Equation (4) has linear computational", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 406, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 406, + 618 + ], + "score": 1.0, + "content": "complexity, as we can reformulate the term using the sum of the gradients.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 631, + 202, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 203, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 203, + 646 + ], + "score": 1.0, + "content": "6 Related Works", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Model-free offline RL A popular approach for offline RL is to regularize the learned policy to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "close to the behavior policy where the offline dataset was collected. BCQ [11] uses a generative", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "model to produce actions with high similarity to the dataset and trains a restricted policy to choose", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "the best action from the neighborhood of the generated actions. 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Note that for the Adroit tasks, we could not reproduce the CQL", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 146, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 161 + ], + "score": 1.0, + "content": "results from the paper completely. 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Task NameBCSACREMCQL (Paper)CQL (Reproduced)SAC-N (Ours)EDAC (Ours)
pen-human25.8±8.84.3±3.85.4±4.355.835.2±6.69.5±1.152.1±8.6
hammer-human3.1±3.20.2±0.00.3±0.02.10.6±0.50.3±0.00.8±0.4
door-human2.8±0.7-0.3±0.0-0.3±0.09.11.2±1.8-0.3±0.010.7±6.8
relocate-human0.0±0.0-0.3±0.0-0.3±0.00.350.0±0.0-0.1±0.10.1±0.1
pen-cloned38.3±11.9-0.8±3.2-1.0±0.140.327.2±11.364.1±8.768.2±7.3
hammer-cloned0.7±0.30.1±0.1-0.3±0.05.71.4±2.10.2±0.20.3±0.0
door-cloned0.0±0.0-0.3±0.1-0.3±0.03.52.4±2.4-0.3±0.09.6±8.3
relocate-cloned0.1±0.0-0.1±0.1-0.2±0.2-0.10.0±0.00.0±0.00.0±0.0
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For pen-", + "type": "text" + }, + { + "bbox": [ + 341, + 339, + 347, + 347 + ], + "score": 0.76, + "content": "^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "tasks, where the considered algorithms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "achieve meaningful performance, EDAC outperforms or matches with the previous state-of-the-art.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 309, + 371 + ], + "score": 1.0, + "content": "Especially, for pen-cloned, both EDAC and SAC-", + "type": "text" + }, + { + "bbox": [ + 309, + 359, + 319, + 369 + ], + "score": 0.64, + "content": ". 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However, our method with ensemble diversification successfully overcomes this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 147, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 147, + 416 + ], + "score": 1.0, + "content": "difficulty.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 337, + 506, + 416 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 265, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 267, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 267, + 441 + ], + "score": 1.0, + "content": "5.3 Computational cost comparison", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 336, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 336, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 336, + 458 + ], + "score": 1.0, + "content": "We compared the computational cost of our methods with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 337, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 337, + 470 + ], + "score": 1.0, + "content": "vanilla SAC and CQL on hopper-medium-v2, where our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 337, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 337, + 480 + ], + "score": 1.0, + "content": "methods require the largest number of Q-networks. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 337, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 337, + 492 + ], + "score": 1.0, + "content": "each method, we measure the runtime per training epoch", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 338, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 338, + 503 + ], + "score": 1.0, + "content": "(1000 gradient steps) along with GPU memory consump-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 337, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 337, + 513 + ], + "score": 1.0, + "content": "tion. We run our experiments on a single machine with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 512, + 328, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 328, + 524 + ], + "score": 1.0, + "content": "one RTX 3090 GPU and provide the results in Table 3.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 446, + 338, + 524 + ] + }, + { + "type": "table", + "bbox": [ + 348, + 476, + 498, + 554 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 343, + 447, + 504, + 469 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 342, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 342, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "Table 3: Computational costs of each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 343, + 458, + 379, + 469 + ], + "spans": [ + { + "bbox": [ + 343, + 458, + 379, + 469 + ], + "score": 1.0, + "content": "method.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 348, + 476, + 498, + 554 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 348, + 476, + 498, + 554 + ], + "spans": [ + { + "bbox": [ + 348, + 476, + 498, + 554 + ], + "score": 0.975, + "html": "
Runtime (s/epoch)GPU Mem. (GB)
SAC21.41.3
CQL38.21.4
SAC-50044.15.1
EDAC30.81.8
", + "type": "table", + "image_path": "f1c501c86effbba26588f9524287241b6686582cec6f2ed75bb579342e8318c4.jpg" + } + ] + } + ], + "index": 31.0, + "virtual_lines": [ + { + "bbox": [ + 348, + 476, + 498, + 515.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 348, + 515.0, + 498, + 554.0 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "index": 26.75 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 336, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 337, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 337, + 540 + ], + "score": 1.0, + "content": "As the result shows, our method EDAC runs faster than", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 337, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 337, + 551 + ], + "score": 1.0, + "content": "CQL with comparable memory consumption. Note that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 336, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 336, + 562 + ], + "score": 1.0, + "content": "CQL is about twice as slower than vanilla SAC due to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "additional computations for Q-value regularization (e.g., dual update and approximate logsumexp", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 434, + 584 + ], + "score": 1.0, + "content": "via sampling). Meanwhile, the inference to the Q-network ensemble in SAC-", + "type": "text" + }, + { + "bbox": [ + 434, + 572, + 444, + 582 + ], + "score": 0.62, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "and EDAC is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "embarrassingly parallelizable, minimizing the runtime increase with the number of Q-networks.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Also, we emphasize that our gradient diversification term in Equation (4) has linear computational", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 406, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 406, + 618 + ], + "score": 1.0, + "content": "complexity, as we can reformulate the term using the sum of the gradients.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 528, + 337, + 562 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 617 + ], + "lines": [], + "index": 36, + "bbox_fs": [ + 105, + 561, + 506, + 618 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 631, + 202, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 203, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 203, + 646 + ], + "score": 1.0, + "content": "6 Related Works", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Model-free offline RL A popular approach for offline RL is to regularize the learned policy to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "close to the behavior policy where the offline dataset was collected. BCQ [11] uses a generative", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "model to produce actions with high similarity to the dataset and trains a restricted policy to choose", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "the best action from the neighborhood of the generated actions. Another line of work, such as BEAR", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "[15] or BRAC [28], stabilizes policy learning by penalizing the divergence from the dataset measured", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "score": 1.0, + "content": "by KL divergence or MMD. While these policy-constraint methods demonstrate high performance on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "datasets from expert behavior policies, they fail to find optimal policies from datasets with suboptimal", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "policies due to the strict policy constraints [9]. Also, these methods require an accurate estimation of", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "the behavior policy, which might be difficult in complex settings with multiple behavior sources or", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "score": 1.0, + "content": "high-dimensional environments. To address these issues, CQL [16] directly regularizes Q-functions", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "by introducing a term that minimizes the Q-values for out-of-distribution actions and maximizes the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "score": 1.0, + "content": "Q-values for in-distribution actions. Without such explicit regularizations, REM [2] proposes to use a", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "random convex combination of Q-network ensembles on environments with discrete action spaces", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 124, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 124, + 162 + ], + "score": 1.0, + "content": "[4].", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 655, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "datasets from expert behavior policies, they fail to find optimal policies from datasets with suboptimal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "policies due to the strict policy constraints [9]. Also, these methods require an accurate estimation of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "the behavior policy, which might be difficult in complex settings with multiple behavior sources or", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "score": 1.0, + "content": "high-dimensional environments. To address these issues, CQL [16] directly regularizes Q-functions", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "by introducing a term that minimizes the Q-values for out-of-distribution actions and maximizes the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "score": 1.0, + "content": "Q-values for in-distribution actions. Without such explicit regularizations, REM [2] proposes to use a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "random convex combination of Q-network ensembles on environments with discrete action spaces", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 124, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 124, + 162 + ], + "score": 1.0, + "content": "[4].", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Estimation bias in Q-learning While Q-learning is one of the most popular algorithms in reinforce-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 429, + 200 + ], + "score": 1.0, + "content": "ment learning, it suffers from overestimation bias due to the maximum operation", + "type": "text" + }, + { + "bbox": [ + 429, + 187, + 505, + 199 + ], + "score": 0.81, + "content": "\\mathrm { m a x } _ { \\mathbf { a } ^ { \\prime } \\in \\mathcal { A } } Q ( \\mathbf { s } ^ { \\prime } , \\mathbf { a } ^ { \\prime } )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 507, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 507, + 212 + ], + "score": 1.0, + "content": "used during Q-function updates [10, 25]. This overestimation bias, together with the bootstrapping,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "can lead to a catastrophic build-up of errors during the Q-learning process. To resolve this issue, TD3", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "[10] introduces a clipped version of Double Q-learning [25] that takes the minimum value of two", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "critics. Subsequently, Maxmin Q-learning [17] theoretically shows that the overestimation bias can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "be controlled by the number of ensembles in the clipped Q-learning. The overestimation problem in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "Q-learning can be exacerbated in the offline setting since the extrapolation error cannot be corrected", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "with further interactions with the environment, and existing offline RL algorithms handle the bias by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 499, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 499, + 287 + ], + "score": 1.0, + "content": "introducing constrained policy optimization [11, 15] or conservative Q-learning frameworks [16].", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "Uncertainty measures in RL Uncertainty estimates have been widely used in RL for various", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "purposes including exploration, Q-learning, and planning. Bootstrapped DQN [21] leverages an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "ensemble of Q-functions to quantify the uncertainty of the Q-value, and utilizes it for efficient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "exploration. Following this work, the UCB exploration algorithm [5] constructs an upper confidence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "bound [3] of the Q-values using the empirical mean and standard deviation of Q-ensembles, which is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "used to promote efficient exploration by applying the principle of optimism in the face of uncertainty", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "[7]. Osband et al. 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The", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 88, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 88, + 635, + 100, + 645 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "linear contextual bandit problem has also been extended to richer classes of parametric bandits such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 89, + 643, + 398, + 656 + ], + "spans": [ + { + "bbox": [ + 89, + 646, + 99, + 655 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 104, + 643, + 398, + 656 + ], + "score": 1.0, + "content": "as the generalized linear bandits [24, 35] and kernelised bandits [44, 15].", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 90, + 660, + 504, + 715 + ], + "lines": [ + { + "bbox": [ + 89, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 89, + 662, + 100, + 671 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "With the prevalence of deep neural networks (DNNs) and their phenomenal performances in many", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 671, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 89, + 673, + 100, + 682 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 106, + 671, + 504, + 682 + ], + "score": 1.0, + "content": "machine learning tasks [32, 25], there has emerged a line of work that employs DNNs to increase the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 89, + 684, + 100, + 693 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "representation power of contextual bandit algorithms [5, 38, 17, 49, 52, 20, 51]. 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Compared with existing neural", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 365, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 93, + 368, + 99, + 376 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 141, + 365, + 470, + 378 + ], + "score": 1.0, + "content": "contextual bandit algorithms, our approach is computationally much more efficient", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 377, + 436, + 389 + ], + "spans": [ + { + "bbox": [ + 90, + 379, + 100, + 387 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 142, + 377, + 436, + 389 + ], + "score": 1.0, + "content": "since it only needs to explore in the last layer of the deep neural network.", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 11.5, + "bbox_fs": [ + 90, + 276, + 470, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 406, + 191, + 420 + ], + "lines": [ + { + "bbox": [ + 87, + 406, + 192, + 422 + ], + "spans": [ + { + "bbox": [ + 87, + 406, + 192, + 422 + ], + "score": 1.0, + "content": "11 1 Introduction", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "index", + "bbox": [ + 90, + 430, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 90, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 90, + 433, + 99, + 442 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 104, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "Multi-armed bandits (MAB) [9, 8, 30] are a class of online decision-making problems where an", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 89, + 444, + 100, + 454 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "agent needs to learn to maximize its expected cumulative reward while repeatedly interacting with a", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 89, + 455, + 100, + 465 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "partially known environment. Based on a bandit algorithm (also called a strategy or policy), in each", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 88, + 465, + 101, + 477 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "round, the agent adaptively chooses an arm, and then observes and receives a reward associated with", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 477, + 100, + 487 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "that arm. Since only the reward of the chosen arm will be observed (bandit information feedback),", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 89, + 488, + 100, + 497 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 104, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "a good bandit algorithm has to deal with the exploration-exploitation dilemma: trade-off between", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 89, + 499, + 100, + 509 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "pulling the best arm based on existing knowledge/history data (exploitation) and trying the arms that", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 507, + 279, + 520 + ], + "spans": [ + { + "bbox": [ + 89, + 510, + 100, + 519 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 507, + 279, + 520 + ], + "score": 1.0, + "content": "have not been fully explored (exploration).", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 89, + 525, + 100, + 535 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "In many real-world applications, the agent will also be able to access detailed contexts associated", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 88, + 537, + 99, + 546 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "with the arms. For example, when a company wants to choose an advertisement to present to a user,", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 545, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 88, + 547, + 100, + 557 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 545, + 507, + 559 + ], + "score": 1.0, + "content": "the recommendation will be much more accurate if the company takes into consideration the contents,", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 557, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 89, + 559, + 100, + 568 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 557, + 507, + 569 + ], + "score": 1.0, + "content": "specifications, and other features of the advertisements in the arm set as well as the profile of the user.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 89, + 569, + 100, + 579 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "score": 1.0, + "content": "To encode the contextual information, contextual bandit models and algorithms have been developed,", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 88, + 580, + 100, + 590 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "and widely studied both in theory and in practice [19, 39, 34, 16, 1]. Most existing contextual bandit", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 88, + 591, + 100, + 601 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "algorithms assume that the expected reward of an arm at a context is a linear function in a known", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 89, + 602, + 99, + 612 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "context-action feature vector, which leads to many useful algorithms such as LinUCB [16], OFUL [1],", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 88, + 613, + 100, + 623 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "etc. The representation power of the linear model can be limited in applications such as marketing,", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 88, + 624, + 100, + 633 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "social networking, clinical studies, etc., where the rewards are usually counts or binary variables. The", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 88, + 635, + 100, + 645 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "linear contextual bandit problem has also been extended to richer classes of parametric bandits such", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 643, + 398, + 656 + ], + "spans": [ + { + "bbox": [ + 89, + 646, + 99, + 655 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 104, + 643, + 398, + 656 + ], + "score": 1.0, + "content": "as the generalized linear bandits [24, 35] and kernelised bandits [44, 15].", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 89, + 662, + 100, + 671 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "With the prevalence of deep neural networks (DNNs) and their phenomenal performances in many", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 671, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 89, + 673, + 100, + 682 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 106, + 671, + 504, + 682 + ], + "score": 1.0, + "content": "machine learning tasks [32, 25], there has emerged a line of work that employs DNNs to increase the", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 89, + 684, + 100, + 693 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "representation power of contextual bandit algorithms [5, 38, 17, 49, 52, 20, 51]. The problems they", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 693, + 504, + 705 + ], + "spans": [ + { + "bbox": [ + 89, + 695, + 100, + 705 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 693, + 504, + 705 + ], + "score": 1.0, + "content": "solve are usually referred to as neural contextual bandits. For example, Zhou et al. [52] developed", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 703, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 89, + 706, + 100, + 715 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 703, + 505, + 717 + ], + "score": 1.0, + "content": "the NeuralUCB algorithm, which can be viewed as a natural extension of LinUCB [16, 1], where they", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "use the output of a deep neural network with the feature vector as input to approximate the reward.", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 88, + 85, + 100, + 96 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "Zhang et al. [51] adapted neural networks in Thompson Sampling [43, 14, 40] for both exploration", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 88, + 96, + 99, + 106 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 94, + 378, + 107 + ], + "score": 1.0, + "content": "and exploitation and proposed NeuralTS . For a fixed time horizon", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 379, + 96, + 387, + 104 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 387, + 94, + 506, + 107 + ], + "score": 1.0, + "content": ", it has been proved that both", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 89, + 109, + 99, + 119 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 106, + 254, + 120 + ], + "score": 1.0, + "content": "NeuralUCB and NeuralTS achieve a", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 254, + 106, + 291, + 119 + ], + "score": 0.93, + "content": "O ( \\widetilde { d } \\sqrt { T } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 292, + 106, + 375, + 120 + ], + "score": 1.0, + "content": "regret bound, where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 375, + 106, + 381, + 117 + ], + "score": 0.86, + "content": "\\hat { d }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 382, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "is the effective dimension of a", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 88, + 120, + 99, + 130 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 118, + 359, + 131 + ], + "score": 1.0, + "content": "neural tangent kernel matrix which can potentially scale with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 359, + 119, + 392, + 131 + ], + "score": 0.92, + "content": "O ( T K )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 392, + 118, + 408, + 131 + ], + "score": 1.0, + "content": "for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 408, + 119, + 419, + 128 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 419, + 118, + 506, + 131 + ], + "score": 1.0, + "content": "-armed bandits. This", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 88, + 132, + 100, + 141 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "high complexity is mainly due to that the exploration is performed over the entire huge neural network", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 141, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 88, + 142, + 99, + 152 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 141, + 505, + 152 + ], + "score": 1.0, + "content": "parameter space, which is inefficient and even infeasible when the number of neurons is large. 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We study a new neural contextual bandit algorithm, which learns a mapping", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 88, + 235, + 99, + 245 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "to transform the raw features associated with each context-action pair using a deep neural network", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 88, + 245, + 99, + 255 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "(deep representation), and then performs an upper confidence bound (UCB)-type exploration over the", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 88, + 256, + 99, + 266 + ], + "score": 1.0, + "content": "53", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "linear output layer of the network (shallow exploration). 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We study a new neural contextual bandit algorithm, which learns a mapping", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 88, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 88, + 235, + 99, + 245 + ], + "score": 1.0, + "content": "51", + "type": "text" + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "to transform the raw features associated with each context-action pair using a deep neural network", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 88, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 88, + 245, + 99, + 255 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "(deep representation), and then performs an upper confidence bound (UCB)-type exploration over the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 88, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 88, + 256, + 99, + 266 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "linear output layer of the network (shallow exploration). We prove a sublinear regret of the proposed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 88, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 88, + 267, + 100, + 278 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 105, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "algorithm by exploiting the UCB exploration techniques in linear contextual bandits [1] and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 89, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 89, + 279, + 99, + 289 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "analysis of deep overparameterized neural networks using neural tangent kernels [27]. Our theory", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 89, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 89, + 290, + 99, + 299 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "confirms the empirically observed effectiveness of decoupling the deep representation learning and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 89, + 298, + 313, + 311 + ], + "spans": [ + { + "bbox": [ + 89, + 301, + 99, + 310 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 313, + 311 + ], + "score": 1.0, + "content": "the UCB exploration in contextual bandits [38, 49].", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 90, + 315, + 417, + 326 + ], + "lines": [ + { + "bbox": [ + 87, + 313, + 419, + 328 + ], + "spans": [ + { + "bbox": [ + 87, + 313, + 419, + 328 + ], + "score": 1.0, + "content": "58 Contributions we summarize the main contributions of this paper as follows.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 97, + 335, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "• We propose a contextual bandit algorithm, Neural-LinUCB, for solving a general class of con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 116, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 116, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "textual bandit problems without knowing the specific reward generating function. The proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 115, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "algorithm learns a deep representation to transform the raw feature vectors and performs UCB-type", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 368, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 115, + 368, + 505, + 379 + ], + "score": 1.0, + "content": "exploration in the last layer of the neural network, which we refer to as deep representation and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 115, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "shallow exploration. Compared with LinUCB [34, 16] and neural bandits such as NeuralUCB [52]", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 116, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "and NeuralTS [51], our algorithm enjoys the best of two worlds: strong expressiveness due to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 116, + 401, + 439, + 413 + ], + "spans": [ + { + "bbox": [ + 116, + 401, + 439, + 413 + ], + "score": 1.0, + "content": "deep representation and computational efficiency due to the shallow exploration.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 91, + 416, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 89, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 89, + 419, + 100, + 428 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 108, + 416, + 376, + 429 + ], + "score": 1.0, + "content": "• Despite the usage of a DNN as the feature mapping, we prove a", + "type": "text" + }, + { + "bbox": [ + 376, + 415, + 408, + 429 + ], + "score": 0.93, + "content": "\\widetilde { O } ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "regret for the proposed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 428, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 89, + 430, + 99, + 440 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 116, + 428, + 506, + 440 + ], + "score": 1.0, + "content": "Neural-LinUCB algorithm, which matches the regret bound of linear contextual bandits [16, 1].", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 89, + 441, + 99, + 450 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 116, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "To the best of our knowledge, this is the first work that theoretically shows the convergence of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 88, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 88, + 452, + 100, + 462 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 115, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "bandits algorithms under the scheme of deep representation and shallow exploration. It is notable", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 89, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 89, + 463, + 99, + 472 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 115, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "that a similar scheme called Neural-Linear was proposed by Riquelme et al. [38] for Thompson", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 88, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 88, + 473, + 99, + 484 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 116, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "sampling algorithms, and they empirically showed that decoupling representation learning and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 89, + 484, + 99, + 494 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 116, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "uncertainty estimation improves the performance. 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Moreover, their algorithms need to access an oracle that", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 89, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "returns the optimal policy in a policy class given a sequence of context and reward vectors, whose", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 89, + 711, + 333, + 723 + ], + "spans": [ + { + "bbox": [ + 89, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 333, + 723 + ], + "score": 1.0, + "content": "regret depends on the VC-dimension of the policy class.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 89, + 73, + 505, + 206 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 88, + 73, + 506, + 208 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 89, + 211, + 505, + 310 + ], + "lines": [], + "index": 16, + "bbox_fs": [ + 88, + 211, + 506, + 311 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 90, + 315, + 417, + 326 + ], + "lines": [ + { + "bbox": [ + 87, + 313, + 419, + 328 + ], + "spans": [ + { + "bbox": [ + 87, + 313, + 419, + 328 + ], + "score": 1.0, + "content": "58 Contributions we summarize the main contributions of this paper as follows.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 87, + 313, + 419, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 335, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "• We propose a contextual bandit algorithm, Neural-LinUCB, for solving a general class of con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 116, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 116, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "textual bandit problems without knowing the specific reward generating function. The proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 115, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "algorithm learns a deep representation to transform the raw feature vectors and performs UCB-type", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 368, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 115, + 368, + 505, + 379 + ], + "score": 1.0, + "content": "exploration in the last layer of the neural network, which we refer to as deep representation and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 115, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "shallow exploration. 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It is notable", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 89, + 463, + 99, + 472 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 115, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "that a similar scheme called Neural-Linear was proposed by Riquelme et al. 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This decoupling of the representation and the exploration will achieve the best of both", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 86, + 315, + 100, + 324 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "worlds: efficient exploration in shallow (linear) models and high expressive power of deep models.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 85, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 100, + 335 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "To learn the unknown feature mapping, we propose to use a neural network to approximate it. In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 85, + 335, + 100, + 346 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "what follows, we will describe a neural contextual bandit algorithm that uses the output of the last", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 85, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 85, + 347, + 100, + 357 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "hidden layer of a neural network to transform the raw feature vectors (deep representation) and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 85, + 357, + 100, + 368 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "performs UCB-type exploration in the last layer of the neural network (shallow exploration). 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Therefore, a natural extension of", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 100, + 220 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "linear contextual bandits is to use a deep neural network to approximate the reward generating", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 218, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 86, + 221, + 100, + 231 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 105, + 218, + 141, + 232 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 141, + 219, + 158, + 231 + ], + "score": 0.89, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 218, + 507, + 232 + ], + "score": 1.0, + "content": ". Nonetheless, DNNs usually have a prohibitively large dimension for weight parameters,", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 229, + 467, + 242 + ], + "spans": [ + { + "bbox": [ + 86, + 232, + 100, + 241 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 105, + 229, + 467, + 242 + ], + "score": 1.0, + "content": "which makes the exploration in neural networks based UCB algorithm inefficient [28, 52].", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 247, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 86, + 248, + 100, + 258 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 247, + 505, + 258 + ], + "score": 1.0, + "content": "In this work, we study a neural contextual bandit algorithm, where the hidden layers of a deep neural", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 86, + 260, + 99, + 269 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "network are used to represent the features and the exploration is only performed in the last layer of the", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 86, + 271, + 99, + 280 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 105, + 268, + 408, + 282 + ], + "score": 1.0, + "content": "neural network. In particular, we assume that the reward generating function", + "type": "text" + }, + { + "bbox": [ + 408, + 268, + 424, + 281 + ], + "score": 0.91, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 86, + 282, + 100, + 291 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "the inner product between a deep represented feature vector and an exploration weight parameter,", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 86, + 293, + 100, + 302 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 289, + 139, + 303 + ], + "score": 1.0, + "content": "namely,", + "type": "text" + }, + { + "bbox": [ + 140, + 290, + 210, + 302 + ], + "score": 0.95, + "content": "\\bar { r ( \\cdot ) } = \\langle \\theta ^ { * } , \\psi ( \\cdot ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 289, + 240, + 303 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 240, + 290, + 276, + 300 + ], + "score": 0.92, + "content": "\\pmb { \\theta } ^ { * } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 289, + 396, + 303 + ], + "score": 1.0, + "content": "is some weight parameter and", + "type": "text" + }, + { + "bbox": [ + 396, + 290, + 415, + 302 + ], + "score": 0.91, + "content": "\\psi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "is an unknown feature", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 85, + 303, + 100, + 313 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 104, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "mapping. This decoupling of the representation and the exploration will achieve the best of both", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 86, + 315, + 100, + 324 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "worlds: efficient exploration in shallow (linear) models and high expressive power of deep models.", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 100, + 335 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "To learn the unknown feature mapping, we propose to use a neural network to approximate it. In", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 85, + 335, + 100, + 346 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "what follows, we will describe a neural contextual bandit algorithm that uses the output of the last", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 85, + 347, + 100, + 357 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "hidden layer of a neural network to transform the raw feature vectors (deep representation) and", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 85, + 357, + 100, + 368 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "performs UCB-type exploration in the last layer of the neural network (shallow exploration). Since", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 86, + 368, + 100, + 379 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "the exploration is performed only in the last linear layer, we call this procedure Neural-LinUCB,", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 378, + 247, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 389 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 247, + 390 + ], + "score": 1.0, + "content": "which is displayed in Algorithm 1.", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 392, + 507, + 407 + ], + "spans": [ + { + "bbox": [ + 85, + 396, + 100, + 405 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 104, + 392, + 196, + 407 + ], + "score": 1.0, + "content": "Specifically, in round", + "type": "text" + }, + { + "bbox": [ + 196, + 395, + 201, + 404 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 392, + 409, + 407 + ], + "score": 1.0, + "content": ", the agent receives an action set with raw features", + "type": "text" + }, + { + "bbox": [ + 410, + 394, + 502, + 406 + ], + "score": 0.91, + "content": "\\mathcal { X } _ { t } = \\{ \\mathbf { x } _ { t , 1 } , . . . , \\mathbf { x } _ { t , K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 392, + 507, + 407 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 405, + 464, + 416 + ], + "spans": [ + { + "bbox": [ + 84, + 405, + 231, + 416 + ], + "score": 1.0, + "content": "155 Then the agent chooses an arm", + "type": "text" + }, + { + "bbox": [ + 232, + 406, + 242, + 416 + ], + "score": 0.85, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 405, + 464, + 416 + ], + "score": 1.0, + "content": "that maximizes the following upper confidence bound:", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + } + ], + "index": 7.5, + "bbox_fs": [ + 86, + 176, + 507, + 242 + ] + }, + { + "type": "index", + "bbox": [ + 85, + 246, + 506, + 389 + ], + "lines": [], + "index": 17, + "bbox_fs": [ + 85, + 247, + 506, + 390 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 393, + 507, + 416 + ], + "lines": [], + "index": 24.5, + "bbox_fs": [ + 84, + 392, + 507, + 416 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 422, + 441, + 448 + ], + "lines": [ + { + "bbox": [ + 169, + 422, + 441, + 448 + ], + "spans": [ + { + "bbox": [ + 169, + 422, + 441, + 448 + ], + "score": 0.94, + "content": "a _ { t } = \\underset { k \\in [ K ] } { \\operatorname { a r g m a x } } \\Big \\{ \\langle \\phi ( \\mathbf { x } _ { t , k } ; 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We will", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 85, + 214, + 508, + 235 + ], + "spans": [ + { + "bbox": [ + 85, + 220, + 100, + 231 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 103, + 214, + 248, + 235 + ], + "score": 1.0, + "content": "discuss more about the initial point", + "type": "text" + }, + { + "bbox": [ + 249, + 218, + 268, + 229 + ], + "score": 0.82, + "content": "\\mathbf { w } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 214, + 465, + 235 + ], + "score": 1.0, + "content": "in the next paragraph. Then Algorithm 2 outputs", + "type": "text" + }, + { + "bbox": [ + 466, + 216, + 487, + 231 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { q } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 214, + 508, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 231, + 100, + 241 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 105, + 229, + 275, + 243 + ], + "score": 1.0, + "content": "we set it as the updated weight parameter", + "type": "text" + }, + { + "bbox": [ + 276, + 231, + 306, + 242 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { H q + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "in Algorithm 1. In the next round, the agent will", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 85, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 85, + 242, + 100, + 253 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 213, + 253 + ], + "score": 1.0, + "content": "receive another action set", + "type": "text" + }, + { + "bbox": [ + 213, + 241, + 235, + 253 + ], + "score": 0.91, + "content": "\\mathcal { X } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "with raw feature vectors and repeat the above steps to choose the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 251, + 371, + 264 + ], + "spans": [ + { + "bbox": [ + 86, + 254, + 100, + 263 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 251, + 371, + 264 + ], + "score": 1.0, + "content": "sub-optimal arm and update estimation for contextual parameters.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 86, + 267, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 86, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 86, + 270, + 99, + 279 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "Initialization: Recall that w is the collection of all hidden layer weight parameters of the neural", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 278, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 86, + 281, + 100, + 290 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 104, + 278, + 506, + 292 + ], + "score": 1.0, + "content": "network. 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For all", + "type": "text" + }, + { + "bbox": [ + 210, + 442, + 236, + 453 + ], + "score": 0.89, + "content": "i \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 440, + 255, + 455 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 255, + 442, + 291, + 454 + ], + "score": 0.92, + "content": "k \\in [ K ]", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 440, + 362, + 455 + ], + "score": 1.0, + "content": ", we assume that", + "type": "text" + }, + { + "bbox": [ + 362, + 442, + 414, + 454 + ], + "score": 0.93, + "content": "\\| \\mathbf { x } _ { i , k } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 440, + 506, + 455 + ], + "score": 1.0, + "content": "and its entries satisfy", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 95, + 450, + 196, + 468 + ], + "spans": [ + { + "bbox": [ + 95, + 450, + 107, + 468 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 107, + 452, + 191, + 466 + ], + "score": 0.91, + "content": "[ { \\bf { x } } _ { i , k } ] _ { j } \\stackrel { - } { = } [ { \\bf { x } } _ { j , k } ] _ { j + d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 450, + 196, + 468 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + } + ], + "index": 23.5, + "bbox_fs": [ + 108, + 343, + 507, + 421 + ] + }, + { + "type": "index", + "bbox": [ + 96, + 441, + 504, + 466 + ], + "lines": [], + "index": 27.5, + "bbox_fs": [ + 94, + 440, + 506, + 468 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 95, + 473, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 93, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 93, + 473, + 191, + 487 + ], + "score": 1.0, + "content": "19 The assumption that", + "type": "text" + }, + { + "bbox": [ + 192, + 474, + 242, + 486 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } _ { i , k } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "is not essential and is only imposed for simplicity, which is also", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 91, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 91, + 484, + 174, + 497 + ], + "score": 1.0, + "content": "20 used in Zou and", + "type": "text" + }, + { + "bbox": [ + 174, + 485, + 189, + 496 + ], + "score": 0.32, + "content": "\\mathrm { G u }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 484, + 437, + 497 + ], + "score": 1.0, + "content": "[53], Zhou et al. 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Based on the above definition, we impose the following assumption on", + "type": "text" + }, + { + "bbox": [ + 470, + 268, + 480, + 278 + ], + "score": 0.68, + "content": "\\mathbf { H }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 92, + 283, + 502, + 306 + ], + "lines": [ + { + "bbox": [ + 90, + 281, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 90, + 281, + 443, + 297 + ], + "score": 1.0, + "content": "43 Assumption 4.3. 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Moreover, it is shown that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 84, + 338, + 100, + 348 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Assumption 4.3 can be easily derived from Assumption 4.1 for two-layer ReLU networks [37, 53].", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 84, + 349, + 100, + 359 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Therefore, Assumption 4.3 is mild or even negligible given the non-degeneration assumption on the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 86, + 361, + 99, + 370 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 106, + 358, + 260, + 371 + ], + "score": 1.0, + "content": "feature vectors. 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It is easy to extend the definition of", + "type": "text" + }, + { + "bbox": [ + 468, + 370, + 479, + 379 + ], + "score": 0.6, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 84, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 84, + 382, + 100, + 392 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 105, + 379, + 344, + 393 + ], + "score": 1.0, + "content": "NTK matrix defined on all layers including the output layer", + "type": "text" + }, + { + "bbox": [ + 345, + 381, + 352, + 390 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 379, + 505, + 393 + ], + "score": 1.0, + "content": ", which would also be positive definite", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 392, + 290, + 404 + ], + "spans": [ + { + "bbox": [ + 86, + 393, + 99, + 403 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 106, + 392, + 290, + 404 + ], + "score": 1.0, + "content": "by Assumption 4.3 and the recursion in (4.2).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 86, + 407, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 86, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 86, + 409, + 100, + 419 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 104, + 406, + 506, + 421 + ], + "score": 1.0, + "content": "Before we present the regret analysis of the neural contextual bandit, we need to modify the regret", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 429 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "defined in (2.1) to account for the randomness of the neural network initialization. For a fixed time", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 429, + 340, + 442 + ], + "spans": [ + { + "bbox": [ + 86, + 431, + 100, + 441 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 104, + 429, + 139, + 442 + ], + "score": 1.0, + "content": "horizon", + "type": "text" + }, + { + "bbox": [ + 140, + 430, + 148, + 439 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 429, + 340, + 442 + ], + "score": 1.0, + "content": ", we define the regret of Algorithm 1 as follows.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 447, + 393, + 480 + ], + "lines": [ + { + "bbox": [ + 218, + 447, + 393, + 480 + ], + "spans": [ + { + "bbox": [ + 218, + 447, + 393, + 480 + ], + "score": 0.95, + "content": "R _ { T } = \\mathbb { E } \\bigg [ \\sum _ { t = 1 } ^ { T } \\big ( \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } ^ { * } } ) - \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } } ) \\big ) \\big | \\mathbf { w } ^ { ( 0 ) } \\bigg ] ,", + "type": "interline_equation", + "image_path": "e138cc6ba15066bbaecd1cf6411285fc128bb09ba373e687791499a7b87b9e82.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 447, + 393, + 463.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 218, + 463.5, + 393, + 480.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 89, + 486, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 85, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 85, + 486, + 427, + 498 + ], + "score": 1.0, + "content": "256 where the expectation is taken over the randomness of the reward noise. Note that", + "type": "text" + }, + { + "bbox": [ + 427, + 487, + 442, + 497 + ], + "score": 0.89, + "content": "R _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "defined in (4.3)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 497, + 453, + 509 + ], + "spans": [ + { + "bbox": [ + 85, + 497, + 453, + 509 + ], + "score": 1.0, + "content": "257 is still a random variable since the initialization of Algorithm 2 is randomly generated.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 92, + 513, + 399, + 525 + ], + "lines": [ + { + "bbox": [ + 88, + 511, + 399, + 528 + ], + "spans": [ + { + "bbox": [ + 88, + 511, + 399, + 528 + ], + "score": 1.0, + "content": "258 Now we are going to present the regret bound of the proposed algorithm.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 95, + 528, + 504, + 552 + ], + "lines": [ + { + "bbox": [ + 92, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 92, + 531, + 100, + 539 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 104, + 527, + 411, + 541 + ], + "score": 1.0, + "content": "Theorem 4.4. Suppose Assumptions 4.1, 4.2 and 4.3 hold. Assume that", + "type": "text" + }, + { + "bbox": [ + 411, + 528, + 464, + 541 + ], + "score": 0.95, + "content": "\\lVert \\pmb { \\theta } ^ { * } \\rVert _ { 2 } \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 94, + 539, + 443, + 552 + ], + "spans": [ + { + "bbox": [ + 94, + 542, + 101, + 550 + ], + "score": 1.0, + "content": "0", + "type": "text" + }, + { + "bbox": [ + 104, + 539, + 176, + 552 + ], + "score": 1.0, + "content": "positive constant", + "type": "text" + }, + { + "bbox": [ + 176, + 540, + 206, + 550 + ], + "score": 0.9, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 539, + 243, + 552 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 244, + 540, + 284, + 551 + ], + "score": 0.94, + "content": "\\delta \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 539, + 341, + 552 + ], + "score": 1.0, + "content": ", let us choose", + "type": "text" + }, + { + "bbox": [ + 342, + 541, + 352, + 550 + ], + "score": 0.86, + "content": "\\alpha _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 539, + 443, + 552 + ], + "score": 1.0, + "content": "in Neural-LinUCB as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 557, + 429, + 579 + ], + "lines": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "spans": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "score": 0.92, + "content": "\\alpha _ { t } = \\nu \\sqrt { 2 \\big ( d \\log ( 1 + t \\log ( H K ) / \\lambda ) + \\log ( 1 / \\delta ) \\big ) } + \\lambda ^ { 1 / 2 } M .", + "type": "interline_equation", + "image_path": "1ccf3a022ad345f8b9e1e117135312146f7c078729730e4c25e7b626c7abe5a8.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 583, + 288, + 595 + ], + "lines": [ + { + "bbox": [ + 85, + 582, + 288, + 597 + ], + "spans": [ + { + "bbox": [ + 85, + 582, + 203, + 597 + ], + "score": 1.0, + "content": "261 We choose the step size", + "type": "text" + }, + { + "bbox": [ + 203, + 586, + 213, + 596 + ], + "score": 0.86, + "content": "\\eta _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 582, + 288, + 597 + ], + "score": 1.0, + "content": "of Algorithm 2 as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 601, + 384, + 619 + ], + "lines": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "spans": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "score": 0.9, + "content": "\\eta _ { q } \\leq C _ { 0 } \\big ( d ^ { 2 } m n T ^ { 5 . 5 } L ^ { 6 } \\log ( T K / \\delta ) \\big ) ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "1a0c80ba7aa5ba80ea8ffcf392b283294329a6a569c484aefbd52aab86c2c2b9.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 624, + 506, + 646 + ], + "lines": [ + { + "bbox": [ + 84, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 84, + 624, + 286, + 638 + ], + "score": 1.0, + "content": "262 and the width of the neural network satisfies", + "type": "text" + }, + { + "bbox": [ + 286, + 625, + 433, + 637 + ], + "score": 0.89, + "content": "m = \\mathrm { p o l y } ( L , d , 1 / \\delta , H , \\log ( T K / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 624, + 505, + 638 + ], + "score": 1.0, + "content": ". With probability", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 84, + 634, + 465, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 137, + 647 + ], + "score": 1.0, + "content": "263 at least", + "type": "text" + }, + { + "bbox": [ + 137, + 635, + 160, + 645 + ], + "score": 0.9, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 634, + 465, + 647 + ], + "score": 1.0, + "content": "over the randomness of the initialization of the neural network, it holds that", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 650, + 477, + 681 + ], + "lines": [ + { + "bbox": [ + 133, + 650, + 477, + 681 + ], + "spans": [ + { + "bbox": [ + 133, + 650, + 477, + 681 + ], + "score": 0.93, + "content": "R _ { T } \\leq C _ { 1 } \\alpha _ { T } \\sqrt { T d \\log \\left( 1 + \\frac { T G ^ { 2 } } { \\lambda d } \\right) } + \\frac { C _ { 2 } \\ell _ { \\mathrm { L i p } } L ^ { 3 } d ^ { 5 / 2 } T \\sqrt { \\log m \\log ( \\frac { 1 } { \\delta } ) \\log ( \\frac { T K } { \\delta } ) } \\| \\mathbf { r } - 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Based on the above definition, we impose the following assumption on", + "type": "text" + }, + { + "bbox": [ + 470, + 268, + 480, + 278 + ], + "score": 0.68, + "content": "\\mathbf { H }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 92, + 267, + 480, + 281 + ] + }, + { + "type": "index", + "bbox": [ + 92, + 283, + 502, + 306 + ], + "lines": [ + { + "bbox": [ + 90, + 281, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 90, + 281, + 443, + 297 + ], + "score": 1.0, + "content": "43 Assumption 4.3. 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Moreover, it is shown that", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 84, + 338, + 100, + 348 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Assumption 4.3 can be easily derived from Assumption 4.1 for two-layer ReLU networks [37, 53].", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 84, + 349, + 100, + 359 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Therefore, Assumption 4.3 is mild or even negligible given the non-degeneration assumption on the", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 86, + 361, + 99, + 370 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 106, + 358, + 260, + 371 + ], + "score": 1.0, + "content": "feature vectors. Also note that matrix", + "type": "text" + }, + { + "bbox": [ + 260, + 359, + 271, + 369 + ], + "score": 0.34, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 358, + 398, + 371 + ], + "score": 1.0, + "content": "is only defined based on layers", + "type": "text" + }, + { + "bbox": [ + 399, + 359, + 450, + 370 + ], + "score": 0.91, + "content": "l = 1 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "of the neural", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 84, + 371, + 100, + 381 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 106, + 370, + 310, + 381 + ], + "score": 1.0, + "content": "network, and does not depend on the output layer", + "type": "text" + }, + { + "bbox": [ + 310, + 370, + 317, + 379 + ], + "score": 0.57, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 370, + 468, + 381 + ], + "score": 1.0, + "content": ". It is easy to extend the definition of", + "type": "text" + }, + { + "bbox": [ + 468, + 370, + 479, + 379 + ], + "score": 0.6, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 84, + 382, + 100, + 392 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 105, + 379, + 344, + 393 + ], + "score": 1.0, + "content": "NTK matrix defined on all layers including the output layer", + "type": "text" + }, + { + "bbox": [ + 345, + 381, + 352, + 390 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 379, + 505, + 393 + ], + "score": 1.0, + "content": ", which would also be positive definite", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 392, + 290, + 404 + ], + "spans": [ + { + "bbox": [ + 86, + 393, + 99, + 403 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 106, + 392, + 290, + 404 + ], + "score": 1.0, + "content": "by Assumption 4.3 and the recursion in (4.2).", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 86, + 409, + 100, + 419 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 104, + 406, + 506, + 421 + ], + "score": 1.0, + "content": "Before we present the regret analysis of the neural contextual bandit, we need to modify the regret", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 429 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "defined in (2.1) to account for the randomness of the neural network initialization. For a fixed time", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 429, + 340, + 442 + ], + "spans": [ + { + "bbox": [ + 86, + 431, + 100, + 441 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 104, + 429, + 139, + 442 + ], + "score": 1.0, + "content": "horizon", + "type": "text" + }, + { + "bbox": [ + 140, + 430, + 148, + 439 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 429, + 340, + 442 + ], + "score": 1.0, + "content": ", we define the regret of Algorithm 1 as follows.", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + } + ], + "index": 14.5, + "bbox_fs": [ + 88, + 281, + 504, + 307 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 314, + 505, + 403 + ], + "lines": [], + "index": 19.5, + "bbox_fs": [ + 84, + 315, + 506, + 404 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 407, + 505, + 441 + ], + "lines": [], + "index": 25, + "bbox_fs": [ + 86, + 406, + 506, + 442 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 447, + 393, + 480 + ], + "lines": [ + { + "bbox": [ + 218, + 447, + 393, + 480 + ], + "spans": [ + { + "bbox": [ + 218, + 447, + 393, + 480 + ], + "score": 0.95, + "content": "R _ { T } = \\mathbb { E } \\bigg [ \\sum _ { t = 1 } ^ { T } \\big ( \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } ^ { * } } ) - \\widehat { r } ( \\mathbf { x } _ { t , a _ { t } } ) \\big ) \\big | \\mathbf { w } ^ { ( 0 ) } \\bigg ] ,", + "type": "interline_equation", + "image_path": "e138cc6ba15066bbaecd1cf6411285fc128bb09ba373e687791499a7b87b9e82.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 447, + 393, + 463.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 218, + 463.5, + 393, + 480.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "index", + "bbox": [ + 89, + 486, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 85, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 85, + 486, + 427, + 498 + ], + "score": 1.0, + "content": "256 where the expectation is taken over the randomness of the reward noise. Note that", + "type": "text" + }, + { + "bbox": [ + 427, + 487, + 442, + 497 + ], + "score": 0.89, + "content": "R _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "defined in (4.3)", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 497, + 453, + 509 + ], + "spans": [ + { + "bbox": [ + 85, + 497, + 453, + 509 + ], + "score": 1.0, + "content": "257 is still a random variable since the initialization of Algorithm 2 is randomly generated.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + } + ], + "index": 29.5, + "bbox_fs": [ + 85, + 486, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 92, + 513, + 399, + 525 + ], + "lines": [ + { + "bbox": [ + 88, + 511, + 399, + 528 + ], + "spans": [ + { + "bbox": [ + 88, + 511, + 399, + 528 + ], + "score": 1.0, + "content": "258 Now we are going to present the regret bound of the proposed algorithm.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 88, + 511, + 399, + 528 + ] + }, + { + "type": "index", + "bbox": [ + 95, + 528, + 504, + 552 + ], + "lines": [ + { + "bbox": [ + 92, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 92, + 531, + 100, + 539 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 104, + 527, + 411, + 541 + ], + "score": 1.0, + "content": "Theorem 4.4. Suppose Assumptions 4.1, 4.2 and 4.3 hold. Assume that", + "type": "text" + }, + { + "bbox": [ + 411, + 528, + 464, + 541 + ], + "score": 0.95, + "content": "\\lVert \\pmb { \\theta } ^ { * } \\rVert _ { 2 } \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 539, + 443, + 552 + ], + "spans": [ + { + "bbox": [ + 94, + 542, + 101, + 550 + ], + "score": 1.0, + "content": "0", + "type": "text" + }, + { + "bbox": [ + 104, + 539, + 176, + 552 + ], + "score": 1.0, + "content": "positive constant", + "type": "text" + }, + { + "bbox": [ + 176, + 540, + 206, + 550 + ], + "score": 0.9, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 539, + 243, + 552 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 244, + 540, + 284, + 551 + ], + "score": 0.94, + "content": "\\delta \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 539, + 341, + 552 + ], + "score": 1.0, + "content": ", let us choose", + "type": "text" + }, + { + "bbox": [ + 342, + 541, + 352, + 550 + ], + "score": 0.86, + "content": "\\alpha _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 539, + 443, + 552 + ], + "score": 1.0, + "content": "in Neural-LinUCB as", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 92, + 527, + 505, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 557, + 429, + 579 + ], + "lines": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "spans": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "score": 0.92, + "content": "\\alpha _ { t } = \\nu \\sqrt { 2 \\big ( d \\log ( 1 + t \\log ( H K ) / \\lambda ) + \\log ( 1 / \\delta ) \\big ) } + \\lambda ^ { 1 / 2 } M .", + "type": "interline_equation", + "image_path": "1ccf3a022ad345f8b9e1e117135312146f7c078729730e4c25e7b626c7abe5a8.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 182, + 557, + 429, + 579 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 583, + 288, + 595 + ], + "lines": [ + { + "bbox": [ + 85, + 582, + 288, + 597 + ], + "spans": [ + { + "bbox": [ + 85, + 582, + 203, + 597 + ], + "score": 1.0, + "content": "261 We choose the step size", + "type": "text" + }, + { + "bbox": [ + 203, + 586, + 213, + 596 + ], + "score": 0.86, + "content": "\\eta _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 582, + 288, + 597 + ], + "score": 1.0, + "content": "of Algorithm 2 as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 85, + 582, + 288, + 597 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 601, + 384, + 619 + ], + "lines": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "spans": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "score": 0.9, + "content": "\\eta _ { q } \\leq C _ { 0 } \\big ( d ^ { 2 } m n T ^ { 5 . 5 } L ^ { 6 } \\log ( T K / \\delta ) \\big ) ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "1a0c80ba7aa5ba80ea8ffcf392b283294329a6a569c484aefbd52aab86c2c2b9.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 225, + 601, + 384, + 619 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "index", + "bbox": [ + 84, + 624, + 506, + 646 + ], + "lines": [ + { + "bbox": [ + 84, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 84, + 624, + 286, + 638 + ], + "score": 1.0, + "content": "262 and the width of the neural network satisfies", + "type": "text" + }, + { + "bbox": [ + 286, + 625, + 433, + 637 + ], + "score": 0.89, + "content": "m = \\mathrm { p o l y } ( L , d , 1 / \\delta , H , \\log ( T K / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 624, + 505, + 638 + ], + "score": 1.0, + "content": ". With probability", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 634, + 465, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 137, + 647 + ], + "score": 1.0, + "content": "263 at least", + "type": "text" + }, + { + "bbox": [ + 137, + 635, + 160, + 645 + ], + "score": 0.9, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 634, + 465, + 647 + ], + "score": 1.0, + "content": "over the randomness of the initialization of the neural network, it holds that", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + } + ], + "index": 37.5, + "bbox_fs": [ + 84, + 624, + 505, + 647 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 650, + 477, + 681 + ], + "lines": [ + { + "bbox": [ + 133, + 650, + 477, + 681 + ], + "spans": [ + { + "bbox": [ + 133, + 650, + 477, + 681 + ], + "score": 0.93, + "content": "R _ { T } \\leq C _ { 1 } \\alpha _ { T } \\sqrt { T d \\log \\left( 1 + \\frac { T G ^ { 2 } } { \\lambda d } \\right) } + \\frac { C _ { 2 } \\ell _ { \\mathrm { L i p } } L ^ { 3 } d ^ { 5 / 2 } T \\sqrt { \\log m \\log ( \\frac { 1 } { \\delta } ) \\log ( \\frac { T K } { \\delta } ) } \\| \\mathbf { r } - \\widetilde { \\mathbf { r } } \\| _ { \\mathbf { H } ^ { - 1 } } } { m ^ { 1 / 6 } } ,", + "type": "interline_equation", + "image_path": "12ee80d826958d4079dc3a4c3a914f1252140256056fcf795f509a38f52cb3e7.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 133, + 650, + 477, + 660.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 133, + 660.3333333333334, + 477, + 670.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 133, + 670.6666666666667, + 477, + 681.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 685, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 83, + 682, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 83, + 682, + 137, + 699 + ], + "score": 1.0, + "content": "264 where", + "type": "text" + }, + { + "bbox": [ + 138, + 685, + 186, + 697 + ], + "score": 0.9, + "content": "\\{ C _ { i } \\} _ { i = 0 , 1 , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 682, + 479, + 699 + ], + "score": 1.0, + "content": "are absolute constants independent of the problem parameters,", + "type": "text" + }, + { + "bbox": [ + 479, + 686, + 505, + 696 + ], + "score": 0.81, + "content": "\\begin{array} { r l } { \\mathbf { r } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 693, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 82, + 693, + 106, + 712 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 106, + 696, + 264, + 709 + ], + "score": 0.85, + "content": "( r ( \\mathbf { x } _ { 1 } ) , r ( \\mathbf { x } _ { 2 } ) , \\ldots , r ( \\mathbf { x } _ { T K } ) ) ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { T K }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 693, + 285, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 696, + 505, + 710 + ], + "score": 0.85, + "content": "\\widetilde { \\textbf { r } } = ( f ( \\mathbf { x } _ { 1 } ; \\pmb { \\theta } _ { 0 } , \\mathbf { w } _ { 0 } ) , \\dots , f ( \\mathbf { x } _ { T K } ; \\pmb { \\theta } _ { T - 1 } , \\mathbf { w } _ { T - 1 } ) ) ^ { \\top } \\ \\in", + "type": "inline_equation" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 82, + 706, + 224, + 724 + ], + "spans": [ + { + "bbox": [ + 82, + 706, + 106, + 724 + ], + "score": 1.0, + "content": "266", + "type": "text" + }, + { + "bbox": [ + 106, + 710, + 128, + 721 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { T K }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 706, + 149, + 724 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 150, + 709, + 219, + 723 + ], + "score": 0.9, + "content": "\\| \\mathbf { r } \\| _ { \\mathbf { A } } = \\sqrt { \\mathbf { r } ^ { \\top } \\mathbf { A } \\mathbf { r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 706, + 224, + 724 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "267", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 71, + 506, + 86 + ], + "score": 1.0, + "content": "Remark 4.5. Theorem 4.4 shows that the regret of Algorithm 1 can be bounded by two parts: the", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 83, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 85, + 87, + 100, + 97 + ], + "score": 1.0, + "content": "268", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 84, + 187, + 98 + ], + "score": 1.0, + "content": "first part is of order", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 187, + 83, + 218, + 97 + ], + "score": 0.93, + "content": "\\widetilde { O } ( \\sqrt { T } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 219, + 84, + 506, + 98 + ], + "score": 1.0, + "content": ", which resembles the regret bound of linear contextual bandits [1]; the", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 97, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 86, + 101, + 99, + 110 + ], + "score": 1.0, + "content": "269", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 97, + 199, + 112 + ], + "score": 1.0, + "content": "second part is of order", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 199, + 97, + 345, + 111 + ], + "score": 0.94, + "content": "\\widetilde { O } ( m ^ { - 1 / 6 } T \\sqrt { ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 345, + 97, + 506, + 112 + ], + "score": 1.0, + "content": ", which depends on the estimation error", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 110, + 480, + 122 + ], + "spans": [ + { + "bbox": [ + 85, + 111, + 100, + 121 + ], + "score": 1.0, + "content": "270", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 110, + 194, + 122 + ], + "score": 1.0, + "content": "of the neural network", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 195, + 110, + 201, + 121 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 202, + 110, + 342, + 122 + ], + "score": 1.0, + "content": "e efor the reward generating function", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 342, + 112, + 348, + 119 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 348, + 110, + 467, + 122 + ], + "score": 1.0, + "content": "and the neural tangent kernel", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 467, + 110, + 477, + 119 + ], + "score": 0.26, + "content": "\\mathbf { H }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 478, + 110, + 480, + 122 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + } + ], + "index": 43, + "bbox_fs": [ + 82, + 682, + 505, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 505, + 121 + ], + "lines": [ + { + "bbox": [ + 85, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "267", + "type": "text" + }, + { + "bbox": [ + 104, + 71, + 506, + 86 + ], + "score": 1.0, + "content": "Remark 4.5. Theorem 4.4 shows that the regret of Algorithm 1 can be bounded by two parts: the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 83, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 85, + 87, + 100, + 97 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 105, + 84, + 187, + 98 + ], + "score": 1.0, + "content": "first part is of order", + "type": "text" + }, + { + "bbox": [ + 187, + 83, + 218, + 97 + ], + "score": 0.93, + "content": "\\widetilde { O } ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 84, + 506, + 98 + ], + "score": 1.0, + "content": ", which resembles the regret bound of linear contextual bandits [1]; the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 97, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 86, + 101, + 99, + 110 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 104, + 97, + 199, + 112 + ], + "score": 1.0, + "content": "second part is of order", + "type": "text" + }, + { + "bbox": [ + 199, + 97, + 345, + 111 + ], + "score": 0.94, + "content": "\\widetilde { O } ( m ^ { - 1 / 6 } T \\sqrt { ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - \\widetilde { \\mathbf { r } } ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 97, + 506, + 112 + ], + "score": 1.0, + "content": ", which depends on the estimation error", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 110, + 480, + 122 + ], + "spans": [ + { + "bbox": [ + 85, + 111, + 100, + 121 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 110, + 194, + 122 + ], + "score": 1.0, + "content": "of the neural network", + "type": "text" + }, + { + "bbox": [ + 195, + 110, + 201, + 121 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 110, + 342, + 122 + ], + "score": 1.0, + "content": "e efor the reward generating function", + "type": "text" + }, + { + "bbox": [ + 342, + 112, + 348, + 119 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 110, + 467, + 122 + ], + "score": 1.0, + "content": "and the neural tangent kernel", + "type": "text" + }, + { + "bbox": [ + 467, + 110, + 477, + 119 + ], + "score": 0.26, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 110, + 480, + 122 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 102, + 131, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 506, + 144 + ], + "score": 1.0, + "content": "It is worth noting that our theoretical analysis depends on the reward structure assumption that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 107, + 142, + 178, + 154 + ], + "score": 0.92, + "content": "r ( \\cdot ) = \\langle \\theta \\ast , \\psi ( \\cdot ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 141, + 340, + 155 + ], + "score": 1.0, + "content": ". However, the linear structure between", + "type": "text" + }, + { + "bbox": [ + 341, + 142, + 353, + 153 + ], + "score": 0.85, + "content": "\\pmb { \\theta } \\ast", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 141, + 371, + 155 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 142, + 391, + 154 + ], + "score": 0.92, + "content": "\\psi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "is not essential. As long as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "the deep representation of the feature vector and the uncertainty weight parameter can be decoupled,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "Algorithm 1 can be easily extended to settings with milder assumptions on the reward structure", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "such as generalized linear models [41, 24, 35, 28]. For more general bandit models where no", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "assumption is imposed to the reward generating function, it is still unclear whether the decoupled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "deep representation and shallow exploration would work especially in cases a thorough exploration", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 171, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 171, + 221 + ], + "score": 1.0, + "content": "may be needed.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 105, + 223, + 437, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 438, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 438, + 237 + ], + "score": 1.0, + "content": "Based on the result in Theorem 4.4, we can easily verify the following conclusion:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 238, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 506, + 254 + ], + "score": 1.0, + "content": "Corollary 4.6. Under the same conditions of Theorem 4.4, if we choose a sufficiently overpa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 262, + 264 + ], + "score": 1.0, + "content": "rameterized neural network mapping", + "type": "text" + }, + { + "bbox": [ + 262, + 251, + 281, + 263 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 250, + 323, + 264 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 324, + 251, + 361, + 262 + ], + "score": 0.92, + "content": "m \\geq T ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 250, + 506, + 264 + ], + "score": 1.0, + "content": ", then the regret of Algorithm 1 is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 262, + 264, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 261, + 277 + ], + "score": 0.91, + "content": "R _ { T } = { \\widetilde { O } } ( { \\sqrt { T } } { \\sqrt { ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 263, + 264, + 277 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 327, + 293 + ], + "score": 1.0, + "content": "Remark 4.7. For the ease of presentation, let us denote", + "type": "text" + }, + { + "bbox": [ + 328, + 281, + 399, + 293 + ], + "score": 0.93, + "content": "\\mathcal { E } : = \\| \\mathbf { r } - \\widetilde { \\mathbf { r } } \\| _ { \\mathbf { H } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 280, + 446, + 293 + ], + "score": 1.0, + "content": ". If we have", + "type": "text" + }, + { + "bbox": [ + 446, + 281, + 487, + 293 + ], + "score": 0.92, + "content": "\\mathcal { E } = O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 255, + 307 + ], + "score": 1.0, + "content": "total regret in Theorem 4.4 becomes", + "type": "text" + }, + { + "bbox": [ + 255, + 292, + 288, + 306 + ], + "score": 0.94, + "content": "\\widetilde { O } ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "which matches the regret of linear contextual bandits", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 437, + 318 + ], + "score": 1.0, + "content": "[1]. We remark that there is a similar assumption in [52] where they assume that", + "type": "text" + }, + { + "bbox": [ + 438, + 305, + 475, + 316 + ], + "score": 0.91, + "content": "\\mathbf { r } ^ { \\top } \\mathbf { H } ^ { - 1 } \\mathbf { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 489, + 329 + ], + "score": 1.0, + "content": "upper bounded by a constant. They show that this term can be bounded by the RKHS norm of", + "type": "text" + }, + { + "bbox": [ + 489, + 318, + 496, + 327 + ], + "score": 0.45, + "content": "\\mathbf { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 317, + 506, + 329 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 451, + 340 + ], + "score": 1.0, + "content": "it belongs to the RKHS induced by the neural tangent kernel [6, 7, 33]. In addition,", + "type": "text" + }, + { + "bbox": [ + 451, + 328, + 459, + 338 + ], + "score": 0.75, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "here is the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "difference between the true reward function and the neural network function, which can also be small", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 450, + 362 + ], + "score": 1.0, + "content": "if the deep neural network function well approximates the reward generating function", + "type": "text" + }, + { + "bbox": [ + 450, + 349, + 466, + 362 + ], + "score": 0.92, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 349, + 470, + 362 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 88, + 379, + 191, + 393 + ], + "lines": [ + { + "bbox": [ + 84, + 378, + 192, + 396 + ], + "spans": [ + { + "bbox": [ + 84, + 378, + 192, + 396 + ], + "score": 1.0, + "content": "290 5 Experiments", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 86, + 405, + 100, + 581 + ], + "lines": [ + { + "bbox": [ + 86, + 408, + 99, + 419 + ], + "spans": [ + { + "bbox": [ + 86, + 408, + 99, + 419 + ], + "score": 1.0, + "content": "291", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 419, + 100, + 430 + ], + "spans": [ + { + "bbox": [ + 86, + 419, + 100, + 430 + ], + "score": 1.0, + "content": "292", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 430, + 100, + 441 + ], + "spans": [ + { + "bbox": [ + 86, + 430, + 100, + 441 + ], + "score": 1.0, + "content": "293", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 441, + 100, + 452 + ], + "spans": [ + { + "bbox": [ + 86, + 441, + 100, + 452 + ], + "score": 1.0, + "content": "294", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 452, + 100, + 463 + ], + "spans": [ + { + "bbox": [ + 86, + 452, + 100, + 463 + ], + "score": 1.0, + "content": "295", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 463, + 99, + 473 + ], + "spans": [ + { + "bbox": [ + 86, + 463, + 99, + 473 + ], + "score": 1.0, + "content": "296", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 474, + 99, + 484 + ], + "spans": [ + { + "bbox": [ + 86, + 474, + 99, + 484 + ], + "score": 1.0, + "content": "297", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 484, + 100, + 496 + ], + "spans": [ + { + "bbox": [ + 86, + 484, + 100, + 496 + ], + "score": 1.0, + "content": "298", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 495, + 100, + 506 + ], + "spans": [ + { + "bbox": [ + 86, + 495, + 100, + 506 + ], + "score": 1.0, + "content": "299", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 506, + 100, + 518 + ], + "spans": [ + { + "bbox": [ + 86, + 506, + 100, + 518 + ], + "score": 1.0, + "content": "300", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 518, + 99, + 528 + ], + "spans": [ + { + "bbox": [ + 86, + 518, + 99, + 528 + ], + "score": 1.0, + "content": "301", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 86, + 527, + 100, + 540 + ], + "spans": [ + { + "bbox": [ + 86, + 527, + 100, + 540 + ], + "score": 1.0, + "content": "302", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 85, + 539, + 100, + 550 + ], + "spans": [ + { + "bbox": [ + 85, + 539, + 100, + 550 + ], + "score": 1.0, + "content": "303", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 86, + 550, + 100, + 562 + ], + "spans": [ + { + "bbox": [ + 86, + 550, + 100, + 562 + ], + "score": 1.0, + "content": "304", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 86, + 566, + 100, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 100, + 577 + ], + "score": 1.0, + "content": "305", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 102, + 406, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "In this section, we provide empirical evaluations of Neural-LinUCB on real-world datasets. As", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "we have discussed in Section 3, Neural-LinUCB could be viewed as an instantiation of the Neural-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "Linear scheme studied in Riquelme et al. [38] except that we use the UCB exploration instead of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "posterior sampling exploration therein. Note that there has been an extensive comparison [38] of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "Neural-Linear methods with many other baselines such as greedy algorithms, Variational Inference,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Expectation-Propagation, Bayesian Non-parametrics and so on. Therefore, we do not seek a thorough", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "empirical comparison of Neural-LinUCB with all existing bandits algorithms. We refer readers who", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "are interested in the performance of Neural-Linear methods with deep representation and shallow", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "exploration compared with a vast of baselines in the literature to the benchmark study by Riquelme", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "et al. [38]. In this experiment, we only aim to show the advantages of our algorithm over the following", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "baselines: (1) Neural-Linear [38]; (2) LinUCB [16], which does not have a deep representation of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "feature vectors; and (3) NeuralUCB [52], which performs UCB exploration on all the parameters of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "the neural network instead of the shallow exploration used in our paper. All numerical experiments", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 548, + 429, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 375, + 560 + ], + "score": 1.0, + "content": "were run on a workstation with Intel(R) Xeon(R) CPU E5-2637 v4", + "type": "text" + }, + { + "bbox": [ + 376, + 549, + 385, + 558 + ], + "score": 0.53, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 548, + 429, + 560 + ], + "score": 1.0, + "content": "3.50GHz.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 38.0 + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "Datasets: we evaluate the performances of all algorithms on bandit problems created from real-world", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "data. Specifically, following the experimental setting in Zhou et al. [52],we use datasets (Shuttle)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "Statlog, Magic and Covertype from UCI machine learning repository [23], and the MINST dataset", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "from LeCun et al. [31]. The details of these datasets are presented in Table 1. In Table 1, each", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 246, + 619 + ], + "score": 1.0, + "content": "instance represents a feature vector", + "type": "text" + }, + { + "bbox": [ + 246, + 607, + 278, + 618 + ], + "score": 0.92, + "content": "\\mathbf { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 608, + 409, + 619 + ], + "score": 1.0, + "content": "that is associated with one of the", + "type": "text" + }, + { + "bbox": [ + 410, + 608, + 420, + 618 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "arms, and dimension", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 107, + 619, + 290, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 113, + 629 + ], + "score": 0.72, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 619, + 290, + 631 + ], + "score": 1.0, + "content": "is the number of attributes in each instance.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 55.5 + }, + { + "type": "table", + "bbox": [ + 178, + 658, + 431, + 711 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 640, + 493, + 653 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 637, + 495, + 655 + ], + "spans": [ + { + "bbox": [ + 111, + 637, + 495, + 655 + ], + "score": 1.0, + "content": "Table 1: Specifications of datasets from the UCI machine learning repository used in this paper.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 59 + }, + { + "type": "table_body", + "bbox": [ + 178, + 658, + 431, + 711 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 178, + 658, + 431, + 711 + ], + "spans": [ + { + "bbox": [ + 178, + 658, + 431, + 711 + ], + "score": 0.977, + "html": "
StatlogMagicCovertypeMNIST
Number of attributes91154784
Number of arms72710
Number of instances58,00019,020581,01260,000
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However, the linear structure between", + "type": "text" + }, + { + "bbox": [ + 341, + 142, + 353, + 153 + ], + "score": 0.85, + "content": "\\pmb { \\theta } \\ast", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 141, + 371, + 155 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 142, + 391, + 154 + ], + "score": 0.92, + "content": "\\psi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "is not essential. As long as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "the deep representation of the feature vector and the uncertainty weight parameter can be decoupled,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "Algorithm 1 can be easily extended to settings with milder assumptions on the reward structure", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "such as generalized linear models [41, 24, 35, 28]. For more general bandit models where no", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "assumption is imposed to the reward generating function, it is still unclear whether the decoupled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "deep representation and shallow exploration would work especially in cases a thorough exploration", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 171, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 171, + 221 + ], + "score": 1.0, + "content": "may be needed.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 130, + 506, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 223, + 437, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 438, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 438, + 237 + ], + "score": 1.0, + "content": "Based on the result in Theorem 4.4, we can easily verify the following conclusion:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 222, + 438, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 238, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 506, + 254 + ], + "score": 1.0, + "content": "Corollary 4.6. Under the same conditions of Theorem 4.4, if we choose a sufficiently overpa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 262, + 264 + ], + "score": 1.0, + "content": "rameterized neural network mapping", + "type": "text" + }, + { + "bbox": [ + 262, + 251, + 281, + 263 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 250, + 323, + 264 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 324, + 251, + 361, + 262 + ], + "score": 0.92, + "content": "m \\geq T ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 250, + 506, + 264 + ], + "score": 1.0, + "content": ", then the regret of Algorithm 1 is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 262, + 264, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 261, + 277 + ], + "score": 0.91, + "content": "R _ { T } = { \\widetilde { O } } ( { \\sqrt { T } } { \\sqrt { ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) ^ { \\top } \\mathbf { H } ^ { - 1 } ( \\mathbf { r } - { \\widetilde { \\mathbf { r } } } ) } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 263, + 264, + 277 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 238, + 506, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 327, + 293 + ], + "score": 1.0, + "content": "Remark 4.7. For the ease of presentation, let us denote", + "type": "text" + }, + { + "bbox": [ + 328, + 281, + 399, + 293 + ], + "score": 0.93, + "content": "\\mathcal { E } : = \\| \\mathbf { r } - \\widetilde { \\mathbf { r } } \\| _ { \\mathbf { H } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 280, + 446, + 293 + ], + "score": 1.0, + "content": ". If we have", + "type": "text" + }, + { + "bbox": [ + 446, + 281, + 487, + 293 + ], + "score": 0.92, + "content": "\\mathcal { E } = O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 255, + 307 + ], + "score": 1.0, + "content": "total regret in Theorem 4.4 becomes", + "type": "text" + }, + { + "bbox": [ + 255, + 292, + 288, + 306 + ], + "score": 0.94, + "content": "\\widetilde { O } ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "which matches the regret of linear contextual bandits", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 437, + 318 + ], + "score": 1.0, + "content": "[1]. We remark that there is a similar assumption in [52] where they assume that", + "type": "text" + }, + { + "bbox": [ + 438, + 305, + 475, + 316 + ], + "score": 0.91, + "content": "\\mathbf { r } ^ { \\top } \\mathbf { H } ^ { - 1 } \\mathbf { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 489, + 329 + ], + "score": 1.0, + "content": "upper bounded by a constant. They show that this term can be bounded by the RKHS norm of", + "type": "text" + }, + { + "bbox": [ + 489, + 318, + 496, + 327 + ], + "score": 0.45, + "content": "\\mathbf { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 317, + 506, + 329 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 451, + 340 + ], + "score": 1.0, + "content": "it belongs to the RKHS induced by the neural tangent kernel [6, 7, 33]. 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As", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "we have discussed in Section 3, Neural-LinUCB could be viewed as an instantiation of the Neural-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "Linear scheme studied in Riquelme et al. [38] except that we use the UCB exploration instead of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "posterior sampling exploration therein. Note that there has been an extensive comparison [38] of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "Neural-Linear methods with many other baselines such as greedy algorithms, Variational Inference,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Expectation-Propagation, Bayesian Non-parametrics and so on. Therefore, we do not seek a thorough", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "empirical comparison of Neural-LinUCB with all existing bandits algorithms. We refer readers who", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "are interested in the performance of Neural-Linear methods with deep representation and shallow", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "exploration compared with a vast of baselines in the literature to the benchmark study by Riquelme", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "et al. [38]. In this experiment, we only aim to show the advantages of our algorithm over the following", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "baselines: (1) Neural-Linear [38]; (2) LinUCB [16], which does not have a deep representation of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "feature vectors; and (3) NeuralUCB [52], which performs UCB exploration on all the parameters of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "the neural network instead of the shallow exploration used in our paper. All numerical experiments", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 548, + 429, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 375, + 560 + ], + "score": 1.0, + "content": "were run on a workstation with Intel(R) Xeon(R) CPU E5-2637 v4", + "type": "text" + }, + { + "bbox": [ + 376, + 549, + 385, + 558 + ], + "score": 0.53, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 548, + 429, + 560 + ], + "score": 1.0, + "content": "3.50GHz.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 38.0, + "bbox_fs": [ + 105, + 405, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "Datasets: we evaluate the performances of all algorithms on bandit problems created from real-world", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "data. Specifically, following the experimental setting in Zhou et al. [52],we use datasets (Shuttle)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "Statlog, Magic and Covertype from UCI machine learning repository [23], and the MINST dataset", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "from LeCun et al. [31]. The details of these datasets are presented in Table 1. 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StatlogMagicCovertypeMNIST
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Number of arms72710
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For neural network based algorithms such as NeuralUCB, Neural-Linear and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 85, + 227, + 100, + 238 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 106, + 226, + 405, + 238 + ], + "score": 1.0, + "content": "Neural-LinUCB, we use a ReLU neural network defined as in (2.2) with", + "type": "text" + }, + { + "bbox": [ + 406, + 226, + 434, + 236 + ], + "score": 0.9, + "content": "L = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "and 2000 for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 234, + 502, + 250 + ], + "spans": [ + { + "bbox": [ + 85, + 239, + 100, + 249 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 103, + 234, + 444, + 250 + ], + "score": 1.0, + "content": "UCI datasets (Statlog, Magic, Covertype). Thus the neural network weights are", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 502, + 248 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 245, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 85, + 249, + 100, + 259 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 106, + 247, + 163, + 259 + ], + "score": 0.93, + "content": "\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { k \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 245, + 186, + 262 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 186, + 247, + 220, + 258 + ], + "score": 0.91, + "content": "\\pmb { \\theta } \\in \\mathbb { R } ^ { \\widetilde { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 245, + 304, + 262 + ], + "score": 1.0, + "content": "respectively, where", + "type": "text" + }, + { + "bbox": [ + 304, + 248, + 342, + 258 + ], + "score": 0.84, + "content": "k = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 245, + 346, + 262 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 346, + 248, + 392, + 258 + ], + "score": 0.86, + "content": "m = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 245, + 414, + 262 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 415, + 248, + 421, + 258 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 245, + 506, + 262 + ], + "score": 1.0, + "content": "is the dimension of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 85, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 85, + 260, + 100, + 271 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "features in the corresponding task. Since the problem size of the MNIST dataset is larger, inspired", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 85, + 271, + 100, + 282 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 269, + 372, + 282 + ], + "score": 1.0, + "content": "by Hinton and Salakhutdinov [26], we use a deeper NN and set", + "type": "text" + }, + { + "bbox": [ + 372, + 270, + 401, + 280 + ], + "score": 0.86, + "content": "L = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 269, + 405, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 405, + 270, + 443, + 280 + ], + "score": 0.84, + "content": "k = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 269, + 462, + 282 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 270, + 502, + 280 + ], + "score": 0.88, + "content": "m = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 85, + 278, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 85, + 282, + 100, + 293 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 104, + 278, + 162, + 294 + ], + "score": 1.0, + "content": "with weights", + "type": "text" + }, + { + "bbox": [ + 162, + 280, + 218, + 291 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 278, + 222, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 280, + 282, + 291 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { m \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 278, + 286, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 286, + 280, + 343, + 291 + ], + "score": 0.92, + "content": "\\mathbf { W _ { 3 } } \\in \\mathbb { R } ^ { k \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 278, + 365, + 294 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 366, + 280, + 398, + 291 + ], + "score": 0.91, + "content": "\\pmb \\theta \\in \\mathbb { R } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 278, + 507, + 294 + ], + "score": 1.0, + "content": ". We set the time horizon", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 85, + 293, + 100, + 303 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 106, + 291, + 161, + 303 + ], + "score": 0.87, + "content": "T = 1 5 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 291, + 505, + 303 + ], + "score": 1.0, + "content": ", which is the total number of rounds for each algorithm on each dataset. We use", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 86, + 304, + 100, + 314 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 302, + 368, + 315 + ], + "score": 1.0, + "content": "gradient decent to optimize the network weights, with a step size", + "type": "text" + }, + { + "bbox": [ + 369, + 303, + 408, + 315 + ], + "score": 0.9, + "content": "\\eta _ { q } = 1 \\mathrm { e } { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "and maximum iteration", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 85, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 85, + 315, + 99, + 325 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 104, + 312, + 140, + 327 + ], + "score": 1.0, + "content": "number", + "type": "text" + }, + { + "bbox": [ + 140, + 313, + 187, + 324 + ], + "score": 0.8, + "content": "n = 1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 312, + 505, + 327 + ], + "score": 1.0, + "content": ". To speed up the training process, the network parameter w is updated every", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 85, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 100, + 336 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 106, + 324, + 145, + 335 + ], + "score": 0.91, + "content": "H = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "rounds starting from round 2000. We also apply early stopping when the loss difference", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 335, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 86, + 337, + 100, + 347 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 335, + 410, + 347 + ], + "score": 1.0, + "content": "of two consecutive iterations is smaller than a threshold of 1e-6. We set", + "type": "text" + }, + { + "bbox": [ + 411, + 335, + 439, + 345 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 335, + 459, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 335, + 504, + 346 + ], + "score": 0.9, + "content": "\\alpha _ { t } = 0 . 0 2", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 85, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 85, + 347, + 100, + 358 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 345, + 183, + 359 + ], + "score": 1.0, + "content": "for all algorithms,", + "type": "text" + }, + { + "bbox": [ + 184, + 346, + 216, + 358 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 345, + 506, + 359 + ], + "score": 1.0, + "content": ". Following the setting in Riquelme et al. [38], we use round-robin to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 86, + 359, + 99, + 368 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "independently select each arm for 3 times at the beginning of each algorithm. For NeuralUCB, since", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 99, + 379 + ], + "score": 1.0, + "content": "326", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 506, + 380 + ], + "score": 1.0, + "content": "it is computationally unaffordable to perform the original UCB exploration as displayed in Zhou et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 377, + 508, + 393 + ], + "spans": [ + { + "bbox": [ + 86, + 383, + 99, + 391 + ], + "score": 1.0, + "content": "327", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 367, + 393 + ], + "score": 1.0, + "content": "[52], we follow their experimental setting to replace the matrix", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 447, + 391 + ], + "score": 0.92, + "content": "\\mathbf { Z } _ { t } \\in \\mathbb { R } ^ { ( d + \\widetilde { p } ) \\times \\widetilde { ( } d + \\widetilde { p } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 377, + 508, + 393 + ], + "score": 1.0, + "content": "in Zhou et al.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 391, + 226, + 403 + ], + "spans": [ + { + "bbox": [ + 86, + 393, + 100, + 402 + ], + "score": 1.0, + "content": "328", + "type": "text" + }, + { + "bbox": [ + 106, + 391, + 226, + 403 + ], + "score": 1.0, + "content": "[52] with its diagonal matrix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Results: we plot the cumulative regret of all algorithms versus round in Figures 1(a), 1(b) and 1(c)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "for UCI datasets and in Figure 1(d) for MNIST. The results are reported based on the average of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "10 repetitions over different random shuffles of the datasets. It can be seen that algorithms based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "on neural network representations (NeuralUCB, Neural-Linear and Neural-LinUCB) consistently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "outperform the linear contextual bandit method LinUCB, which shows that linear models may", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "lack representation power and find biased estimates for the underlying reward generating function.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "Furthermore, our proposed Neural-LinUCB achieves a comparable regret with NeuralUCB in all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "experiments despite the fact that our algorithm only explores in the output layer of the neural network,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "score": 1.0, + "content": "which is more computationally efficient as we will show in the sequel.The results in our experiment", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "are well aligned with our theory that deep representation and shallow exploration are sufficient to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "guarantee a good performance of neural contextual bandit algorithms, which is also consistent with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "the findings in existing literature [38] that decoupling the representation learning and uncertainty", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 538, + 259, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 259, + 551 + ], + "score": 1.0, + "content": "estimation improves the performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "We also conducted experiments to study the effects of different widths of deep neural networks on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "the regret performance and to show the computational efficiency of Neural-LinUCB compared with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 576, + 480, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 480, + 588 + ], + "score": 1.0, + "content": "existing neural bandit algorithms. Due to the space limit, we defer the results to Appendix A.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 101, + 597, + 187, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 190, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 190, + 613 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 105, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "In this paper, we propose a new neural contextual bandit algorithm called Neural-LinUCB, which uses", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "the hidden layers of a ReLU neural network as a deep representation of the raw feature vectors and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "performs UCB type exploration on the last layer of the neural network. By incorporating techniques", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "in liner contextual bandits and neural tangent kernels, we prove that the proposed algorithm achieves", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "a sublinear regret when the width of the network is sufficiently large. This is the first regret analysis", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "of neural contextual bandit algorithms with deep representation and shallow exploration, which have", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "been observed in practice to work well on many benchmark bandit problems [38]. We also conducted", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "experiments on real-world datasets to demonstrate the advantage of the proposed algorithm over", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 340, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 340, + 723 + ], + "score": 1.0, + "content": "LinUCB and existing neural contextual bandit algorithms.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 74, + 504, + 159 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 74, + 504, + 159 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 74, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 74, + 504, + 159 + ], + "score": 0.961, + "type": "image", + "image_path": "943bd96058f4439657901f76b8c00684d91994e2afa724f14d33961644a23aaf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 74, + 504, + 102.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 102.33333333333333, + 504, + 130.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 130.66666666666666, + 504, + 159.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 168, + 503, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "Figure 1: The cumulative regrets of LinUCB, NeuralUCB, Neural-Linear and Neural-LinUCB over", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 180, + 355, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 355, + 192 + ], + "score": 1.0, + "content": "15, 000 rounds. Experiments are averaged over 10 repetitions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "index", + "bbox": [ + 88, + 204, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 85, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 85, + 205, + 100, + 216 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "Implementations: for LinUCB, we follow the setting in Li et al. [34] to use disjoint models", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 85, + 216, + 100, + 227 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "for different arms. For neural network based algorithms such as NeuralUCB, Neural-Linear and", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 85, + 227, + 100, + 238 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 106, + 226, + 405, + 238 + ], + "score": 1.0, + "content": "Neural-LinUCB, we use a ReLU neural network defined as in (2.2) with", + "type": "text" + }, + { + "bbox": [ + 406, + 226, + 434, + 236 + ], + "score": 0.9, + "content": "L = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "and 2000 for the", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 234, + 502, + 250 + ], + "spans": [ + { + "bbox": [ + 85, + 239, + 100, + 249 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 103, + 234, + 444, + 250 + ], + "score": 1.0, + "content": "UCI datasets (Statlog, Magic, Covertype). Thus the neural network weights are", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 502, + 248 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 245, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 85, + 249, + 100, + 259 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 106, + 247, + 163, + 259 + ], + "score": 0.93, + "content": "\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { k \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 245, + 186, + 262 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 186, + 247, + 220, + 258 + ], + "score": 0.91, + "content": "\\pmb { \\theta } \\in \\mathbb { R } ^ { \\widetilde { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 245, + 304, + 262 + ], + "score": 1.0, + "content": "respectively, where", + "type": "text" + }, + { + "bbox": [ + 304, + 248, + 342, + 258 + ], + "score": 0.84, + "content": "k = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 245, + 346, + 262 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 346, + 248, + 392, + 258 + ], + "score": 0.86, + "content": "m = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 245, + 414, + 262 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 415, + 248, + 421, + 258 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 245, + 506, + 262 + ], + "score": 1.0, + "content": "is the dimension of", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 85, + 260, + 100, + 271 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "features in the corresponding task. Since the problem size of the MNIST dataset is larger, inspired", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 85, + 271, + 100, + 282 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 269, + 372, + 282 + ], + "score": 1.0, + "content": "by Hinton and Salakhutdinov [26], we use a deeper NN and set", + "type": "text" + }, + { + "bbox": [ + 372, + 270, + 401, + 280 + ], + "score": 0.86, + "content": "L = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 269, + 405, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 405, + 270, + 443, + 280 + ], + "score": 0.84, + "content": "k = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 269, + 462, + 282 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 270, + 502, + 280 + ], + "score": 0.88, + "content": "m = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 278, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 85, + 282, + 100, + 293 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 104, + 278, + 162, + 294 + ], + "score": 1.0, + "content": "with weights", + "type": "text" + }, + { + "bbox": [ + 162, + 280, + 218, + 291 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 278, + 222, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 280, + 282, + 291 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { m \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 278, + 286, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 286, + 280, + 343, + 291 + ], + "score": 0.92, + "content": "\\mathbf { W _ { 3 } } \\in \\mathbb { R } ^ { k \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 278, + 365, + 294 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 366, + 280, + 398, + 291 + ], + "score": 0.91, + "content": "\\pmb \\theta \\in \\mathbb { R } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 278, + 507, + 294 + ], + "score": 1.0, + "content": ". We set the time horizon", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 85, + 293, + 100, + 303 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 106, + 291, + 161, + 303 + ], + "score": 0.87, + "content": "T = 1 5 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 291, + 505, + 303 + ], + "score": 1.0, + "content": ", which is the total number of rounds for each algorithm on each dataset. We use", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 86, + 304, + 100, + 314 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 302, + 368, + 315 + ], + "score": 1.0, + "content": "gradient decent to optimize the network weights, with a step size", + "type": "text" + }, + { + "bbox": [ + 369, + 303, + 408, + 315 + ], + "score": 0.9, + "content": "\\eta _ { q } = 1 \\mathrm { e } { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "and maximum iteration", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 85, + 315, + 99, + 325 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 104, + 312, + 140, + 327 + ], + "score": 1.0, + "content": "number", + "type": "text" + }, + { + "bbox": [ + 140, + 313, + 187, + 324 + ], + "score": 0.8, + "content": "n = 1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 312, + 505, + 327 + ], + "score": 1.0, + "content": ". To speed up the training process, the network parameter w is updated every", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 85, + 325, + 100, + 336 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 106, + 324, + 145, + 335 + ], + "score": 0.91, + "content": "H = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "rounds starting from round 2000. We also apply early stopping when the loss difference", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 335, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 86, + 337, + 100, + 347 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 335, + 410, + 347 + ], + "score": 1.0, + "content": "of two consecutive iterations is smaller than a threshold of 1e-6. We set", + "type": "text" + }, + { + "bbox": [ + 411, + 335, + 439, + 345 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 335, + 459, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 335, + 504, + 346 + ], + "score": 0.9, + "content": "\\alpha _ { t } = 0 . 0 2", + "type": "inline_equation" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 85, + 347, + 100, + 358 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 345, + 183, + 359 + ], + "score": 1.0, + "content": "for all algorithms,", + "type": "text" + }, + { + "bbox": [ + 184, + 346, + 216, + 358 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 345, + 506, + 359 + ], + "score": 1.0, + "content": ". Following the setting in Riquelme et al. [38], we use round-robin to", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 86, + 359, + 99, + 368 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "independently select each arm for 3 times at the beginning of each algorithm. For NeuralUCB, since", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 99, + 379 + ], + "score": 1.0, + "content": "326", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 506, + 380 + ], + "score": 1.0, + "content": "it is computationally unaffordable to perform the original UCB exploration as displayed in Zhou et al.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 377, + 508, + 393 + ], + "spans": [ + { + "bbox": [ + 86, + 383, + 99, + 391 + ], + "score": 1.0, + "content": "327", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 367, + 393 + ], + "score": 1.0, + "content": "[52], we follow their experimental setting to replace the matrix", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 447, + 391 + ], + "score": 0.92, + "content": "\\mathbf { Z } _ { t } \\in \\mathbb { R } ^ { ( d + \\widetilde { p } ) \\times \\widetilde { ( } d + \\widetilde { p } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 377, + 508, + 393 + ], + "score": 1.0, + "content": "in Zhou et al.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 391, + 226, + 403 + ], + "spans": [ + { + "bbox": [ + 86, + 393, + 100, + 402 + ], + "score": 1.0, + "content": "328", + "type": "text" + }, + { + "bbox": [ + 106, + 391, + 226, + 403 + ], + "score": 1.0, + "content": "[52] with its diagonal matrix.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 85, + 204, + 508, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Results: we plot the cumulative regret of all algorithms versus round in Figures 1(a), 1(b) and 1(c)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "for UCI datasets and in Figure 1(d) for MNIST. The results are reported based on the average of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "10 repetitions over different random shuffles of the datasets. It can be seen that algorithms based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "on neural network representations (NeuralUCB, Neural-Linear and Neural-LinUCB) consistently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "outperform the linear contextual bandit method LinUCB, which shows that linear models may", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "lack representation power and find biased estimates for the underlying reward generating function.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "Furthermore, our proposed Neural-LinUCB achieves a comparable regret with NeuralUCB in all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "experiments despite the fact that our algorithm only explores in the output layer of the neural network,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "score": 1.0, + "content": "which is more computationally efficient as we will show in the sequel.The results in our experiment", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "are well aligned with our theory that deep representation and shallow exploration are sufficient to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "guarantee a good performance of neural contextual bandit algorithms, which is also consistent with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "the findings in existing literature [38] that decoupling the representation learning and uncertainty", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 538, + 259, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 259, + 551 + ], + "score": 1.0, + "content": "estimation improves the performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 407, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "We also conducted experiments to study the effects of different widths of deep neural networks on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "the regret performance and to show the computational efficiency of Neural-LinUCB compared with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 576, + 480, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 480, + 588 + ], + "score": 1.0, + "content": "existing neural bandit algorithms. Due to the space limit, we defer the results to Appendix A.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 554, + 505, + 588 + ] + }, + { + "type": "title", + "bbox": [ + 101, + 597, + 187, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 190, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 190, + 613 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 105, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "In this paper, we propose a new neural contextual bandit algorithm called Neural-LinUCB, which uses", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "the hidden layers of a ReLU neural network as a deep representation of the raw feature vectors and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "performs UCB type exploration on the last layer of the neural network. By incorporating techniques", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "in liner contextual bandits and neural tangent kernels, we prove that the proposed algorithm achieves", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "a sublinear regret when the width of the network is sufficiently large. This is the first regret analysis", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "of neural contextual bandit algorithms with deep representation and shallow exploration, which have", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "been observed in practice to work well on many benchmark bandit problems [38]. 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For all authors...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 146, + 108, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 145, + 107, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 145, + 107, + 505, + 120 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 161, + 118, + 288, + 131 + ], + "spans": [ + { + "bbox": [ + 161, + 118, + 288, + 131 + ], + "score": 1.0, + "content": "contributions and scope? [Yes]", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 146, + 131, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 146, + 131, + 506, + 143 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] We discussed the limitation of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 162, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 162, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "the assumptions made in this paper. We also admit in the experiment that the theory", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 160, + 153, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 160, + 153, + 506, + 165 + ], + "score": 1.0, + "content": "maybe conservative since our experiment does not require a very wide neural network", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 163, + 282, + 177 + ], + "spans": [ + { + "bbox": [ + 161, + 163, + 282, + 177 + ], + "score": 1.0, + "content": "to achieve good performance.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 147, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 147, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 162, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 162, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "focuses on a general methodology in bandit problems and its theoretical analysis. It", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 162, + 199, + 333, + 210 + ], + "spans": [ + { + "bbox": [ + 162, + 199, + 333, + 210 + ], + "score": 1.0, + "content": "does not cause any negative social impact.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 146, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 146, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 161, + 221, + 214, + 235 + ], + "spans": [ + { + "bbox": [ + 161, + 221, + 214, + 235 + ], + "score": 1.0, + "content": "them? [Yes]", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 131, + 237, + 302, + 249 + ], + "lines": [ + { + "bbox": [ + 129, + 236, + 304, + 250 + ], + "spans": [ + { + "bbox": [ + 129, + 236, + 304, + 250 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 146, + 252, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 145, + 251, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 145, + 251, + 506, + 265 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 162, + 263, + 289, + 274 + ], + "spans": [ + { + "bbox": [ + 162, + 263, + 289, + 274 + ], + "score": 1.0, + "content": "assumptions listed in Section 4", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 144, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 144, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [Yes] Proofs are provided in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 161, + 285, + 219, + 300 + ], + "spans": [ + { + "bbox": [ + 161, + 285, + 219, + 300 + ], + "score": 1.0, + "content": "the appendix.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 131, + 302, + 241, + 313 + ], + "lines": [ + { + "bbox": [ + 128, + 300, + 243, + 316 + ], + "spans": [ + { + "bbox": [ + 128, + 300, + 243, + 316 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 146, + 316, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 146, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 146, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 162, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 162, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] We provide", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 162, + 339, + 307, + 351 + ], + "spans": [ + { + "bbox": [ + 162, + 339, + 307, + 351 + ], + "score": 1.0, + "content": "them in the supplementary material.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 146, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 146, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 161, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 161, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "were chosen)? [Yes] We specify all the details in the Implementations paragraph of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 162, + 373, + 205, + 385 + ], + "spans": [ + { + "bbox": [ + 162, + 373, + 205, + 385 + ], + "score": 1.0, + "content": "Section 5.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 146, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 146, + 386, + 506, + 398 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 162, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 162, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "ments multiple times)? [Yes] All the figures are plotted with the standard error with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 162, + 408, + 283, + 420 + ], + "spans": [ + { + "bbox": [ + 162, + 408, + 283, + 420 + ], + "score": 1.0, + "content": "respect to random repetitions.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 146, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 146, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 162, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 162, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [Yes] We stated the type of workstation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 162, + 443, + 342, + 454 + ], + "spans": [ + { + "bbox": [ + 162, + 443, + 342, + 454 + ], + "score": 1.0, + "content": "at the end of the first paragraph of Section 5.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 131, + 457, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 128, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 128, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "4. 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[N/A] This work", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 162, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "focuses on a general methodology in bandit problems and its theoretical analysis. It", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 162, + 199, + 333, + 210 + ], + "spans": [ + { + "bbox": [ + 162, + 199, + 333, + 210 + ], + "score": 1.0, + "content": "does not cause any negative social impact.", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 146, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 221, + 214, + 235 + ], + "spans": [ + { + "bbox": [ + 161, + 221, + 214, + 235 + ], + "score": 1.0, + "content": "them? 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[Yes] See the", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 263, + 289, + 274 + ], + "spans": [ + { + "bbox": [ + 162, + 263, + 289, + 274 + ], + "score": 1.0, + "content": "assumptions listed in Section 4", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 144, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 144, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [Yes] Proofs are provided in", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 285, + 219, + 300 + ], + "spans": [ + { + "bbox": [ + 161, + 285, + 219, + 300 + ], + "score": 1.0, + "content": "the appendix.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + } + ], + "index": 14.5, + "bbox_fs": [ + 144, + 251, + 506, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 302, + 241, + 313 + ], + "lines": [ + { + "bbox": [ + 128, + 300, + 243, + 316 + ], + "spans": [ + { + "bbox": [ + 128, + 300, + 243, + 316 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 128, + 300, + 243, + 316 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 316, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 146, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 146, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 162, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] We provide", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 162, + 339, + 307, + 351 + ], + "spans": [ + { + "bbox": [ + 162, + 339, + 307, + 351 + ], + "score": 1.0, + "content": "them in the supplementary material.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 146, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 161, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "were chosen)? [Yes] We specify all the details in the Implementations paragraph of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 162, + 373, + 205, + 385 + ], + "spans": [ + { + "bbox": [ + 162, + 373, + 205, + 385 + ], + "score": 1.0, + "content": "Section 5.", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 146, + 386, + 506, + 398 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 162, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "ments multiple times)? 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All", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 162, + 494, + 285, + 506 + ], + "spans": [ + { + "bbox": [ + 162, + 494, + 285, + 506 + ], + "score": 1.0, + "content": "the assets were properly cited.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 146, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "(b) Did you mention the license of the assets? [N/A] All the codes and datasets are", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 163, + 520, + 215, + 531 + ], + "spans": [ + { + "bbox": [ + 163, + 520, + 215, + 531 + ], + "score": 1.0, + "content": "open-source.", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 146, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 542, + 402, + 554 + ], + "spans": [ + { + "bbox": [ + 162, + 542, + 402, + 554 + ], + "score": 1.0, + "content": "We include our code in the supplementary for reproduction.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 146, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "(d) Did you discuss whether and how consent was obtained from people whose data you’re", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 566, + 253, + 578 + ], + "spans": [ + { + "bbox": [ + 162, + 566, + 253, + 578 + ], + "score": 1.0, + "content": "using/curating? 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[N/A] The data does not contain any personally", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 600, + 343, + 612 + ], + "spans": [ + { + "bbox": [ + 162, + 600, + 343, + 612 + ], + "score": 1.0, + "content": "identifiable information or offensive content.", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + } + ], + "index": 36.5, + "bbox_fs": [ + 146, + 472, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 615, + 432, + 627 + ], + "lines": [ + { + "bbox": [ + 128, + 614, + 433, + 629 + ], + "spans": [ + { + "bbox": [ + 128, + 614, + 433, + 629 + ], + "score": 1.0, + "content": "5. If you used crowdsourcing or conducted research with human subjects...", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 128, + 614, + 433, + 629 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 630, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 145, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 145, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "(a) Did you include the full text of instructions given to participants and screenshots, if", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 641, + 237, + 653 + ], + "spans": [ + { + "bbox": [ + 161, + 641, + 237, + 653 + ], + "score": 1.0, + "content": "applicable? 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sha256:308170fc40b36d2c8f2b7050807d05323826a9fd937e26a883fb59e0ce8f0f22 +size 36710 diff --git a/parse/train/uY-XMIbyXec/images/f08787cde0f4c01aad9875cd47aeb4e530e12fac12dbe8e97041bb38e110875e.jpg b/parse/train/uY-XMIbyXec/images/f08787cde0f4c01aad9875cd47aeb4e530e12fac12dbe8e97041bb38e110875e.jpg new file mode 100644 index 0000000000000000000000000000000000000000..afb0f2d147bce648229bfde1c1914dd34d5fe6a9 --- /dev/null +++ b/parse/train/uY-XMIbyXec/images/f08787cde0f4c01aad9875cd47aeb4e530e12fac12dbe8e97041bb38e110875e.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1f5a13b279443400837b677227e7204ca5aa14784414b2112eb8aa05c094c9e7 +size 89434 diff --git a/parse/train/wXgk_iCiYGo/wXgk_iCiYGo.md b/parse/train/wXgk_iCiYGo/wXgk_iCiYGo.md new file mode 100644 index 0000000000000000000000000000000000000000..1269dfbfa2806c5668b506d535047192f1f69926 --- /dev/null +++ b/parse/train/wXgk_iCiYGo/wXgk_iCiYGo.md @@ -0,0 +1,639 @@ +# A DIFFUSION THEORY FOR DEEP LEARNING DYNAMICS: STOCHASTIC GRADIENT DESCENT EXPONENTIALLY FAVORS FLAT MINIMA + +Zeke Xie1,2, Issei Sato 1,2, and Masashi Sugiyama2,1 + +1The University of Tokyo 2RIKEN Center for AIP xie@ms.k.u-tokyo.ac.jp {sato,sugi}@k.u-tokyo.ac.jp + +# ABSTRACT + +Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks in practice. SGD is known to find a flat minimum that often generalizes well. However, it is mathematically unclear how deep learning can select a flat minimum among so many minima. To answer the question quantitatively, we develop a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. To the best of our knowledge, we are the first to theoretically and empirically prove that, benefited from the Hessian-dependent covariance of stochastic gradient noise, SGD favors flat minima exponentially more than sharp minima, while Gradient Descent (GD) with injected white noise favors flat minima only polynomially more than sharp minima. We also reveal that either a small learning rate or large-batch training requires exponentially many iterations to escape from minima in terms of the ratio of the batch size and learning rate. Thus, large-batch training cannot search flat minima efficiently in a realistic computational time. + +# 1 INTRODUCTION + +In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various application areas. Due to the over-parametrization and the highly complex loss landscape of deep networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995; 1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al., 2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al., 2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al., 2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b), and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a). However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat minimum. + +The diffusion theory is an important theoretical tool to understand how deep learning dynamics works. It helps us model the diffusion process of probability densities of parameters instead of model parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics (SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrz˛ebski et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented a Beyesian perspective on generalization of SGD. Wu et al. (2018) studied the escape problems of + +SGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained a related proposition that describes the mean escape time in terms of a free energy that depends on the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise. + +However, the related papers mainly analyzed the diffusion process under parameter-independent and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four contributions: + +• The proposed theory formulates the fundamental roles of gradient noise, batch size, the learning rate, and the Hessian in minima selection. +The SGN covariance is approximately proportional to the Hessian and inverse to batch size. Either a small learning rate or large-batch training requires exponentially many iterations to escape minima in terms of ratio of batch size and learning rate. +• To the best of our knowledge, we are the first to theoretically and empirically reveal that SGD favors flat minima exponentially more than sharp minima. + +# 2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS + +We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We denote the data samples as $\{ x _ { j } \} _ { j = 1 } ^ { m }$ , the model parameters as $\theta$ and the loss function over data samples $x$ as $L ( \theta , x )$ . For simplicity, we denote the training loss as $L ( \theta )$ . Following Mandt et al. (2017), we may write SGD dynamics as + +$$ +\theta _ { t + 1 } = \theta _ { t } - \eta \frac { \partial \hat { L } ( \theta _ { t } ) } { \partial \theta _ { t } } = \theta _ { t } - \eta \frac { \partial L ( \theta _ { t } ) } { \partial \theta _ { t } } + \eta C ( \theta _ { t } ) ^ { \frac { 1 } { 2 } } \zeta _ { t } , +$$ + +where $\hat { L } ( \theta )$ is the loss of one minibatch, $\zeta _ { t } \sim \mathcal { N } ( 0 , I )$ , and $C ( \theta )$ represents the gradient noise covariance matrix. The classic approach is to model SGN by Gaussian noise, ${ \mathcal { N } } ( 0 , { \overline { { C } } } ( \theta ) )$ (Mandt et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018). + +Stochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch training, $\begin{array} { r } { C ( \theta _ { t } ) ^ { \frac { 1 } { 2 } } \zeta _ { t } = \frac { \partial L ( \theta _ { t } ) } { \partial \theta _ { t } } - \frac { \partial \hat { L } ( \theta _ { t } ) } { \partial \theta _ { t } } } \end{array}$ , which is the difference between gradient descent and stochastic gradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the mean of many infinite-variance random variables converges to a stable distribution, while the mean of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in practice, we believe the Gaussian approximation of SGN is reasonable. + +Simsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise. They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed distribution looks closer to a stable distribution than a Gaussian distribution. However, this research line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed “SGN” across $n$ model parameters and regarded “SGN" as $n$ samples drawn from a single-variant distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic Gaussian noise. In our paper, SGN computed over different minibatches obeys a $n$ -variant Gaussian distribution, which can be parameter-dependent and anisotropic. + +In Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that the contradicted observations are from the different formulations of gradient noise. Simsekli et al. (2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a $n$ -variant distribution. Our empirical analysis in Figure 1 holds well at least when the batch size $B$ is larger than 16, which is common in practice. Similar empirical evidence can be observed for training ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C. + +![](images/9f25e21da6eaf1f4dcc5fb4e55195174a32550f9f088693c21380f6b59439613.jpg) +Figure 1: The Stochastic Gradient Noise Analysis. The histogram of the norm of the gradient noises computed with the three-layer fully-connected network on MNIST (LeCun, 1998). (a) and (c): the histograms of the norms of two kinds of gradient noise: (a) “SGN” is computed over parameters, which is actually stochastic gradient rather than SGN; (c) SGN is computed over minibatches. (b) and (d): the histograms of the norms of (scaled) Gaussian noise and Lévy noise. Based on (a) and (b), Simsekli et al. (2019) argued that gradient noise across parameters is heavy-tailed Lévy noise. Based on (c) and (d), we show that SGN without the isotropic restriction is approximately Gaussian. + +Panigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN is best described as Gaussian at-least in the early phases of training. Comparing our results with Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors. First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network matters. Because SGN on some layers/networks is more Gaussian than other layers/networks. + +The isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameterdependent and anisotropic SGN brings a large improvement over existing parameter-independent and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under parameter-independent anisotropic heavy-tailed noise, when the batch size is too small $( B \sim 1 )$ to apply Central Limit Theorem. We will leave it as future work. + +SGD Dynamics. Let us replace $\eta$ by $d t$ as unit time. Then the continuous-time dynamics of SGD (Coffey & Kalmykov, 2012) is written as + +$$ +d \theta = - \frac { \partial L ( \theta ) } { \partial \theta } d t + [ 2 D ( \theta ) ] ^ { \frac { 1 } { 2 } } d W _ { t } , +$$ + +where $d W _ { t } \sim { \mathcal { N } } ( 0 , I d t )$ and $\begin{array} { r } { D ( \theta ) = \frac { \eta } { 2 } C ( \theta ) } \end{array}$ . We note that the dynamical time $t$ in the continuoustime dynamics is equal to the product of the number of iterations $T$ and the learning rate $\eta$ : $t = \eta T$ . The associated Fokker-Planck Equation is written as + +$$ +\begin{array} { r l r } { { \frac { \partial P ( \theta , t ) } { \partial t } = \nabla \cdot [ P ( \theta , t ) \nabla L ( \theta ) ] + \nabla \cdot \nabla D ( \theta ) P ( \theta , t ) } } \\ & { } & { = \sum _ { i } \frac { \partial } { \partial \theta _ { i } } [ P ( \theta , t ) \frac { \partial L ( \theta ) } { \partial \theta _ { i } } ] + \sum _ { i } \sum _ { j } \frac { \partial ^ { 2 } } { \partial \theta _ { i } \partial \theta _ { j } } D _ { i j } ( \theta ) P ( \theta , t ) , } \end{array} +$$ + +where $\nabla$ is a nabla operator, and $D _ { i j }$ is the element in the ith row and $j$ th column of $D$ . In standard SGLD, the injected gradient noise is fixed and isotropic Gaussian, $D = I$ . + +![](images/084198cb7082c38ce670ef6ab05aa9b80fc9eb6c2d5851d7717d8d63c4a5ddfd.jpg) +Figure 2: We empirically verified MNIST (LeCun, 1998). The pret $\begin{array} { r } { C ( \theta ) = \frac { H ( \theta ) } { B } } \end{array}$ by using three-layer fully-connected network ons are usually near critical points, while randomly Initialized Models are far from critical points. We display all elements $H _ { ( i , j ) } \in [ 1 e - 4 , 0 . 5 ]$ of the Hessian matrix and the corresponding elements $C _ { ( i , j ) }$ of gradient noise covariance matrix in the space spanned by the eigenvectors of Hessian. Another supplementary experiment on Avila Dataset (De Stefano et al., 2018) in Appendix C reports $\hat { C } _ { a v i l a } \approx 1 . 0 0 4 \frac { H } { B }$ . The small difference factor between the empirical result and the ideal Equation is mainly because the pretrained network is not perfectly located at a critical point. + +The next question is how to formulate the SGN covariance $C ( \theta )$ for SGD? Based on Smith & Le (2018), we can express the SGN covariance as + +$$ +\boldsymbol { \Sigma } ( \theta ) = \frac { 1 } { B } \left[ \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \boldsymbol { \nabla } L ( \theta , x _ { j } ) \boldsymbol { \nabla } L ( \theta , x _ { j } ) ^ { \top } - \boldsymbol { \nabla } L ( \theta ) \boldsymbol { \nabla } L ( \theta ) ^ { \top } \right] \approx \frac { 1 } { B m } \sum _ { j = 1 } ^ { m } \boldsymbol { \nabla } L ( \theta , x _ { j } ) \boldsymbol { \nabla } L ( \theta , x _ { j } ) ^ { \top } . +$$ + +The approximation is true near critical points, due to the fact that the gradient noise variance dominates the gradient mean near critical points. We know the observed fisher information matrix satisfies $\operatorname { F I M } ( \theta ) \approx H ( \theta )$ near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrz˛ebski et al. (2017); Zhu et al. (2019), we obtain + +$$ +C ( \theta ) \approx \frac { 1 } { B m } \sum _ { j = 1 } ^ { m } \nabla L ( \theta , x _ { j } ) \nabla L ( \theta , x _ { j } ) ^ { \top } = \frac { 1 } { B } \mathrm { F I M } ( \theta ) \approx \frac { 1 } { B } H ( \theta ) , +$$ + +which approximately gives + +$$ +D ( \theta ) = \frac { \eta } { 2 } C ( \theta ) = \frac { \eta } { 2 B } H ( \theta ) +$$ + +near minima. It indicates that the SGN covariance $C ( \theta )$ is approximately proportional to the Hessian $H ( \theta )$ and inverse to the batch size $B$ . Obviously, we can generalize Equation 7 by $\begin{array} { r } { D ( \theta ) ~ = ~ \frac { \eta C ( \theta ) } { 2 } ~ = ~ \frac { \eta } { 2 B } [ H ( \theta ) ] ^ { + } } \end{array}$ near critical points, when there exist negative eigenvalues in $H$ along some directions. We use $[ \cdot ] ^ { + }$ to denote the positive semidefinite transformation of a symmetric matrix: if we have the eigendecomposation $H = U \mathrm { d i a g } ( H _ { 1 } , \cdot \cdot \cdot , H _ { n - 1 } , H _ { n } ) U ^ { \top }$ , then $[ H ] ^ { + } = U \mathrm { d i a g } ( | H _ { 1 } | , \cdots , | H _ { n - 1 } | , | H _ { n } | ) U ^ { \dagger }$ . + +We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a followup paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that, the relation still approximately holds for even the randomly network, which is far from critical points. The correlation is especially high along the flat directions with small-magnitude eigenvalues of the Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian noise approximation all failed to capture this core relation in deep learning dynamics. + +# 3 SGD DIFFUSION THEORY + +We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We assume there are two valleys, Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ , seen in Figure 3. Also Col b is the boundary between two valleys. What is the mean escape time for a particle governed by Equation 2 to escape from Sharp Valley $a _ { 1 }$ to Flat Valley $a _ { 2 }$ ? The mean escape time is widely used in related statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019). + +![](images/73689560624558dfdf080463c0fd2eec0e7953c645f73e038906c281f99c7fba.jpg) +Figure 3: Kramers Escape Problem. $a _ { 1 }$ and $ { \boldsymbol { a } } _ { a }$ are minima of two neighboring valleys. $b$ is the saddle point separating the two valleys. $c$ locates outside of Valley $a _ { 1 }$ . + +Gauss’s Divergence Theorem (Arfken & Weber, 1999; Lipschutz et al., 2009) states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. We respectively denote the mean escape time as $\tau$ , the escape rate as $\gamma$ , and the probability current as $J$ . We apply Gauss’s Divergence Theorem to the Fokker-Planck Equation resulting in + +$$ +\nabla \cdot \left[ P ( \theta , t ) \nabla L ( \theta ) \right] + \nabla \cdot \nabla D ( \theta ) P ( \theta , t ) = \frac { \partial P ( \theta , t ) } { \partial t } = - \nabla \cdot J ( \theta , t ) . +$$ + +The mean escape time is expressed (Van Kampen, 1992) as + +$$ +\tau = { \frac { 1 } { \gamma } } = { \frac { P ( \theta \in V _ { a } ) } { \int _ { S _ { a } } J \cdot d S } } , +$$ + +where $\begin{array} { r } { P ( \theta \in V _ { a } ) = \int _ { V _ { a } } P ( \theta ) d V } \end{array}$ is the current probability inside Valley a, $J$ is the probability current produced by the probability source $P ( \theta \in V _ { a } )$ , $\begin{array} { r } { j = \int _ { S _ { a } } J \cdot d S } \end{array}$ is the probability flux (surface integrals of probability current), $S _ { a }$ is the surface (boundary) surrounding Valley a, and $V _ { a }$ is the volume surrounded by $S _ { a }$ . We have $j = J$ in the case of one-dimensional escape. + +Classical Assumptions. We state three classical assumptions first for the density diffusion theory. Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring, 1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012), and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers (Jastrz˛ebski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape Problems. + +Assumption 1 (The Second Order Taylor Approximation). The loss function around critical points $\theta ^ { \star }$ can be approximately written as + +$$ +L ( \theta ) = L ( \theta ^ { \star } ) + g ( \theta ^ { \star } ) ( \theta - \theta ^ { \star } ) + \frac { 1 } { 2 } ( \theta - \theta ^ { \star } ) ^ { \top } H ( \theta ^ { \star } ) ( \theta - \theta ^ { \star } ) . +$$ + +Assumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima. +Assumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature). + +We will dive into these two assumptions deeper than previous papers for SGD dynamics. Assumptions 2 and 3 both mean that our diffusion theory can better describe the escape processes that cost more iterations. As this class of “slow” escape processes takes main computational time compared with “fast” escape processes, this class of “slow” escape process is more interesting for training of deep neural networks. Our empirical analysis in Section 4 supports that the escape processes in the wide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus, Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B. + +Escape paths. We generalize the concept of critical points into critical paths as the path where 1) the gradient perpendicular to the path direction must be zero, and 2) the second order directional derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs) for escaping must be critical paths. The most possible escape direction at one point must be the direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs, which are strictly flat, is ignorable according to our following analysis. + +The boundary between Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ is the saddle point $b$ . The Hessian at $b$ , $H _ { b }$ , must have only one negative eigenvalue and the corresponding eigenvector is the escape direction. Without losing generality, we first assume that there is only one most possible path through $\operatorname { C o l } b$ existing between Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ . + +SGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers are interested in SGLD, when the injected noise dominates SGN as $\eta 0$ in final epochs. Because SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is usually simplified as Gradient Descent with injected white noise, whose behavior is identical to Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a mixture of injected white noise and SGN. + +Theorem 3.1 (SGLD Escapes Minima). The loss function $L ( \theta )$ is of class $C ^ { 2 }$ and $n$ -dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the outside of Valley a is + +$$ +\tau = \frac { 1 } { \gamma } = 2 \pi \sqrt { \frac { - \operatorname * { d e t } ( H _ { b } ) } { \operatorname * { d e t } ( H _ { a } ) } } \frac { 1 } { | H _ { b e } | } \exp \left( \frac { \Delta L } { D } \right) . +$$ + +We denote that $H _ { a }$ and $H _ { b }$ are the Hessians of the loss function at the minimum a and the saddle point $b$ , $\Delta L = L ( b ) - L ( a )$ is the loss barrier height, e indicates the escape direction, and $H _ { b e }$ is the eigenvalue of the Hessian $H _ { b }$ corresponding to the escape direction. The diffusion coefficient $D$ is usually set to 1 in SGLD. + +SGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD is far from the Gibs-Boltzmann distribution, $\begin{array} { r } { P ( \theta ) = \frac { 1 } { Z } \exp \left( - \frac { L ( \theta ) } { D } \right) } \end{array}$ . These different characteristics make SGD diffusion behave differently from known physical dynamical systems and much less studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN and injected white noise, as long as the eigenvectors of $D ( \theta )$ are closely aligned with the eigenvectors of $H ( \theta )$ . + +Theorem 3.2 (SGD Escapes Minima). The loss function $L ( \theta )$ is of class $C ^ { 2 }$ and $n$ -dimensional. Only one most possible path exists between Valley a and the outside of Valley $^ { a }$ . If Assumption $I$ , 2, and 3 hold, and the dynamics is governed by $S G D$ , then the mean escape time from Valley a to the outside of Valley a is + +$$ +\tau = 2 \pi \frac { 1 } { | H _ { b e } | } \exp \left[ \frac { 2 B \Delta L } { \eta } \left( \frac { s } { H _ { a e } } + \frac { ( 1 - s ) } { | H _ { b e } | } \right) \right] , +$$ + +where $s \in ( 0 , 1 )$ is a path-dependent parameter, and $H _ { a e }$ and $H _ { b e }$ are, respectively, the eigenvalues of the Hessians at the minimum a and the saddle point $b$ corresponding to the escape direction e. + +Multiple-path escape. Each escape path contributes to the total escape rate. Multiple paths combined together have a total escape rate. If there are multiple parallel from the start valley to the end valley, we can compute the total escape rate easily based on the following computation rule. The computation rule is based on the fact that the probability flux integrals are additive. We can easily generalize the mean escape time analysis into the cases that there are multiple parallel escape paths indexed by $p$ As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the dimensionality directly, higher dimensionality may increase the number of escape paths and loss valleys, and change the spectrum of the Hessians. + +![](images/40850afd82cda33fe28c5aa92bf1019fad755c726e83948318f7a64db61eaf3d.jpg) +Figure 4: The mean escape time analysis of SGD by using Styblinski-Tang Function. The Pearson Correlation is higher than 0.99. Left Column: Sharpness. Middle Column: Batch Size. Right Column: Learning Rate. + +Rule 1. If there are multiple MPPs between the start valley and the end valley, then $\begin{array} { r } { \gamma _ { t o t a l } = \sum _ { p } \gamma _ { p } } \end{array}$ + +Thus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and analyze the escape rates. + +Minima selection. Now, we may formulate the probability of minima selection as Proposition 1. We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and the landscape contain many good modes and bad modes. SGD transits from one mode to another mode during training. The mean escape time of one mode corresponds to the number of iterations which SGD spends on this mode during training, which is naturally proportional to the probability of selecting this mode after training. + +Proposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by + +$$ +P ( \theta \in V _ { a } ) = \frac { \tau _ { a } } { \sum _ { v } \tau _ { v } } , +$$ + +where v is the index of valleys, and $\tau _ { v }$ is the mean escape time from Valley v to the outside of Valley $v$ + +# 4 EMPIRICAL ANALYSIS + +In this section, we try to directly validate the escape formulas on real-world datasets. Each escape process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100 times under various gradient noise scales, batch sizes, learning rates, and sharpness. + +How to compare the escape rates under the same settings with various minima sharpness? Our√ method is to multiply a rescaling factor $\sqrt { k }$ to each parameter, and the Hessian will be proportionally√ rescaled by a factor $k$ . If we let $L ( \theta ) = f ( \theta ) L ( \theta ) = f ( { \sqrt { k } } \theta )$ , then $H ( \theta ) = \nabla ^ { 2 } f ( \theta ) \to H ( \theta ) =$ $k \nabla ^ { 2 } f ( \theta )$ . Thus, we can use $k$ to indicate the minima sharpness. The theoretical relations of SGD we try to validate can be formulated as: $( 1 ) - \log ( \gamma ) = \bar { \mathcal { O } } ( \textstyle { \frac { 1 } { k } } )$ , $2 ) - \log ( \gamma ) = \mathcal { O } ( B )$ , and (3) $- \log ( \gamma ) = \mathcal { O } ( \textstyle { \frac { 1 } { \eta } } )$ . + +The mean escape time analysis of SGD. Styblinski-Tang Function, which has multiple minima and saddle points, is a common test function for nonconvex optimization. We conduct an intuitional 10-dimensional experiment, where the simulations start from a given minimum and terminate when reaching the boundary of the loss valley. The number of iterations is recorded for calculating the escape rate. We also train fully connected networks on four real-world datasets, including a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis (De Stefano et al., 2018; Dua & Graff, 2017). Figure 4 and Figure 5 clearly verifies that the escape rate exponentially depends on the minima sharpness (reflected by $k$ ), the batch size, and the learning rate on both test functions and real-world training, which fully supports our theoretical results. + +![](images/8bd372226d5c3c86bb1d2b2130b67041a3b2fb52d4924db07d0de0674e8493ba.jpg) +Figure 5: The mean escape time analysis of SGD by training neural networks on Avila Dataset. Left Column: Sharpness. Middle Column:Batch Size. Right Column: Learning Rate. We leave the results on Banknote Authentication, Cardiotocography, and Sensorless Drive Diagnosis in Appendix D. + +![](images/1f2c25c1998116a59e897d7ca8015655aad9a6ac9d7ea655f81156edacd74762.jpg) +Figure 6: The mean escape time analysis of SGLD. Subfigure (a) and (b): Styblinski-Tang Function. Subfigure (c) and (d): Neural Network. + +Model architecture and details: We used fully-connected networks with the depth 2 and the width 10 in Figure 5. The experiments using Logistic Regression and Fully-connected networks with the depth 3 are presented in Appendix E. We leave more experimental details and results in Appendix D.1 and Appendix E. + +The mean escape time analysis of SGLD. We try to validate $\gamma = \mathcal { O } ( k )$ and $- \log ( \gamma ) = \mathcal { O } ( \frac { 1 } { D } ) .$ for SGLD (dominated by injected Gaussian noise). Figure 6 shows that SGLD only favors flat minima polynomially more than sharp minima as Theorem 3.1 indicates. Figure 6 also verifies that the injected gradient noise scale exponentially affects flat minima selection. + +# 5 DISCUSSION + +SGD favors flat minima exponentially more than sharp minima. We can discover a few interesting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially depends on the eigenvalue of the Hessian at minima along the escape direction, $H _ { a e }$ . Thus, SGD favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD comes from the exponential relation of the mean escape time and the minima sharpness. The measure of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness” is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly more than Adam. + +The ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains why large-batch training can easily get trapped near sharp minima, and increasing the learning rate proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause is large-batch training expects exponentially longer time to escape minima. Note that, as the mean escape time in the theorems is equivalent to the product of the learning rate and the number of iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the batch size and the learning rate. The practical computational time in large-batch training is usually too short to search many enough flat minima. We conjecture that exponentially increasing training iterations may be helpful for large batch training, while this is often too expensive in practice. + +Low dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of overparametrized deep networks are close to zero, while only a small number of eigenvalues are large (Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018). + +As the escape rate exponentially depends the corresponding eigenvalues, a small number of large eigenvalues means that the process of minima selection mainly happens in the relatively low dimensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics hardly depends on those “meaningless” dimensions with small second order directional derivatives. This novel characteristic of SGD significantly reduces the explorable parameter space around one minimum into a much lower dimensional space. + +High-order effects. As we have applied the second-order Taylor approximation near critical points, our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this paper. However, we also argue that the third-order effect is much smaller than the second-order effect under the low temperature assumption in Kramers Escape Problems. We will leave the more refined high-order theory as future work. + +# 6 CONCLUSION + +In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to formulate the exponential relation of minima selection to the minima sharpness, the batch size, and the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory not only helps us understand how SGD selects flat minima, but also will provide researchers a powerful theoretical tool to analyze more learning behaviors and design better optimizers in future. + +# ACKNOWLEDGEMENT + +We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study. + +# REFERENCES + +Alessandro Achille and Stefano Soatto. Where is the information in a deep neural network? arXiv preprint arXiv:1905.12213, 2019. + +George B Arfken and Hans J Weber. 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We decompose the proof into two steps: 1) compute the probability of locating in valley a, $P ( \theta \in V _ { a } )$ , and 2) compute the probability flux $\begin{array} { r } { j = \int _ { S _ { a } } \bar { \boldsymbol { J } } \cdot d \boldsymbol { S } } \end{array}$ . + +Without losing generality, we first prove the one-dimensional case. + +Step 1: Under Assumption 1, the stationary distribution around minimum a is $\begin{array} { r l } { P ( \theta ) } & { { } = } \end{array}$ $\begin{array} { r } { P ( a ) \exp [ - \frac { L ( \theta ) - L ( a ) } { T } ] } \end{array}$ , where $T = D$ . Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. We use the $T$ notation as the temperature parameter in the stationary distribution, and use the $D$ notation as the diffusion coefficient in the dynamics, for their different roles. + +$$ +\begin{array} { r l } & { \quad P ( \theta \in V _ { a } ) } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( \theta ) d V } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( a ) \exp \left[ - \frac { L ( \theta ) - L ( a ) } { T } \right] d \theta } \\ & { = P ( a ) \displaystyle \int _ { \theta \in V _ { a } } \exp \left[ - \frac { \frac { 1 } { 2 } ( \theta - a ) ^ { \top } H _ { a } ( \theta - a ) + \mathcal { O } ( \Delta \theta ^ { 3 } ) } { T } \right] d \theta } \\ & { = P ( a ) \displaystyle \frac { ( 2 \pi T ) ^ { \frac { 1 } { 2 } } } { H ^ { \frac { 1 } { 2 } } } . } \end{array} +$$ + +Step 2: + +$$ +\begin{array} { l } { \displaystyle J = P ( \theta ) \nabla L ( \theta ) + P ( \theta ) \nabla D + D \nabla P ( \theta ) } \\ { \displaystyle J = P ( \theta ) \left( \nabla L ( \theta ) + \nabla D - \displaystyle \frac { D } { T } \nabla L ( \theta ) \right) } \\ { \displaystyle \nabla D = \left( \displaystyle \frac { D } { T } - 1 \right) \nabla L } \end{array} +$$ + +Apply this result to the Fokker-Planck Equation 4, we have + +$$ +\begin{array} { r l } & { \nabla \cdot \nabla [ D ( \theta ) P ( \theta , t ) ] } \\ & { = \nabla \cdot D \nabla P ( \theta , t ) + \nabla \cdot \left[ \left( \displaystyle \frac { D } { T } - 1 \right) \nabla L ( \theta ) \right] P ( \theta , t ) } \end{array} +$$ + +And thus we obtain the Smoluchowski equation and a new form of J + +$$ +\begin{array} { r l r } & { } & { \displaystyle { \frac { \partial P ( \theta , t ) } { \partial t } = \nabla \cdot \left[ D \left( \frac { 1 } { T } \nabla L ( \theta ) + \nabla \right) P ( \theta , t ) \right] = - \nabla \cdot J ( \theta , t ) , } } \\ & { } & { \displaystyle { J ( \theta ) = D \exp \left( \frac { - L ( \theta ) } { T } \right) \nabla \left[ \exp \left( \frac { L ( \theta ) } { T } \right) P ( \theta ) \right] . } } \end{array} +$$ + +We note that the probability density outside Valley a must be zero, $P ( c ) = 0$ . As we want to compute the probability flux escaping from Valley a in the proof, the probability flux escaping from other valleys into Valley a should be ignored. Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path + +$$ +\begin{array} { r } { \displaystyle \int _ { a } ^ { c } \frac { \partial } { \partial \theta } \left[ \exp \left( \frac { L ( \theta ) } { T } \right) P ( \theta ) \right] d \theta = \int _ { a } ^ { c } - \frac { J } { D } \exp \left( \frac { L ( \theta ) } { T } \right) d \theta } \\ { \displaystyle \exp \left( \frac { L ( \theta ) } { T } \right) P ( \theta ) | _ { a } ^ { c } = - \frac { J } { D } \int _ { a } ^ { c } \exp \left( \frac { L ( \theta ) } { T } \right) d \theta } \\ { \displaystyle 0 - \exp \left( \frac { L ( a ) } { T } \right) P ( a ) = - \frac { J } { D } \int _ { a } ^ { c } \exp \left( \frac { L ( \theta ) } { T } \right) d \theta } \\ { \displaystyle J = \frac { D \exp \left( \frac { L ( a ) } { T } \right) P ( a ) } { \int _ { a } ^ { c } \exp \left( \frac { L ( \theta ) } { T } \right) d \theta } . } \end{array} +$$ + +We move $J$ to the outside of integral based on Gauss’s Divergence Theorem, because $J$ is fixed on the escape path from one minimum to another. As there is no field source on the escape path, $\begin{array} { r } { \int _ { V } \nabla \cdot \boldsymbol { J } ( \boldsymbol { \theta } ) d \boldsymbol { \dot { V } } = \boldsymbol { 0 } } \end{array}$ . Then $\nabla J ( \theta ) = 0$ . Obviously, only minima are probability sources in deep learning. Under Assumption 3 and the second-order Taylor approximation, we have + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { a } ^ { c } \exp \left( \frac { L ( \theta ) } { T } \right) d \theta } \\ & { = \displaystyle \int _ { a } ^ { c } \exp \left[ \frac { L ( b ) + \frac { 1 } { 2 } ( \theta - b ) ^ { \top } H _ { b } ( \theta - b ) + \mathcal { O } ( \Delta \theta ^ { 3 } ) } { T } \right] d \theta } \\ & { \approx \exp \left( \frac { L ( b ) } { T } \right) \displaystyle \int _ { - \infty } ^ { + \infty } \exp \left[ \frac { \frac { 1 } { 2 } ( \theta - b ) ^ { \top } H _ { b } ( \theta - b ) } { T } \right] d \theta } \\ & { = \exp \left( \frac { L ( b ) } { T } \right) \sqrt { \frac { 2 \pi T } { | H _ { b } | } } . } \end{array} +$$ + +Based on the results of Step 1 and Step 2, we obtain + +$$ +\begin{array} { r l } & { \gamma = \frac { \displaystyle \int _ { S _ { a } } J \cdot d S } { \displaystyle P ( \theta \in V _ { a } ) } = \frac { J } { P \left( \theta \in V _ { a } \right) } } \\ & { \quad = \frac { \displaystyle P P \left( a \right) \exp \left( \frac { L ( a ) } { T } \right) } { \displaystyle \exp \left( \frac { L ( b ) } { T } \right) \sqrt { \frac { 2 \pi T } { | R _ { b } | } } } \frac { 1 } { P \left( a \right) \sqrt { \frac { 2 \pi T } { H _ { a } } } } } \\ & { \quad = \frac { \displaystyle \frac { D \sqrt { H _ { a } } \| H _ { b } \| } { 2 \pi T } } { \displaystyle 2 \pi T } \exp \left( - \frac { \Delta L _ { a b } } { T } \right) } \\ & { \quad = \frac { \displaystyle \sqrt { H _ { a } } | H _ { b } | } { \displaystyle 2 \pi } \exp \left( - \frac { \Delta L _ { a b } } { D } \right) } \end{array} +$$ + +We generalize the proof of one-dimensional diffusion to high-dimensional diffusion + +Step 1: + +$$ +\begin{array} { r l } & { \quad P ( \theta \in V _ { a } ) } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( \theta ) d V } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( a ) \exp \left[ - \frac { L ( \theta ) - L ( a ) } { T } \right] d V } \\ & { = P ( a ) \displaystyle \int _ { \theta \in V _ { a } } \exp \left[ - \frac { \frac { 1 } { 2 } ( \theta - a ) ^ { \top } H _ { a } ( \theta - a ) + \mathcal { O } ( \Delta \theta ^ { 3 } ) } { T } \right] d V } \\ & { = P ( a ) \displaystyle \frac { ( 2 \pi T ) ^ { \frac { n } { 2 } } } { \mathrm { d e t } ( H _ { a } ) ^ { \frac { 1 } { 2 } } } } \end{array} +$$ + +Step 2: Based on the formula of the one-dimensional probability current and flux, we obtain + +So we have + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { S _ { b } } J \cdot d S } \\ & { = \displaystyle J _ { b } \int _ { S _ { b } } \exp \left[ - \frac { \frac { 1 } { 2 } ( \theta - b ) ^ { \top } H _ { b } ^ { + } ( \theta - b ) } { T } \right] d S } \\ & { = \displaystyle J _ { b } \frac { ( 2 \pi T ) ^ { \frac { n - 1 } { 2 } } } { ( \prod _ { i = 1 } ^ { n - 1 } H _ { b i } ) ^ { \frac { 1 } { 2 } } } } \end{array} +$$ + +$$ +\begin{array} { c } { { \tau = 2 \pi \sqrt { \displaystyle \frac { \prod _ { i = 1 } ^ { n - 1 } H _ { b i } } { \operatorname * { d e t } ( H _ { a } ) | H _ { b e } | } } \exp \left( \displaystyle \frac { \Delta L } { T } \right) } } \\ { { = 2 \pi \sqrt { \displaystyle \frac { - \operatorname * { d e t } ( H _ { b } ) } { \operatorname * { d e t } ( H _ { a } ) } } \displaystyle \frac { 1 } { | H _ { b e } | } \exp \left( \displaystyle \frac { \Delta L } { D } \right) . } } \end{array} +$$ + +# A.2 PROOF OF THEOREM 3.2 + +Proof. We decompose the proof into two steps and analyze the one-dimensional case like before. The following proof is similar to the proof of SGLD except that we make $T _ { a }$ the temperature near the minimum a and $T _ { b }$ the temperature near the saddle point b. + +One-dimensional SGD Diffusion: + +Step 1: Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. + +$$ +\begin{array} { r l } & { \quad P ( \theta \in V _ { a } ) } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( \theta ) d V } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( a ) \exp \left[ - \frac { L ( \theta ) - L ( a ) } { T _ { a } } \right] d V } \\ & { = P ( a ) \displaystyle \int _ { \theta \in V _ { a } } \exp \left[ - \frac { \frac { 1 } { 2 } ( \theta - a ) ^ { \top } H _ { a } ( \theta - a ) + \mathcal { O } ( \Delta \theta ^ { 3 } ) } { T _ { a } } \right] d \theta } \\ & { = P ( a ) \displaystyle \frac { ( 2 \pi T _ { a } ) ^ { \frac { 1 } { 2 } } } { H ^ { \frac { 1 } { 2 } } } } \end{array} +$$ + +Step 2: + +$$ +\begin{array} { l } { \displaystyle J = P ( \boldsymbol { \theta } ) \nabla L ( \boldsymbol { \theta } ) + P ( \boldsymbol { \theta } ) \nabla D + D \nabla P ( \boldsymbol { \theta } ) } \\ { \displaystyle J = P ( \boldsymbol { \theta } ) \left[ \nabla L ( \boldsymbol { \theta } ) + \nabla D - \frac { D } { T } \nabla L ( \boldsymbol { \theta } ) - D L ( \boldsymbol { \theta } ) \nabla \left( \frac { 1 } { T } \right) \right] } \end{array} +$$ + +According to Equation 7, $\nabla \left( { \frac { 1 } { T } } \right)$ is ignorable near the minimum a and the col $\mathbf { b }$ , thus + +$$ +\nabla D = \left( \frac { D } { T } - 1 \right) \nabla L . +$$ + +Apply this result to the Fokker-Planck Equation 4, we have + +$$ +\begin{array} { r l } & { \nabla \cdot \nabla [ D ( \theta ) P ( \theta , t ) ] } \\ & { = \nabla \cdot D \nabla P ( \theta , t ) + \nabla \cdot \left[ \left( \displaystyle \frac { D } { T } - 1 \right) \nabla L ( \theta ) \right] P ( \theta , t ) } \end{array} +$$ + +And thus we obtain the Smoluchowski equation and a new form of J + +$$ +\begin{array} { r } { \frac { \partial P ( \theta , t ) } { \partial t } = \nabla \cdot \left[ D \left( \frac { 1 } { T } \nabla L ( \theta ) + \nabla \right) P ( \theta , t ) \right] = - \nabla \cdot J , } \\ { J = D \exp \left( \frac { - L ( \theta ) } { T } \right) \nabla \left[ \exp \left( \frac { L ( \theta ) } { T } \right) P ( \theta ) \right] . } \end{array} +$$ + +We note that the Smoluchowski equation is true only near critical points. We assume the point s is the midpoint on the most possible path between a and $\mathbf { b }$ , where $L ( \bar { s } ) = ( 1 - s ) L ( a ) + s \bar { L } ( b )$ . The temperature $T _ { a }$ dominates the path $a s$ , while temperature $T _ { b }$ dominates the path $s \to b$ . So we have + +$$ +\nabla \left[ \exp \left( \frac { L ( \theta ) - L ( s ) } { T } \right) P ( \theta ) \right] = J D ^ { - 1 } \exp \left( \frac { L ( \theta ) - L ( s ) } { T } \right) . +$$ + +Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path + +$$ +\begin{array} { l } { { L e f t = \int _ { a } ^ { c } \frac { \partial } { \partial \theta } [ \exp \left( \displaystyle \frac { L ( \theta ) - L ( s ) } { T } \right) P ( \theta ) ] d \theta } } \\ { { \ = \int _ { a } ^ { s } \frac { \partial } { \partial \theta } \left[ \exp \left( \displaystyle \frac { L ( \theta ) - L ( s ) } { T _ { a } } \right) P ( \theta ) \right] d \theta } } \\ { { \ ~ + \int _ { s } ^ { c } \frac { \partial } { \partial \theta } \left[ \exp \left( \displaystyle \frac { L ( \theta ) - L ( s ) } { T _ { b } } \right) P ( \theta ) \right] d \theta } } \\ { { \ = [ P ( s ) - \exp \left( \displaystyle \frac { L ( a ) - L ( s ) } { T _ { a } } \right) P ( a ) ] + [ 0 - P ( s ) ] } } \\ { { \ ~ } } \\ { { \ = - \exp \left( \displaystyle \frac { L ( a ) - L ( s ) } { T _ { a } } \right) P ( a ) } } \end{array} +$$ + +$$ +R i g h t = - \ J \int _ { a } ^ { c } D ^ { - 1 } \exp \left( \frac { L ( \theta ) - L ( s ) } { T } \right) d \theta +$$ + +We move $J$ to the outside of integral based on Gauss’s Divergence Theorem, because $J$ is fixed on the escape path from one minimum to another. As there is no field source on the escape path, $\begin{array} { r } { \int _ { V } \nabla \cdot \boldsymbol { J } ( \boldsymbol { \theta } ) \dot { d V } = 0 } \end{array}$ and $\nabla J ( \theta ) = 0$ . Obviously, only minima are probability sources in deep learning. So we obtain + +$$ +J = \frac { \exp \left( \frac { L ( a ) - L ( s ) } { T _ { a } } \right) P ( a ) } { \int _ { a } ^ { c } D ^ { - 1 } \exp \left( \frac { L ( \theta ) - L ( s ) } { T } \right) d \theta } . +$$ + +Under Assumption 3, we have + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { a } ^ { c } D ^ { - 1 } \exp \left( \frac { L ( \theta ) - L ( s ) } { T } \right) d \theta } \\ & { \approx \displaystyle \int _ { a } ^ { c } D ^ { - 1 } \exp \left[ \frac { L ( b ) - L ( s ) + \frac { 1 } { 2 } ( \theta - b ) ^ { \top } H _ { b } ( \theta - b ) } { T b } \right] d \theta } \\ & { \approx D _ { b } ^ { - 1 } \displaystyle \int _ { - \infty } ^ { + \infty } \exp \left[ \frac { L ( b ) - L ( s ) + \frac { 1 } { 2 } ( \theta - b ) ^ { \top } H _ { b } ( \theta - b ) } { T b } \right] d \theta } \\ & { = D _ { b } ^ { - 1 } \exp \left( \frac { L ( b ) - L ( s ) } { T _ { b } } \right) \sqrt { \frac { 2 \pi T _ { b } } { | H _ { b } | } } . } \end{array} +$$ + +Based on the results of Step 1 and Step 2, we have + +$$ +\begin{array} { r l } & { \gamma = \frac { \displaystyle \int _ { S _ { a } } J \cdot d S } { \displaystyle P ( \theta \in V _ { a } ) } = \frac { J } { \displaystyle P ( \theta \in V _ { a } ) } } \\ & { \quad = \frac { \displaystyle P ( a ) \exp \Big ( \frac { L ( a ) - L ( s ) } { T _ { a } } \Big ) } { \displaystyle D _ { b } ^ { - 1 } \exp \Big ( \frac { L ( b ) - L ( s ) } { T _ { b } } \Big ) \sqrt { \frac { 2 \pi T _ { b } } { | t _ { b } | } } P ( a ) \sqrt { \frac { 2 \pi T _ { a } } { H _ { a } } } } } \\ & { \quad = \frac { \displaystyle \sqrt { T _ { b } H _ { a } } \big | H _ { b } \big | } { \displaystyle 2 \pi \sqrt { T _ { a } } } \exp \left( - \frac { L ( s ) - L ( a ) } { T _ { a } } - \frac { L ( b ) - L ( s ) } { T _ { b } } \right) } \\ & { \quad = \frac { \displaystyle \sqrt { T _ { b } H _ { a } } \big | H _ { b } \big | } { \displaystyle 2 \pi \sqrt { T _ { a } } } \exp \left( - \frac { s \Delta L } { T _ { a } } - \frac { ( 1 - s ) \Delta L } { T _ { b } } \right) } \end{array} +$$ + +So we have + +$$ +\tau = \frac { 1 } { \gamma } = 2 \pi \sqrt { \frac { T _ { a } } { T _ { b } H _ { a } | H _ { b } | } } \exp \left( { \frac { s \Delta L } { T _ { a } } + \frac { ( 1 - s ) \Delta L } { T _ { b } } } \right) . +$$ + +In the case of pure SGN, $\begin{array} { r } { T _ { a } = \frac { \eta } { 2 B } H _ { a } } \end{array}$ and $\begin{array} { r } { T _ { b } = - \frac { \eta } { 2 B } H _ { b } } \end{array}$ gives + +$$ +\tau = \frac { 1 } { \gamma } = 2 \pi \frac { 1 } { | H _ { b } | } \exp \left[ \frac { 2 B \Delta L } { \eta } ( \frac { s } { H _ { a } } + \frac { ( 1 - s ) } { | H _ { b } | } ) \right] . +$$ + +We generalize the proof above into the high-dimensional SGD diffusion. + +Step 1: + +$$ +\begin{array} { r l } & { \quad P ( \theta \in V _ { a } ) } \\ & { = \displaystyle \int _ { \theta \in V _ { a } } P ( \theta ) d V } \\ & { = P ( a ) \int _ { \theta \in V _ { a } } \exp \left[ - \frac { 1 } { 2 } ( \theta - a ) ^ { \top } ( D _ { a } ^ { - \frac { 1 } { 2 } } H _ { a } D _ { a } ^ { - \frac { 1 } { 2 } } ) ( \theta - a ) \right] d V } \\ & { = P ( a ) \frac { ( 2 \pi ) ^ { \frac { n } { 2 } } } { \operatorname* { d e t } ( D _ { a } ^ { - 1 } H _ { a } ) ^ { \frac { 1 } { 2 } } } } \end{array} +$$ + +Step 2: Based on the formula of the one-dimensional probability current and flux, we obtain the high-dimensional flux escaping through Col b: + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { S _ { b } } J \cdot d S } \\ & { = J _ { 1 d } \int _ { S _ { b } } \exp \left[ - \frac { 1 } { 2 } ( \theta - b ) ^ { \top } [ D _ { b } ^ { - \frac { 1 } { 2 } } H _ { b } D _ { b } ^ { - \frac { 1 } { 2 } } ] ^ { \perp e } ( \theta - b ) \right] d S } \\ & { = J _ { 1 d } \frac { ( 2 \pi ) ^ { \frac { n - 1 } { 2 } } } { ( \prod _ { i \neq e } ( D _ { b i } ^ { - 1 } H _ { b i } ) ) ^ { \frac { 1 } { 2 } } } , } \end{array} +$$ + +where $[ \cdot ] ^ { \perp e }$ indicates the directions perpendicular to the escape direction $e$ . So we have + +$$ +\gamma = { \frac { 1 } { 2 \pi } } { \sqrt { \frac { \operatorname* { d e t } ( H _ { a } D _ { a } ^ { - 1 } ) } { - \operatorname* { d e t } ( H _ { b } D _ { b } ^ { - 1 } ) } } } | H _ { b e } | \exp \left( - { \frac { s \Delta L } { T _ { a } } } - { \frac { ( 1 - s ) \Delta L } { T _ { b } } } \right) +$$ + +$T _ { a }$ and $T _ { b }$ are the eigenvalues of $H _ { a } ^ { - 1 } D _ { a }$ and $H _ { b } ^ { - 1 } D _ { b }$ corresponding to the escape direction. We know $\begin{array} { r } { D _ { a } \ = \ \frac { \eta } { 2 B } \mathbf { \bar { { H } } } _ { a } } \end{array}$ and $\begin{array} { r } { D _ { b } ~ = ~ \frac { \eta } { 2 B } [ H _ { b } ] ^ { + } } \end{array}$ . As $D$ must be positive semidefinite, we replace $H _ { b } \ = \ U _ { b } ^ { \top } d i a g ( H _ { b 1 } , \cdot \cdot \cdot , H _ { b ( n - 1 ) } , H _ { b e } ) U _ { b }$ by its positive semidefinite analog $[ H _ { b } ] ^ { + } =$ $U _ { b } ^ { \top } d i a g ( H _ { b 1 } , \cdot \cdot \cdot , H _ { b ( n - 1 ) } , | H _ { b e } | ) U _ { b }$ . Thus, we have + +$$ +\tau = \frac { 1 } { \gamma } = 2 \pi \frac { 1 } { \left| H _ { b e } \right| } \exp \left[ \frac { 2 B \Delta L } { \eta } \left( \frac { s } { H _ { a e } } + \frac { \left( 1 - s \right) } { \left| H _ { b e } \right| } \right) \right] . +$$ + +# A.3 PROOF OF PROPOSITION 1 + +Proof. A stationary distribution must have a balanced probability flux between valleys. So the probability flux of each valley must be equivalent, + +$$ +P ( \theta \in V _ { 1 } ) \gamma _ { 1 2 } = P ( \theta \in V _ { 2 } ) \gamma _ { 2 1 } +$$ + +As $\tau = \gamma ^ { - 1 }$ , it leads to $P ( \theta \in V _ { v } ) \propto \tau _ { v }$ . We normalize the total probability to 1, then we obtain the result. □ + +# B ASSUMPTIONS + +Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily near saddle points. It means that $\begin{array} { r } { \frac { \partial P ( \theta , t ) } { \partial t } = - \nabla \cdot J ( \theta , t ) \approx 0 } \end{array}$ holds near minima $a _ { 1 }$ and $a _ { 2 }$ , but not necessarily holds near saddle point $b$ . Quasi-Equilibrium Assumption is actually weaker but more useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt et al., 2017). Under Assumption 2, the probability density $P$ can behave like a stationary distribution only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about density motion can be obtained under the stationary assumption. + +Low Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jastrz˛ebski et al., 2017), and is always justified when $\frac { \eta } { B }$ is small. Under Assumption 3, the probability densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor approximation, Assumption 1, even more reasonable in SGD diffusion. + +Here, we try to provide a more intuitive explanation about Low Temperature Assumption in the domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The temperature can be interpreted as a real number $D$ . In SGD, we have the temperature as $\begin{array} { r } { D = \frac { \eta } { 2 B } H } \end{array}$ In statistical physics, if $\scriptstyle { \frac { \Delta L } { D } }$ is large, then we call it Low Temperature Approximation. Note that $\scriptstyle { \frac { \Delta L } { D } }$ appears insides an exponential function in the theoretical analysis. People usually believe that, numerically, $\begin{array} { r } { \frac { \Delta L } { D } > 6 } \end{array}$ can make a good approximation, for a similar reason of the 6-sigma rule in statistics. In the final training phase of deep networks, a common setting is $\eta = 0 . 0 1$ and $B = 1 2 8$ . $\textstyle { \frac { \Delta L } { H } } > 2 . 3 \times \mathbf { \dot { 1 } } 0 ^ { - 4 }$ ly apply Assumption 3 to th. Empirically, the condition $\begin{array} { r } { \frac { \Delta L } { H } > 2 . 3 \times 1 0 ^ { - 4 } } \end{array}$ h satisfy the very mild conditionholds well in SGD dynamics. It also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain barrier heights. + +# C THE STOCHASTIC GRADIENT NOISE ANALYSIS + +Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet with $B = 5 0$ on CIFAR-10. We also note that the SGN on ResNet seems less Gaussian than SGN on + +![](images/6304cb3730f61ed48a4694c4774da457ddc5e526dfda5ae7dfe6a42b50aee709.jpg) +Figure 7: The Gradient Noise Analysis. The histogram of the norm of the gradient noises computed with ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009). + +![](images/c79c12433ca079002710133471885496c0dcf563a1d82028bab379f639344eb0.jpg) +Figure 8: The plot of the SGN covariance and the Hessian by training fully-connected network on MNIST. We display all elements $H _ { ( i , j ) } \in [ - 0 . 0 3 , 0 . 0 3 ]$ of the Hessian matrix and the corresponding elements in gradient noise covariance matrix in the original coordinates. + +![](images/8d5acc2b749c95f09c4f662adbdfc17fc43a0921f1406e52bdae5c7bae9ae0f4.jpg) +Figure 9: The plot of the SGN covariance and the Hessian by training fully-connected network on Avila. We display all elements $H _ { ( i , j ) } \in [ 1 e - 4 , 0 . 5 ]$ of the Hessian matrix and the corresponding elements in gradient noise covariance matrix in the space spanned by the eigenvectors of Hessians. + +fully-connected networks with the same batch size. Panigrahi et al. (2019) presented more results on the Gaussianity of SGN under various conditions. + +By Figure 8, we validate $\begin{array} { r } { C = \frac { H } { B } } \end{array}$ in the original coordinates on MNIST. By Figure 9, we also validate $\begin{array} { r } { C = \frac { H } { B } } \end{array}$ on another dataset, Avila, in the space spanned by the eigenvectors of Hessian. The relationcan still be observed in these two cases. $\begin{array} { r } { C = \frac { H } { B } } \end{array}$ + +Data Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on MNIST. We leave the preprocessing of Avila in D. Model: Fully-connected networks. + +# D MAIN EXPERIMENTS + +Figure 10, 11, and 12 respectively validate that the exponential relation of the escape rate with the Hessian, the batch size and the learning rate. + +# D.1 EXPERIMENTAL SETTINGS + +Datasets: a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis. + +Data Precessing: We perform per-pixel zero-mean and unit-variance normalization on input data. For simplicity, we also transform multi-class problems into binary-class problems by grouping labels, although this is unnecessary. + +Model: Two-layer fully-connected networks with one hidden layer and 10 neurons per hidden layer. + +Initializations: To ensure the initialized models are near minima, we first pretrain models with 200-1000 epochs to fit each data set as well as possible. We set the pretrained models’ parameters as the initialized $\theta _ { t = 0 }$ . + +Valleys’ Boundary: In principle, any small neighborhood around $\theta _ { t = 0 }$ can be regarded as the inside of the start valleys. In our experiments, we set each dimension’s distance from $\theta _ { t = 0 }$ should be less than 0.05, namely $| \Delta \theta _ { i } | \le 0 . 0 5$ for each dimension $i$ . If we rescale the landscape by a factor $k$ , the neighborhood will also be rescaled by $k$ . Although we don’t know which loss valleys exist inside the neighborhood, we know the landscape of the neighborhood is invariant in each simulation. + +Hyperparameters: In Figure 10: (a) $\eta = 0 . 0 0 1 , B = 1$ , (b) $\eta = 0 . 0 1 5 , B = 1 .$ , (c) $\eta = 0 . 0 0 5 , B =$ 1, (d) $\eta = 0 . 0 0 0 5 , B = 1$ . In Figure 11: (a) $\eta = 0 . 0 2$ , (b) $\eta = 0 . 6$ , (c) $\eta = 0 . 1 8$ , (d) $\eta = 0 . 0 1$ . In Figure 12: (a) $B = 1$ , (b) $B = 1$ , (c) $B = 1$ , (d) $B = 1$ . In Figure 13: (a) $\eta = 0 . 0 0 0 2 , B = 1 0 0$ , (b) $\eta = 0 . 0 0 1 , B = 1 0 0$ , (c) $\eta = 0 . 0 0 0 2 , B = 1 0 0$ , (d) $\eta = 0 . 0 0 0 1 , B = 1 0 0$ . In Figure 14: (a) $\eta = 0 . 0 0 0 2$ , $B = 1 0 0 , D = 0 . 0 0 0 2$ , (b) $\eta = 0 . 0 0 1 , B = 1 0 0 , D = 0 . 0 0 0 1$ , (c) $\eta = 0 . 0 0 0 2 , B =$ $1 0 0 , D = 0 . 0 0 0 5$ , (d) $\eta = 0 . 0 0 0 1 , B = 1 0 0 , D = 0 . 0 0 0 3$ . We note that the hyperparameters need be tuned for each initialized pretrained models, due to the stochastic property of deep learning. + +![](images/e611bd5ee1c1627c6eebbe465cd800ad1aa2a077166549a38354ad6ff51c8c74.jpg) +Figure 10: The escape rate exponentially depends on the “path Hessians” in the dynamics of SGD. $- \log ( \gamma )$ is linear with $\textstyle { \frac { 1 } { k } }$ . The “path Hessians” indicates the eigenvalues of Hessians corresponding to the escape directions. + +![](images/cc24938c36d0e50a1712f00debfae83e0ef961a9118b02da5c6b68dace8eb7b0.jpg) +Figure 11: The escape rate exponentially depends on the batch size in the dynamics of SGD. $- \log ( \gamma )$ is linear with $B$ . + +![](images/95d8cee65da16f84618d2949f3e4ec958b0cee119bbb90092d89784746e231aa.jpg) +Figure 12: The escape rate exponentially depends on the learning rate in the dynamics of SGD. $- \log ( \gamma )$ is linear with $\frac { 1 } { \eta }$ . The estimated escape rate has incorporated $\eta$ as the time unit. + +![](images/32b28be96d16487646694dc86dedef57fef4d0cb3660ae3df14106d0d9ee0727.jpg) +Figure 13: The relation of the escape rate and the isotropic diffusion coefficient D. The escape formula that − $\log ( \gamma )$ is linear with $\dot { \frac { 1 } { D } }$ is validated. + +According to our experience, we can always find the hyperparameters to discover the quantitative relations as long as the pretrained model fits the data set well enough. The fined-tuned requirement can be avoided in Section E, because the models in Section E are artificially initialized. + +Observation: we observe the number of iterations from the initialized position to the terminated position. We repeat experiments 100 times to estimate the escape rate $\gamma$ and the mean escape time $\tau$ . As the escape time is a random variable obeying an exponential distribution, $t \sim E x p o n e n t i a l ( \gamma )$ , the estimated escape rate can be written as + +$$ +\hat { \gamma } = \frac { 1 0 0 - 2 } { \sum _ { i = 1 } ^ { 1 0 0 } t _ { i } } . +$$ + +The $9 5 \%$ confidence interval of this estimator is + +$$ +\hat { \gamma } ( 1 - \frac { 1 . 9 6 } { \sqrt { 1 0 0 } } ) \leq \hat { \gamma } \leq \hat { \gamma } ( 1 + \frac { 1 . 9 6 } { \sqrt { 1 0 0 } } ) . +$$ + +# D.2 EXPERIMENTS ON SGLD + +Experimental Results: Figure 13 shows a highly precise exponential relation of the escape rate and the diffusion coefficient in the figure. Figure 14 shows a proportional relation of the escape rate and the Hessian determinant in the figure. Overall, the empirical results support the density diffusion theory in the dynamics of white noise. In experiments on SGLD, we carefully adjust the injected gradient noise scale in experiment to ensure that $D$ is significantly smaller than the loss barrier’ height and large enough to dominate SGN scale. If $D$ is too large, learning dynamics will be reduced to Free Brownian Motion. + +![](images/83ef704670a42a8d7638e9a3a2075a495a934d06c96343b2f88bb6398afe7b0c.jpg) +Figure 14: The relation of the escape rate and the Hessian determinant in the dynamics of white noise.The escape formula that $\gamma$ is linear with $k$ is validated. + +# E EXPERIMENTS ON MORE MODELS + +We supply experiments of training three models on artificial Gaussian datasets. In these experiments, we can analytically know the locations of the minima, Hessians and loss barriers, as each input feature is Gaussian noise. + +# E.1 EXPERIMENTS SETTINGS + +Data Set: We generate 50000 Gaussian samples and random two-class labels as the training data set, $\{ ( x ^ { ( i ) } , y ^ { ( i ) } ) | x ^ { ( \bar { i } ) } \sim \mathcal { N } ( 0 , I ) , y ^ { ( i ) } \in \{ 0 , 1 \} , i \stackrel { \cdot } { \in } \{ 1 , 2 , \cdot \cdot , 5 0 0 0 0 \} \}$ + +Hyperparameters: In Figure 15: (a) $\eta = 0 . 0 0 0 1 , B = 1 0 0$ , (b) $\eta = 0 . 0 0 1 , B = 1 0 0$ , (c) $\eta =$ 0.0003, $B = 1 0 0$ . In Figure 16: (a) $\eta = 0 . 0 0 0 1 , B = 5 0 , D = 0 . 2$ , (b) $\eta = 0 . 0 0 1 , B = 5 0 , D =$ 0.0005, (c) $\eta = 0 . 0 0 0 3 , B = 1 , D = 0 . 0 0 0 3$ . In Figure 17: (a) $\eta = 0 . 0 0 6 , B = 5 0$ , (b) $\eta =$ 0.05, $, B = 5 0$ , (c) $\eta = 0 . 0 0 5 , B = 1$ . In Figure 18: (a) $\eta = 0 . 0 0 6$ , (b) $\eta = 0 . 0 6$ , (c) $\eta = 0 . 1$ . In Figure 19: (a) $B = 1$ , (b) $B = 1$ , (c) $B = 1$ . We note that the hyperparameters are recommended and needn’t be fine tuned again. The artificially initialized parameters avoids the stochastic property of the initial states. + +Experiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex optimization, written as + +$$ +f ( \theta ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } ( \theta _ { i } ^ { 4 } - 1 6 \theta _ { i } ^ { 2 } + 5 \theta _ { i } ) . +$$ + +We use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as training data. + +$$ +L ( \theta ) = f ( \theta - x ) , +$$ + +where data samples $x \sim \mathcal { N } ( 0 , I )$ . The one-dimensional Styblinski-Tang Function has one global minimum located at $a = - 2 . 9 0 3 5 3 4$ , one local minimum located at $d$ , and one saddle point $b =$ 0.156731 as the boundary separating Valley $a _ { 1 }$ and Valley $a _ { 2 }$ . For a $\mathbf { n }$ -dimensional Styblinski-Tang Function, we initialize parameters $\theta _ { t = 0 } = \textstyle { \frac { 1 } { \sqrt { k } } } ( - 2 . 9 0 3 5 3 4 , \cdot \cdot \cdot , - 2 . 9 0 3 5 3 4 )$ , and set the valley’s boundary as $\begin{array} { r } { \theta _ { i } < \frac { 1 } { \sqrt { k } } 0 . 1 5 6 7 3 1 } \end{array}$ , where $i$ is the dimension index. We record the number of iterations required to escape from the valley to the outside of valley. The setting 1 does not need labels. + +Experiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initialization: $\theta _ { t = 0 } = ( 0 , \cdot \cdot \cdot , 0 )$ . Valley Boundary: $- 0 . 1 < \theta _ { i } < 0 . 1$ . Due to the randomness of training data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin valley group, and study the mean escape time from the group of valleys. + +Experiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization: $\theta _ { t = 0 } = ( 0 . 1 , \cdot \cdot \cdot , 0 . 1 )$ with a small Gaussian noise $\epsilon = ( 0 , 0 . 0 1 I )$ . Valley Boundary: $0 . 0 5 < \theta _ { i } < 0 . 1 5$ . To prevent the gradient disappearance problem of deep learning, we move the starting point from the origin. For symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value. Due to the complex loss landscape of deep networks, we can hardly know the exact information about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether or not the propositions can hold with very-small-batch gradient noise in practice. + +# E.2 EXPERIMENTS RESULTS + +Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. Figure 16 shows the relation of the escape rate and the Hessian determinant in the dynamics of white noise. Figure 17 shows the relation of the escape rate and the second order directional derivative in the dynamics of SGD. Figure 18 shows the relation of the escape rate and the batch size in the dynamics of SGD. Figure 19 shows the relation of the escape rate and the learning rate in the dynamics of SGD. + +![](images/a3af91f70dd76a14cfbf68c4af33655d6324c430f4d0a0f672fcb25cb4546d66.jpg) +Figure 15: The relation of the escape rate and the diffusion coefficient D in the dynamics of SGLD. The escape formula that $- \log ( \gamma )$ is linear with $\textstyle { \frac { 1 } { D } }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. + +![](images/9bdfc4ba7eacbef971e2839103509719b827e0264c02b872459d91774f4fc065.jpg) +Figure 16: The relation of the escape rate and the Hessian determinants in the dynamics of SGLD. The escape formula that $\gamma$ is linear with $k$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. + +![](images/72037af24de8ea4733e366a2c5e96d4046e24eab734ab0312a61c39283d69b88.jpg) +Figure 17: The escape rate exponentially depends on the sharpness in the dynamics of SGD. The escape formula that $- \log ( \gamma )$ is linear with $\frac { 1 } { k }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. + +![](images/6ee2fb941d89c488a872f7dd33c40187fc2f3f42d856655122e3c48cc20c441a.jpg) +Figure 18: The escape rate exponentially depends on the batch size in the dynamics of SGD. The escape formula that $\bar { - } \log ( \gamma )$ is linear with $B$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. + +![](images/f24a88bb49321f7d8fee99556f21f5bdc68661b8c4d02b911e13f9652bb6df38.jpg) +Figure 19: The escape rate exponentially depends on the learning rate in the dynamics of SGD. The escape formula that $- \log ( \gamma )$ is linear with $\frac { \mathbf { i } } { \eta }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. \ No newline at end of file diff --git a/parse/train/wXgk_iCiYGo/wXgk_iCiYGo_content_list.json b/parse/train/wXgk_iCiYGo/wXgk_iCiYGo_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..54b3ad8d0f35206a9ae3ae548121d8dc926e2571 --- /dev/null +++ b/parse/train/wXgk_iCiYGo/wXgk_iCiYGo_content_list.json @@ -0,0 +1,3158 @@ +[ + { + "type": "text", + "text": "A DIFFUSION THEORY FOR DEEP LEARNING DYNAMICS: STOCHASTIC GRADIENT DESCENT EXPONENTIALLY FAVORS FLAT MINIMA ", + "text_level": 1, + "bbox": [ + 174, + 98, + 828, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zeke Xie1,2, Issei Sato 1,2, and Masashi Sugiyama2,1 ", + "bbox": [ + 326, + 199, + 668, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1The University of Tokyo 2RIKEN Center for AIP xie@ms.k.u-tokyo.ac.jp {sato,sugi}@k.u-tokyo.ac.jp ", + "bbox": [ + 408, + 227, + 593, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 325, + 544, + 340 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks in practice. SGD is known to find a flat minimum that often generalizes well. However, it is mathematically unclear how deep learning can select a flat minimum among so many minima. To answer the question quantitatively, we develop a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. To the best of our knowledge, we are the first to theoretically and empirically prove that, benefited from the Hessian-dependent covariance of stochastic gradient noise, SGD favors flat minima exponentially more than sharp minima, while Gradient Descent (GD) with injected white noise favors flat minima only polynomially more than sharp minima. We also reveal that either a small learning rate or large-batch training requires exponentially many iterations to escape from minima in terms of the ratio of the batch size and learning rate. Thus, large-batch training cannot search flat minima efficiently in a realistic computational time. ", + "bbox": [ + 233, + 358, + 766, + 551 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 579, + 336, + 594 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various application areas. Due to the over-parametrization and the highly complex loss landscape of deep networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995; 1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al., 2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al., 2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al., 2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b), and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a). However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat minimum. ", + "bbox": [ + 173, + 611, + 826, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The diffusion theory is an important theoretical tool to understand how deep learning dynamics works. It helps us model the diffusion process of probability densities of parameters instead of model parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics (SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrz˛ebski et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented a Beyesian perspective on generalization of SGD. Wu et al. (2018) studied the escape problems of ", + "bbox": [ + 173, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "SGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained a related proposition that describes the mean escape time in terms of a free energy that depends on the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise. ", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, the related papers mainly analyzed the diffusion process under parameter-independent and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four contributions: ", + "bbox": [ + 174, + 208, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• The proposed theory formulates the fundamental roles of gradient noise, batch size, the learning rate, and the Hessian in minima selection. \nThe SGN covariance is approximately proportional to the Hessian and inverse to batch size. Either a small learning rate or large-batch training requires exponentially many iterations to escape minima in terms of ratio of batch size and learning rate. \n• To the best of our knowledge, we are the first to theoretically and empirically reveal that SGD favors flat minima exponentially more than sharp minima. ", + "bbox": [ + 215, + 318, + 825, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 449, + 655, + 467 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We denote the data samples as $\\{ x _ { j } \\} _ { j = 1 } ^ { m }$ , the model parameters as $\\theta$ and the loss function over data samples $x$ as $L ( \\theta , x )$ . For simplicity, we denote the training loss as $L ( \\theta )$ . Following Mandt et al. (2017), we may write SGD dynamics as ", + "bbox": [ + 174, + 481, + 825, + 540 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/6ef5022f684f592cdaef889ee88891f75529c47375e329ab2cb8554faec9688b.jpg", + "text": "$$\n\\theta _ { t + 1 } = \\theta _ { t } - \\eta \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } = \\theta _ { t } - \\eta \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } + \\eta C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 545, + 676, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\hat { L } ( \\theta )$ is the loss of one minibatch, $\\zeta _ { t } \\sim \\mathcal { N } ( 0 , I )$ , and $C ( \\theta )$ represents the gradient noise covariance matrix. The classic approach is to model SGN by Gaussian noise, ${ \\mathcal { N } } ( 0 , { \\overline { { C } } } ( \\theta ) )$ (Mandt et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018). ", + "bbox": [ + 178, + 589, + 825, + 632 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Stochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch training, $\\begin{array} { r } { C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } = \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } - \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } } \\end{array}$ , which is the difference between gradient descent and stochastic gradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the mean of many infinite-variance random variables converges to a stable distribution, while the mean of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in practice, we believe the Gaussian approximation of SGN is reasonable. ", + "bbox": [ + 173, + 638, + 825, + 729 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Simsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise. They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed distribution looks closer to a stable distribution than a Gaussian distribution. However, this research line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed “SGN” across $n$ model parameters and regarded “SGN\" as $n$ samples drawn from a single-variant distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic Gaussian noise. In our paper, SGN computed over different minibatches obeys a $n$ -variant Gaussian distribution, which can be parameter-dependent and anisotropic. ", + "bbox": [ + 173, + 734, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that the contradicted observations are from the different formulations of gradient noise. Simsekli et al. (2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a $n$ -variant distribution. Our empirical analysis in Figure 1 holds well at least when the batch size $B$ is larger than 16, which is common in practice. Similar empirical evidence can be observed for training ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C. ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/9f25e21da6eaf1f4dcc5fb4e55195174a32550f9f088693c21380f6b59439613.jpg", + "image_caption": [ + "Figure 1: The Stochastic Gradient Noise Analysis. The histogram of the norm of the gradient noises computed with the three-layer fully-connected network on MNIST (LeCun, 1998). (a) and (c): the histograms of the norms of two kinds of gradient noise: (a) “SGN” is computed over parameters, which is actually stochastic gradient rather than SGN; (c) SGN is computed over minibatches. (b) and (d): the histograms of the norms of (scaled) Gaussian noise and Lévy noise. Based on (a) and (b), Simsekli et al. (2019) argued that gradient noise across parameters is heavy-tailed Lévy noise. Based on (c) and (d), we show that SGN without the isotropic restriction is approximately Gaussian. " + ], + "image_footnote": [], + "bbox": [ + 179, + 116, + 808, + 234 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 383, + 825, + 455 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Panigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN is best described as Gaussian at-least in the early phases of training. Comparing our results with Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors. First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network matters. Because SGN on some layers/networks is more Gaussian than other layers/networks. ", + "bbox": [ + 173, + 462, + 825, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameterdependent and anisotropic SGN brings a large improvement over existing parameter-independent and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under parameter-independent anisotropic heavy-tailed noise, when the batch size is too small $( B \\sim 1 )$ to apply Central Limit Theorem. We will leave it as future work. ", + "bbox": [ + 174, + 537, + 825, + 637 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "SGD Dynamics. Let us replace $\\eta$ by $d t$ as unit time. Then the continuous-time dynamics of SGD (Coffey & Kalmykov, 2012) is written as ", + "bbox": [ + 173, + 642, + 823, + 671 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9941a935271e458ce4253fd77fc922fa780d0d1282c06cc73e97eabe4d867726.jpg", + "text": "$$\nd \\theta = - \\frac { \\partial L ( \\theta ) } { \\partial \\theta } d t + [ 2 D ( \\theta ) ] ^ { \\frac { 1 } { 2 } } d W _ { t } ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 690, + 614, + 723 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $d W _ { t } \\sim { \\mathcal { N } } ( 0 , I d t )$ and $\\begin{array} { r } { D ( \\theta ) = \\frac { \\eta } { 2 } C ( \\theta ) } \\end{array}$ . We note that the dynamical time $t$ in the continuoustime dynamics is equal to the product of the number of iterations $T$ and the learning rate $\\eta$ : $t = \\eta T$ . The associated Fokker-Planck Equation is written as ", + "bbox": [ + 174, + 741, + 826, + 785 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d5b05d54a2d68aac5885131414903f159c198a9efbf3dadf579d435dcb96f521.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = \\nabla \\cdot [ P ( \\theta , t ) \\nabla L ( \\theta ) ] + \\nabla \\cdot \\nabla D ( \\theta ) P ( \\theta , t ) } } \\\\ & { } & { = \\sum _ { i } \\frac { \\partial } { \\partial \\theta _ { i } } [ P ( \\theta , t ) \\frac { \\partial L ( \\theta ) } { \\partial \\theta _ { i } } ] + \\sum _ { i } \\sum _ { j } \\frac { \\partial ^ { 2 } } { \\partial \\theta _ { i } \\partial \\theta _ { j } } D _ { i j } ( \\theta ) P ( \\theta , t ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 804, + 732, + 877 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\nabla$ is a nabla operator, and $D _ { i j }$ is the element in the ith row and $j$ th column of $D$ . In standard SGLD, the injected gradient noise is fixed and isotropic Gaussian, $D = I$ . ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/084198cb7082c38ce670ef6ab05aa9b80fc9eb6c2d5851d7717d8d63c4a5ddfd.jpg", + "image_caption": [ + "Figure 2: We empirically verified MNIST (LeCun, 1998). The pret $\\begin{array} { r } { C ( \\theta ) = \\frac { H ( \\theta ) } { B } } \\end{array}$ by using three-layer fully-connected network ons are usually near critical points, while randomly Initialized Models are far from critical points. We display all elements $H _ { ( i , j ) } \\in [ 1 e - 4 , 0 . 5 ]$ of the Hessian matrix and the corresponding elements $C _ { ( i , j ) }$ of gradient noise covariance matrix in the space spanned by the eigenvectors of Hessian. Another supplementary experiment on Avila Dataset (De Stefano et al., 2018) in Appendix C reports $\\hat { C } _ { a v i l a } \\approx 1 . 0 0 4 \\frac { H } { B }$ . The small difference factor between the empirical result and the ideal Equation is mainly because the pretrained network is not perfectly located at a critical point. " + ], + "image_footnote": [], + "bbox": [ + 184, + 116, + 807, + 223 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The next question is how to formulate the SGN covariance $C ( \\theta )$ for SGD? Based on Smith & Le (2018), we can express the SGN covariance as ", + "bbox": [ + 174, + 380, + 823, + 410 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b9d79fb860e473b985959cf631bc674c905731483c5fe79f22f112928253e4b8.jpg", + "text": "$$\n\\boldsymbol { \\Sigma } ( \\theta ) = \\frac { 1 } { B } \\left[ \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } - \\boldsymbol { \\nabla } L ( \\theta ) \\boldsymbol { \\nabla } L ( \\theta ) ^ { \\top } \\right] \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 415, + 828, + 465 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The approximation is true near critical points, due to the fact that the gradient noise variance dominates the gradient mean near critical points. We know the observed fisher information matrix satisfies $\\operatorname { F I M } ( \\theta ) \\approx H ( \\theta )$ near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrz˛ebski et al. (2017); Zhu et al. (2019), we obtain ", + "bbox": [ + 174, + 486, + 825, + 541 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/484eebf70737b8e39684e093ac75f159d42ec466e794097965f95b4e72b7c815.jpg", + "text": "$$\nC ( \\theta ) \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\nabla L ( \\theta , x _ { j } ) \\nabla L ( \\theta , x _ { j } ) ^ { \\top } = \\frac { 1 } { B } \\mathrm { F I M } ( \\theta ) \\approx \\frac { 1 } { B } H ( \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 279, + 546, + 717, + 590 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which approximately gives ", + "bbox": [ + 174, + 597, + 352, + 611 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bca581a28a061cac16042fb3570f808fa0d6498893fcb26a8494b5ab75188d0a.jpg", + "text": "$$\nD ( \\theta ) = \\frac { \\eta } { 2 } C ( \\theta ) = \\frac { \\eta } { 2 B } H ( \\theta )\n$$", + "text_format": "latex", + "bbox": [ + 403, + 616, + 594, + 645 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "near minima. It indicates that the SGN covariance $C ( \\theta )$ is approximately proportional to the Hessian $H ( \\theta )$ and inverse to the batch size $B$ . Obviously, we can generalize Equation 7 by $\\begin{array} { r } { D ( \\theta ) ~ = ~ \\frac { \\eta C ( \\theta ) } { 2 } ~ = ~ \\frac { \\eta } { 2 B } [ H ( \\theta ) ] ^ { + } } \\end{array}$ near critical points, when there exist negative eigenvalues in $H$ along some directions. We use $[ \\cdot ] ^ { + }$ to denote the positive semidefinite transformation of a symmetric matrix: if we have the eigendecomposation $H = U \\mathrm { d i a g } ( H _ { 1 } , \\cdot \\cdot \\cdot , H _ { n - 1 } , H _ { n } ) U ^ { \\top }$ , then $[ H ] ^ { + } = U \\mathrm { d i a g } ( | H _ { 1 } | , \\cdots , | H _ { n - 1 } | , | H _ { n } | ) U ^ { \\dagger }$ . ", + "bbox": [ + 173, + 648, + 825, + 741 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a followup paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that, the relation still approximately holds for even the randomly network, which is far from critical points. The correlation is especially high along the flat directions with small-magnitude eigenvalues of the Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian noise approximation all failed to capture this core relation in deep learning dynamics. ", + "bbox": [ + 173, + 746, + 826, + 844 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 SGD DIFFUSION THEORY ", + "text_level": 1, + "bbox": [ + 176, + 863, + 419, + 880 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We assume there are two valleys, Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ , seen in Figure 3. Also Col b is the boundary between two valleys. What is the mean escape time for a particle governed by Equation 2 to escape from Sharp Valley $a _ { 1 }$ to Flat Valley $a _ { 2 }$ ? The mean escape time is widely used in related statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019). ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/73689560624558dfdf080463c0fd2eec0e7953c645f73e038906c281f99c7fba.jpg", + "image_caption": [ + "Figure 3: Kramers Escape Problem. $a _ { 1 }$ and $ { \\boldsymbol { a } } _ { a }$ are minima of two neighboring valleys. $b$ is the saddle point separating the two valleys. $c$ locates outside of Valley $a _ { 1 }$ . " + ], + "image_footnote": [], + "bbox": [ + 307, + 121, + 689, + 255 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 324, + 825, + 367 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gauss’s Divergence Theorem (Arfken & Weber, 1999; Lipschutz et al., 2009) states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. We respectively denote the mean escape time as $\\tau$ , the escape rate as $\\gamma$ , and the probability current as $J$ . We apply Gauss’s Divergence Theorem to the Fokker-Planck Equation resulting in ", + "bbox": [ + 174, + 373, + 825, + 444 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/81ae893b80fe266236bb47f161db168952995549551e12a490ee4ee02a12db9d.jpg", + "text": "$$\n\\nabla \\cdot \\left[ P ( \\theta , t ) \\nabla L ( \\theta ) \\right] + \\nabla \\cdot \\nabla D ( \\theta ) P ( \\theta , t ) = \\frac { \\partial P ( \\theta , t ) } { \\partial t } = - \\nabla \\cdot J ( \\theta , t ) .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 450, + 730, + 482 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The mean escape time is expressed (Van Kampen, 1992) as ", + "bbox": [ + 173, + 487, + 562, + 502 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ad70caef78bf136711800b537c605309c9d02bbcc9f890c7ef0780c521a61cb9.jpg", + "text": "$$\n\\tau = { \\frac { 1 } { \\gamma } } = { \\frac { P ( \\theta \\in V _ { a } ) } { \\int _ { S _ { a } } J \\cdot d S } } ,\n$$", + "text_format": "latex", + "bbox": [ + 424, + 508, + 573, + 546 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { P ( \\theta \\in V _ { a } ) = \\int _ { V _ { a } } P ( \\theta ) d V } \\end{array}$ is the current probability inside Valley a, $J$ is the probability current produced by the probability source $P ( \\theta \\in V _ { a } )$ , $\\begin{array} { r } { j = \\int _ { S _ { a } } J \\cdot d S } \\end{array}$ is the probability flux (surface integrals of probability current), $S _ { a }$ is the surface (boundary) surrounding Valley a, and $V _ { a }$ is the volume surrounded by $S _ { a }$ . We have $j = J$ in the case of one-dimensional escape. ", + "bbox": [ + 174, + 553, + 825, + 616 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Classical Assumptions. We state three classical assumptions first for the density diffusion theory. Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring, 1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012), and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers (Jastrz˛ebski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape Problems. ", + "bbox": [ + 173, + 621, + 826, + 733 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 1 (The Second Order Taylor Approximation). The loss function around critical points $\\theta ^ { \\star }$ can be approximately written as ", + "bbox": [ + 174, + 737, + 821, + 766 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/51182f3a03e686be47ce14ef11732d6ba605ccb76a1cc4c2d7b85c9425862cd8.jpg", + "text": "$$\nL ( \\theta ) = L ( \\theta ^ { \\star } ) + g ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) + \\frac { 1 } { 2 } ( \\theta - \\theta ^ { \\star } ) ^ { \\top } H ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 772, + 704, + 803 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima. \nAssumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature). ", + "bbox": [ + 173, + 808, + 825, + 843 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We will dive into these two assumptions deeper than previous papers for SGD dynamics. Assumptions 2 and 3 both mean that our diffusion theory can better describe the escape processes that cost more iterations. As this class of “slow” escape processes takes main computational time compared with “fast” escape processes, this class of “slow” escape process is more interesting for training of deep neural networks. Our empirical analysis in Section 4 supports that the escape processes in the wide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus, Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Escape paths. We generalize the concept of critical points into critical paths as the path where 1) the gradient perpendicular to the path direction must be zero, and 2) the second order directional derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs) for escaping must be critical paths. The most possible escape direction at one point must be the direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs, which are strictly flat, is ignorable according to our following analysis. ", + "bbox": [ + 173, + 138, + 825, + 279 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The boundary between Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ is the saddle point $b$ . The Hessian at $b$ , $H _ { b }$ , must have only one negative eigenvalue and the corresponding eigenvector is the escape direction. Without losing generality, we first assume that there is only one most possible path through $\\operatorname { C o l } b$ existing between Sharp Valley $a _ { 1 }$ and Flat Valley $a _ { 2 }$ . ", + "bbox": [ + 174, + 285, + 826, + 342 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "SGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers are interested in SGLD, when the injected noise dominates SGN as $\\eta 0$ in final epochs. Because SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is usually simplified as Gradient Descent with injected white noise, whose behavior is identical to Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a mixture of injected white noise and SGN. ", + "bbox": [ + 173, + 347, + 825, + 445 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.1 (SGLD Escapes Minima). The loss function $L ( \\theta )$ is of class $C ^ { 2 }$ and $n$ -dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the outside of Valley a is ", + "bbox": [ + 173, + 448, + 825, + 505 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/425a3e09bd66d9bf92cb9a14d67ccbb4cc6bbbd38570a307c9f37b27183c2f35.jpg", + "text": "$$\n\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\sqrt { \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\frac { \\Delta L } { D } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 503, + 651, + 545 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We denote that $H _ { a }$ and $H _ { b }$ are the Hessians of the loss function at the minimum a and the saddle point $b$ , $\\Delta L = L ( b ) - L ( a )$ is the loss barrier height, e indicates the escape direction, and $H _ { b e }$ is the eigenvalue of the Hessian $H _ { b }$ corresponding to the escape direction. The diffusion coefficient $D$ is usually set to 1 in SGLD. ", + "bbox": [ + 173, + 546, + 825, + 603 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "SGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD is far from the Gibs-Boltzmann distribution, $\\begin{array} { r } { P ( \\theta ) = \\frac { 1 } { Z } \\exp \\left( - \\frac { L ( \\theta ) } { D } \\right) } \\end{array}$ . These different characteristics make SGD diffusion behave differently from known physical dynamical systems and much less studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN and injected white noise, as long as the eigenvectors of $D ( \\theta )$ are closely aligned with the eigenvectors of $H ( \\theta )$ . ", + "bbox": [ + 173, + 613, + 825, + 736 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.2 (SGD Escapes Minima). The loss function $L ( \\theta )$ is of class $C ^ { 2 }$ and $n$ -dimensional. Only one most possible path exists between Valley a and the outside of Valley $^ { a }$ . If Assumption $I$ , 2, and 3 hold, and the dynamics is governed by $S G D$ , then the mean escape time from Valley a to the outside of Valley a is ", + "bbox": [ + 174, + 738, + 826, + 795 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ec8a128a9aa153b1c19ff5fa460949aff35a421d555e99c40ce59e45380110bc.jpg", + "text": "$$\n\\tau = 2 \\pi \\frac { 1 } { | H _ { b e } | } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } \\left( \\frac { s } { H _ { a e } } + \\frac { ( 1 - s ) } { | H _ { b e } | } \\right) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 801, + 660, + 835 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $s \\in ( 0 , 1 )$ is a path-dependent parameter, and $H _ { a e }$ and $H _ { b e }$ are, respectively, the eigenvalues of the Hessians at the minimum a and the saddle point $b$ corresponding to the escape direction e. ", + "bbox": [ + 174, + 842, + 821, + 871 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Multiple-path escape. Each escape path contributes to the total escape rate. Multiple paths combined together have a total escape rate. If there are multiple parallel from the start valley to the end valley, we can compute the total escape rate easily based on the following computation rule. The computation rule is based on the fact that the probability flux integrals are additive. We can easily generalize the mean escape time analysis into the cases that there are multiple parallel escape paths indexed by $p$ As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the dimensionality directly, higher dimensionality may increase the number of escape paths and loss valleys, and change the spectrum of the Hessians. ", + "bbox": [ + 176, + 881, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/40850afd82cda33fe28c5aa92bf1019fad755c726e83948318f7a64db61eaf3d.jpg", + "image_caption": [ + "Figure 4: The mean escape time analysis of SGD by using Styblinski-Tang Function. The Pearson Correlation is higher than 0.99. Left Column: Sharpness. Middle Column: Batch Size. Right Column: Learning Rate. " + ], + "image_footnote": [], + "bbox": [ + 197, + 116, + 797, + 257 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 340, + 825, + 425 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Rule 1. If there are multiple MPPs between the start valley and the end valley, then $\\begin{array} { r } { \\gamma _ { t o t a l } = \\sum _ { p } \\gamma _ { p } } \\end{array}$ ", + "bbox": [ + 171, + 429, + 823, + 445 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Thus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and analyze the escape rates. ", + "bbox": [ + 176, + 457, + 820, + 486 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Minima selection. Now, we may formulate the probability of minima selection as Proposition 1. We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and the landscape contain many good modes and bad modes. SGD transits from one mode to another mode during training. The mean escape time of one mode corresponds to the number of iterations which SGD spends on this mode during training, which is naturally proportional to the probability of selecting this mode after training. ", + "bbox": [ + 173, + 492, + 826, + 577 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by ", + "bbox": [ + 174, + 582, + 825, + 609 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ed10cf147b3b58efa64c529c4d2fec7e55a1a844d052c80b1fed9db1d4901ac2.jpg", + "text": "$$\nP ( \\theta \\in V _ { a } ) = \\frac { \\tau _ { a } } { \\sum _ { v } \\tau _ { v } } ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 616, + 571, + 647 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where v is the index of valleys, and $\\tau _ { v }$ is the mean escape time from Valley v to the outside of Valley $v$ ", + "bbox": [ + 173, + 654, + 823, + 667 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EMPIRICAL ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 688, + 392, + 704 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we try to directly validate the escape formulas on real-world datasets. Each escape process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100 times under various gradient noise scales, batch sizes, learning rates, and sharpness. ", + "bbox": [ + 174, + 719, + 825, + 762 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How to compare the escape rates under the same settings with various minima sharpness? Our√ method is to multiply a rescaling factor $\\sqrt { k }$ to each parameter, and the Hessian will be proportionally√ rescaled by a factor $k$ . If we let $L ( \\theta ) = f ( \\theta ) L ( \\theta ) = f ( { \\sqrt { k } } \\theta )$ , then $H ( \\theta ) = \\nabla ^ { 2 } f ( \\theta ) \\to H ( \\theta ) =$ $k \\nabla ^ { 2 } f ( \\theta )$ . Thus, we can use $k$ to indicate the minima sharpness. The theoretical relations of SGD we try to validate can be formulated as: $( 1 ) - \\log ( \\gamma ) = \\bar { \\mathcal { O } } ( \\textstyle { \\frac { 1 } { k } } )$ , $2 ) - \\log ( \\gamma ) = \\mathcal { O } ( B )$ , and (3) $- \\log ( \\gamma ) = \\mathcal { O } ( \\textstyle { \\frac { 1 } { \\eta } } )$ . ", + "bbox": [ + 173, + 768, + 825, + 862 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The mean escape time analysis of SGD. Styblinski-Tang Function, which has multiple minima and saddle points, is a common test function for nonconvex optimization. We conduct an intuitional 10-dimensional experiment, where the simulations start from a given minimum and terminate when reaching the boundary of the loss valley. The number of iterations is recorded for calculating the escape rate. We also train fully connected networks on four real-world datasets, including a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis (De Stefano et al., 2018; Dua & Graff, 2017). Figure 4 and Figure 5 clearly verifies that the escape rate exponentially depends on the minima sharpness (reflected by $k$ ), the batch size, and the learning rate on both test functions and real-world training, which fully supports our theoretical results. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/8bd372226d5c3c86bb1d2b2130b67041a3b2fb52d4924db07d0de0674e8493ba.jpg", + "image_caption": [ + "Figure 5: The mean escape time analysis of SGD by training neural networks on Avila Dataset. Left Column: Sharpness. Middle Column:Batch Size. Right Column: Learning Rate. We leave the results on Banknote Authentication, Cardiotocography, and Sensorless Drive Diagnosis in Appendix D. " + ], + "image_footnote": [], + "bbox": [ + 196, + 119, + 797, + 257 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1f2c25c1998116a59e897d7ca8015655aad9a6ac9d7ea655f81156edacd74762.jpg", + "image_caption": [ + "Figure 6: The mean escape time analysis of SGLD. Subfigure (a) and (b): Styblinski-Tang Function. Subfigure (c) and (d): Neural Network. " + ], + "image_footnote": [], + "bbox": [ + 289, + 349, + 696, + 655 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 728, + 825, + 799 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Model architecture and details: We used fully-connected networks with the depth 2 and the width 10 in Figure 5. The experiments using Logistic Regression and Fully-connected networks with the depth 3 are presented in Appendix E. We leave more experimental details and results in Appendix D.1 and Appendix E. ", + "bbox": [ + 174, + 804, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The mean escape time analysis of SGLD. We try to validate $\\gamma = \\mathcal { O } ( k )$ and $- \\log ( \\gamma ) = \\mathcal { O } ( \\frac { 1 } { D } ) .$ for SGLD (dominated by injected Gaussian noise). Figure 6 shows that SGLD only favors flat minima polynomially more than sharp minima as Theorem 3.1 indicates. Figure 6 also verifies that the injected gradient noise scale exponentially affects flat minima selection. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 310, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "SGD favors flat minima exponentially more than sharp minima. We can discover a few interesting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially depends on the eigenvalue of the Hessian at minima along the escape direction, $H _ { a e }$ . Thus, SGD favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD comes from the exponential relation of the mean escape time and the minima sharpness. The measure of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness” is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly more than Adam. ", + "bbox": [ + 174, + 133, + 825, + 272 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains why large-batch training can easily get trapped near sharp minima, and increasing the learning rate proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause is large-batch training expects exponentially longer time to escape minima. Note that, as the mean escape time in the theorems is equivalent to the product of the learning rate and the number of iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the batch size and the learning rate. The practical computational time in large-batch training is usually too short to search many enough flat minima. We conjecture that exponentially increasing training iterations may be helpful for large batch training, while this is often too expensive in practice. ", + "bbox": [ + 174, + 280, + 825, + 419 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Low dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of overparametrized deep networks are close to zero, while only a small number of eigenvalues are large (Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018). ", + "bbox": [ + 174, + 426, + 825, + 496 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As the escape rate exponentially depends the corresponding eigenvalues, a small number of large eigenvalues means that the process of minima selection mainly happens in the relatively low dimensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics hardly depends on those “meaningless” dimensions with small second order directional derivatives. This novel characteristic of SGD significantly reduces the explorable parameter space around one minimum into a much lower dimensional space. ", + "bbox": [ + 174, + 502, + 825, + 601 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "High-order effects. As we have applied the second-order Taylor approximation near critical points, our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this paper. However, we also argue that the third-order effect is much smaller than the second-order effect under the low temperature assumption in Kramers Escape Problems. We will leave the more refined high-order theory as future work. ", + "bbox": [ + 174, + 607, + 825, + 690 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 712, + 318, + 728 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to formulate the exponential relation of minima selection to the minima sharpness, the batch size, and the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory not only helps us understand how SGD selects flat minima, but also will provide researchers a powerful theoretical tool to analyze more learning behaviors and design better optimizers in future. ", + "bbox": [ + 174, + 744, + 825, + 842 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 864, + 356, + 878 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Alessandro Achille and Stefano Soatto. Where is the information in a deep neural network? arXiv preprint arXiv:1905.12213, 2019. ", + "bbox": [ + 176, + 126, + 823, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "George B Arfken and Hans J Weber. Mathematical methods for physicists, 1999. 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", + "bbox": [ + 174, + 247, + 826, + 290 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A PROOFS ", + "text_level": 1, + "bbox": [ + 176, + 318, + 277, + 334 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 PROOF OF THEOREM 3.1 ", + "text_level": 1, + "bbox": [ + 176, + 349, + 388, + 364 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. This proposition is a well known conclusion in statistical physics under Assumption 1, 2 and 3. We still provide an intuitional proof here, and the following proof of SGD Diffusion will closely relate to this proof. We decompose the proof into two steps: 1) compute the probability of locating in valley a, $P ( \\theta \\in V _ { a } )$ , and 2) compute the probability flux $\\begin{array} { r } { j = \\int _ { S _ { a } } \\bar { \\boldsymbol { J } } \\cdot d \\boldsymbol { S } } \\end{array}$ . ", + "bbox": [ + 173, + 376, + 825, + 434 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Without losing generality, we first prove the one-dimensional case. ", + "bbox": [ + 174, + 440, + 609, + 457 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Step 1: Under Assumption 1, the stationary distribution around minimum a is $\\begin{array} { r l } { P ( \\theta ) } & { { } = } \\end{array}$ $\\begin{array} { r } { P ( a ) \\exp [ - \\frac { L ( \\theta ) - L ( a ) } { T } ] } \\end{array}$ , where $T = D$ . Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. We use the $T$ notation as the temperature parameter in the stationary distribution, and use the $D$ notation as the diffusion coefficient in the dynamics, for their different roles. ", + "bbox": [ + 174, + 462, + 825, + 536 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/bd028780dd2307306b0ef1c3cf5571d3f87a0980dc06ebcc80761551d74e5334.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( \\theta \\in V _ { a } ) } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( \\theta ) d V } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( a ) \\exp \\left[ - \\frac { L ( \\theta ) - L ( a ) } { T } \\right] d \\theta } \\\\ & { = P ( a ) \\displaystyle \\int _ { \\theta \\in V _ { a } } \\exp \\left[ - \\frac { \\frac { 1 } { 2 } ( \\theta - a ) ^ { \\top } H _ { a } ( \\theta - a ) + \\mathcal { O } ( \\Delta \\theta ^ { 3 } ) } { T } \\right] d \\theta } \\\\ & { = P ( a ) \\displaystyle \\frac { ( 2 \\pi T ) ^ { \\frac { 1 } { 2 } } } { H ^ { \\frac { 1 } { 2 } } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 541, + 696, + 717 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Step 2: ", + "bbox": [ + 173, + 729, + 222, + 744 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0666c45f1734b833cfd7c2b0ccea7592163e6840ea2cd939c582fa149686af4a.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle J = P ( \\theta ) \\nabla L ( \\theta ) + P ( \\theta ) \\nabla D + D \\nabla P ( \\theta ) } \\\\ { \\displaystyle J = P ( \\theta ) \\left( \\nabla L ( \\theta ) + \\nabla D - \\displaystyle \\frac { D } { T } \\nabla L ( \\theta ) \\right) } \\\\ { \\displaystyle \\nabla D = \\left( \\displaystyle \\frac { D } { T } - 1 \\right) \\nabla L } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 748, + 643, + 842 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Apply this result to the Fokker-Planck Equation 4, we have ", + "bbox": [ + 173, + 847, + 562, + 862 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0117dfdceda6575276c290cb3fb82e11a4ca28f1721fca27fecf9f137cd52096.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla \\cdot \\nabla [ D ( \\theta ) P ( \\theta , t ) ] } \\\\ & { = \\nabla \\cdot D \\nabla P ( \\theta , t ) + \\nabla \\cdot \\left[ \\left( \\displaystyle \\frac { D } { T } - 1 \\right) \\nabla L ( \\theta ) \\right] P ( \\theta , t ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 867, + 673, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "And thus we obtain the Smoluchowski equation and a new form of J ", + "bbox": [ + 173, + 103, + 620, + 119 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9f6572b78d91ef3c9b3382276fe224c086aa3102c807136ee0e359e8c0ef0c46.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\displaystyle { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = \\nabla \\cdot \\left[ D \\left( \\frac { 1 } { T } \\nabla L ( \\theta ) + \\nabla \\right) P ( \\theta , t ) \\right] = - \\nabla \\cdot J ( \\theta , t ) , } } \\\\ & { } & { \\displaystyle { J ( \\theta ) = D \\exp \\left( \\frac { - L ( \\theta ) } { T } \\right) \\nabla \\left[ \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) P ( \\theta ) \\right] . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 128, + 710, + 203 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We note that the probability density outside Valley a must be zero, $P ( c ) = 0$ . As we want to compute the probability flux escaping from Valley a in the proof, the probability flux escaping from other valleys into Valley a should be ignored. Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path ", + "bbox": [ + 176, + 210, + 823, + 267 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9450417ea177aa9b4c0564a50de297f6bf9620ad9d23812bcca6e31f9caff837.jpg", + "text": "$$\n\\begin{array} { r } { \\displaystyle \\int _ { a } ^ { c } \\frac { \\partial } { \\partial \\theta } \\left[ \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) P ( \\theta ) \\right] d \\theta = \\int _ { a } ^ { c } - \\frac { J } { D } \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) d \\theta } \\\\ { \\displaystyle \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) P ( \\theta ) | _ { a } ^ { c } = - \\frac { J } { D } \\int _ { a } ^ { c } \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) d \\theta } \\\\ { \\displaystyle 0 - \\exp \\left( \\frac { L ( a ) } { T } \\right) P ( a ) = - \\frac { J } { D } \\int _ { a } ^ { c } \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) d \\theta } \\\\ { \\displaystyle J = \\frac { D \\exp \\left( \\frac { L ( a ) } { T } \\right) P ( a ) } { \\int _ { a } ^ { c } \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) d \\theta } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 279, + 700, + 440 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We move $J$ to the outside of integral based on Gauss’s Divergence Theorem, because $J$ is fixed on the escape path from one minimum to another. As there is no field source on the escape path, $\\begin{array} { r } { \\int _ { V } \\nabla \\cdot \\boldsymbol { J } ( \\boldsymbol { \\theta } ) d \\boldsymbol { \\dot { V } } = \\boldsymbol { 0 } } \\end{array}$ . Then $\\nabla J ( \\theta ) = 0$ . Obviously, only minima are probability sources in deep learning. Under Assumption 3 and the second-order Taylor approximation, we have ", + "bbox": [ + 174, + 448, + 825, + 506 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/460447d994af14908440a697f4e648d84feea7cea62b0f1145507908139dad58.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { a } ^ { c } \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) d \\theta } \\\\ & { = \\displaystyle \\int _ { a } ^ { c } \\exp \\left[ \\frac { L ( b ) + \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ( \\theta - b ) + \\mathcal { O } ( \\Delta \\theta ^ { 3 } ) } { T } \\right] d \\theta } \\\\ & { \\approx \\exp \\left( \\frac { L ( b ) } { T } \\right) \\displaystyle \\int _ { - \\infty } ^ { + \\infty } \\exp \\left[ \\frac { \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ( \\theta - b ) } { T } \\right] d \\theta } \\\\ & { = \\exp \\left( \\frac { L ( b ) } { T } \\right) \\sqrt { \\frac { 2 \\pi T } { | H _ { b } | } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 515, + 687, + 681 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Based on the results of Step 1 and Step 2, we obtain ", + "bbox": [ + 173, + 698, + 516, + 713 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/353126f9e6f5ea2b0c242d95a77ed41f78dc5e525be66788ebc34b2308c95dcc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\gamma = \\frac { \\displaystyle \\int _ { S _ { a } } J \\cdot d S } { \\displaystyle P ( \\theta \\in V _ { a } ) } = \\frac { J } { P \\left( \\theta \\in V _ { a } \\right) } } \\\\ & { \\quad = \\frac { \\displaystyle P P \\left( a \\right) \\exp \\left( \\frac { L ( a ) } { T } \\right) } { \\displaystyle \\exp \\left( \\frac { L ( b ) } { T } \\right) \\sqrt { \\frac { 2 \\pi T } { | R _ { b } | } } } \\frac { 1 } { P \\left( a \\right) \\sqrt { \\frac { 2 \\pi T } { H _ { a } } } } } \\\\ & { \\quad = \\frac { \\displaystyle \\frac { D \\sqrt { H _ { a } } \\| H _ { b } \\| } { 2 \\pi T } } { \\displaystyle 2 \\pi T } \\exp \\left( - \\frac { \\Delta L _ { a b } } { T } \\right) } \\\\ & { \\quad = \\frac { \\displaystyle \\sqrt { H _ { a } } | H _ { b } | } { \\displaystyle 2 \\pi } \\exp \\left( - \\frac { \\Delta L _ { a b } } { D } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 723, + 620, + 892 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We generalize the proof of one-dimensional diffusion to high-dimensional diffusion ", + "bbox": [ + 169, + 909, + 722, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Step 1: ", + "bbox": [ + 173, + 103, + 222, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b05632ce7e039bb12556394de6977242e47f1284d01fda9536b62dc577384e2a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( \\theta \\in V _ { a } ) } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( \\theta ) d V } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( a ) \\exp \\left[ - \\frac { L ( \\theta ) - L ( a ) } { T } \\right] d V } \\\\ & { = P ( a ) \\displaystyle \\int _ { \\theta \\in V _ { a } } \\exp \\left[ - \\frac { \\frac { 1 } { 2 } ( \\theta - a ) ^ { \\top } H _ { a } ( \\theta - a ) + \\mathcal { O } ( \\Delta \\theta ^ { 3 } ) } { T } \\right] d V } \\\\ & { = P ( a ) \\displaystyle \\frac { ( 2 \\pi T ) ^ { \\frac { n } { 2 } } } { \\mathrm { d e t } ( H _ { a } ) ^ { \\frac { 1 } { 2 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 116, + 700, + 289 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Step 2: Based on the formula of the one-dimensional probability current and flux, we obtain ", + "bbox": [ + 169, + 289, + 776, + 303 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "So we have ", + "bbox": [ + 173, + 420, + 251, + 435 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6d036e0291db65976eaa462fa7b09a7523085fe2d93ddea9ab3e7877252500ab.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { S _ { b } } J \\cdot d S } \\\\ & { = \\displaystyle J _ { b } \\int _ { S _ { b } } \\exp \\left[ - \\frac { \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ^ { + } ( \\theta - b ) } { T } \\right] d S } \\\\ & { = \\displaystyle J _ { b } \\frac { ( 2 \\pi T ) ^ { \\frac { n - 1 } { 2 } } } { ( \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } ) ^ { \\frac { 1 } { 2 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 303, + 645, + 422 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c1fff91c97fdb3a8ca6647cfd560bf9c72d6976c78eaa5818adf5bec6d53555c.jpg", + "text": "$$\n\\begin{array} { c } { { \\tau = 2 \\pi \\sqrt { \\displaystyle \\frac { \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } } { \\operatorname * { d e t } ( H _ { a } ) | H _ { b e } | } } \\exp \\left( \\displaystyle \\frac { \\Delta L } { T } \\right) } } \\\\ { { = 2 \\pi \\sqrt { \\displaystyle \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\displaystyle \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\displaystyle \\frac { \\Delta L } { D } \\right) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 433, + 633, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 PROOF OF THEOREM 3.2 ", + "text_level": 1, + "bbox": [ + 174, + 547, + 390, + 563 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. We decompose the proof into two steps and analyze the one-dimensional case like before. The following proof is similar to the proof of SGLD except that we make $T _ { a }$ the temperature near the minimum a and $T _ { b }$ the temperature near the saddle point b. ", + "bbox": [ + 174, + 574, + 826, + 617 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "One-dimensional SGD Diffusion: ", + "bbox": [ + 174, + 623, + 395, + 638 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Step 1: Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. ", + "bbox": [ + 173, + 645, + 823, + 672 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6b98c14f78c578592e441fa2787324f61b329928cf6f628f881c9858a1d1af0c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( \\theta \\in V _ { a } ) } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( \\theta ) d V } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( a ) \\exp \\left[ - \\frac { L ( \\theta ) - L ( a ) } { T _ { a } } \\right] d V } \\\\ & { = P ( a ) \\displaystyle \\int _ { \\theta \\in V _ { a } } \\exp \\left[ - \\frac { \\frac { 1 } { 2 } ( \\theta - a ) ^ { \\top } H _ { a } ( \\theta - a ) + \\mathcal { O } ( \\Delta \\theta ^ { 3 } ) } { T _ { a } } \\right] d \\theta } \\\\ & { = P ( a ) \\displaystyle \\frac { ( 2 \\pi T _ { a } ) ^ { \\frac { 1 } { 2 } } } { H ^ { \\frac { 1 } { 2 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 672, + 696, + 848 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Step 2: ", + "bbox": [ + 173, + 853, + 222, + 868 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2bcfffbf508f26ff82ad5c8553ed5b1acaa973e3cebdfec530a5aa3888383699.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle J = P ( \\boldsymbol { \\theta } ) \\nabla L ( \\boldsymbol { \\theta } ) + P ( \\boldsymbol { \\theta } ) \\nabla D + D \\nabla P ( \\boldsymbol { \\theta } ) } \\\\ { \\displaystyle J = P ( \\boldsymbol { \\theta } ) \\left[ \\nabla L ( \\boldsymbol { \\theta } ) + \\nabla D - \\frac { D } { T } \\nabla L ( \\boldsymbol { \\theta } ) - D L ( \\boldsymbol { \\theta } ) \\nabla \\left( \\frac { 1 } { T } \\right) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 867, + 689, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "According to Equation 7, $\\nabla \\left( { \\frac { 1 } { T } } \\right)$ is ignorable near the minimum a and the col $\\mathbf { b }$ , thus ", + "bbox": [ + 173, + 102, + 728, + 119 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d810c79ceb48e0266ebd12c4623c15436736a29744f97f7887c38bb77a12f74a.jpg", + "text": "$$\n\\nabla D = \\left( \\frac { D } { T } - 1 \\right) \\nabla L .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 132, + 576, + 169 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Apply this result to the Fokker-Planck Equation 4, we have ", + "bbox": [ + 173, + 180, + 562, + 196 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d64d4a1c707ca1d6bd6fce08bf6f351358dc7c7975fce66023e578f7e37dd6d4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla \\cdot \\nabla [ D ( \\theta ) P ( \\theta , t ) ] } \\\\ & { = \\nabla \\cdot D \\nabla P ( \\theta , t ) + \\nabla \\cdot \\left[ \\left( \\displaystyle \\frac { D } { T } - 1 \\right) \\nabla L ( \\theta ) \\right] P ( \\theta , t ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 207, + 673, + 265 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "And thus we obtain the Smoluchowski equation and a new form of J ", + "bbox": [ + 173, + 275, + 620, + 290 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2355e1ad7e14dcb9231eecc30702b56e26637370df8523a6da6e8f7d2dde8bc2.jpg", + "text": "$$\n\\begin{array} { r } { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = \\nabla \\cdot \\left[ D \\left( \\frac { 1 } { T } \\nabla L ( \\theta ) + \\nabla \\right) P ( \\theta , t ) \\right] = - \\nabla \\cdot J , } \\\\ { J = D \\exp \\left( \\frac { - L ( \\theta ) } { T } \\right) \\nabla \\left[ \\exp \\left( \\frac { L ( \\theta ) } { T } \\right) P ( \\theta ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 303, + 692, + 376 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We note that the Smoluchowski equation is true only near critical points. We assume the point s is the midpoint on the most possible path between a and $\\mathbf { b }$ , where $L ( \\bar { s } ) = ( 1 - s ) L ( a ) + s \\bar { L } ( b )$ . The temperature $T _ { a }$ dominates the path $a s$ , while temperature $T _ { b }$ dominates the path $s \\to b$ . So we have ", + "bbox": [ + 176, + 386, + 823, + 441 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/366fd405fbb644996adac63bce623b92718f347943136bb31f7fe998d784d02e.jpg", + "text": "$$\n\\nabla \\left[ \\exp \\left( \\frac { L ( \\theta ) - L ( s ) } { T } \\right) P ( \\theta ) \\right] = J D ^ { - 1 } \\exp \\left( \\frac { L ( \\theta ) - L ( s ) } { T } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 453, + 712, + 488 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path ", + "bbox": [ + 171, + 501, + 823, + 530 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3db891eae54a203cba2f98d48bf9e74cffd14b189bc5b7960c70ca4facebd309.jpg", + "text": "$$\n\\begin{array} { l } { { L e f t = \\int _ { a } ^ { c } \\frac { \\partial } { \\partial \\theta } [ \\exp \\left( \\displaystyle \\frac { L ( \\theta ) - L ( s ) } { T } \\right) P ( \\theta ) ] d \\theta } } \\\\ { { \\ = \\int _ { a } ^ { s } \\frac { \\partial } { \\partial \\theta } \\left[ \\exp \\left( \\displaystyle \\frac { L ( \\theta ) - L ( s ) } { T _ { a } } \\right) P ( \\theta ) \\right] d \\theta } } \\\\ { { \\ ~ + \\int _ { s } ^ { c } \\frac { \\partial } { \\partial \\theta } \\left[ \\exp \\left( \\displaystyle \\frac { L ( \\theta ) - L ( s ) } { T _ { b } } \\right) P ( \\theta ) \\right] d \\theta } } \\\\ { { \\ = [ P ( s ) - \\exp \\left( \\displaystyle \\frac { L ( a ) - L ( s ) } { T _ { a } } \\right) P ( a ) ] + [ 0 - P ( s ) ] } } \\\\ { { \\ ~ } } \\\\ { { \\ = - \\exp \\left( \\displaystyle \\frac { L ( a ) - L ( s ) } { T _ { a } } \\right) P ( a ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 541, + 692, + 719 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/7487bfc7407640c63c31d76f95b653ba7c56e6fdcc6db0eae8fb054763c1fbb3.jpg", + "text": "$$\nR i g h t = - \\ J \\int _ { a } ^ { c } D ^ { - 1 } \\exp \\left( \\frac { L ( \\theta ) - L ( s ) } { T } \\right) d \\theta\n$$", + "text_format": "latex", + "bbox": [ + 341, + 746, + 656, + 781 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We move $J$ to the outside of integral based on Gauss’s Divergence Theorem, because $J$ is fixed on the escape path from one minimum to another. As there is no field source on the escape path, $\\begin{array} { r } { \\int _ { V } \\nabla \\cdot \\boldsymbol { J } ( \\boldsymbol { \\theta } ) \\dot { d V } = 0 } \\end{array}$ and $\\nabla J ( \\theta ) = 0$ . Obviously, only minima are probability sources in deep learning. So we obtain ", + "bbox": [ + 173, + 801, + 826, + 858 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6da14d3ae6e65097837dfca8bec2cc2ce21ab61b2fa98deb372cf79669ad0551.jpg", + "text": "$$\nJ = \\frac { \\exp \\left( \\frac { L ( a ) - L ( s ) } { T _ { a } } \\right) P ( a ) } { \\int _ { a } ^ { c } D ^ { - 1 } \\exp \\left( \\frac { L ( \\theta ) - L ( s ) } { T } \\right) d \\theta } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 868, + 612, + 921 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Under Assumption 3, we have ", + "bbox": [ + 174, + 103, + 375, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/accc247c2b95d7d1911d106a6e156071cf82fd7c6f08566796963594c9997f53.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { a } ^ { c } D ^ { - 1 } \\exp \\left( \\frac { L ( \\theta ) - L ( s ) } { T } \\right) d \\theta } \\\\ & { \\approx \\displaystyle \\int _ { a } ^ { c } D ^ { - 1 } \\exp \\left[ \\frac { L ( b ) - L ( s ) + \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ( \\theta - b ) } { T b } \\right] d \\theta } \\\\ & { \\approx D _ { b } ^ { - 1 } \\displaystyle \\int _ { - \\infty } ^ { + \\infty } \\exp \\left[ \\frac { L ( b ) - L ( s ) + \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ( \\theta - b ) } { T b } \\right] d \\theta } \\\\ & { = D _ { b } ^ { - 1 } \\exp \\left( \\frac { L ( b ) - L ( s ) } { T _ { b } } \\right) \\sqrt { \\frac { 2 \\pi T _ { b } } { | H _ { b } | } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 121, + 700, + 285 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Based on the results of Step 1 and Step 2, we have ", + "bbox": [ + 173, + 286, + 506, + 301 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b4345f9850f20f8d6b4e66661d7ff54557e935094a880928427b8eebfbc8b586.jpg", + "text": "$$\n\\begin{array} { r l } & { \\gamma = \\frac { \\displaystyle \\int _ { S _ { a } } J \\cdot d S } { \\displaystyle P ( \\theta \\in V _ { a } ) } = \\frac { J } { \\displaystyle P ( \\theta \\in V _ { a } ) } } \\\\ & { \\quad = \\frac { \\displaystyle P ( a ) \\exp \\Big ( \\frac { L ( a ) - L ( s ) } { T _ { a } } \\Big ) } { \\displaystyle D _ { b } ^ { - 1 } \\exp \\Big ( \\frac { L ( b ) - L ( s ) } { T _ { b } } \\Big ) \\sqrt { \\frac { 2 \\pi T _ { b } } { | t _ { b } | } } P ( a ) \\sqrt { \\frac { 2 \\pi T _ { a } } { H _ { a } } } } } \\\\ & { \\quad = \\frac { \\displaystyle \\sqrt { T _ { b } H _ { a } } \\big | H _ { b } \\big | } { \\displaystyle 2 \\pi \\sqrt { T _ { a } } } \\exp \\left( - \\frac { L ( s ) - L ( a ) } { T _ { a } } - \\frac { L ( b ) - L ( s ) } { T _ { b } } \\right) } \\\\ & { \\quad = \\frac { \\displaystyle \\sqrt { T _ { b } H _ { a } } \\big | H _ { b } \\big | } { \\displaystyle 2 \\pi \\sqrt { T _ { a } } } \\exp \\left( - \\frac { s \\Delta L } { T _ { a } } - \\frac { ( 1 - s ) \\Delta L } { T _ { b } } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 304, + 687, + 472 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "So we have ", + "bbox": [ + 173, + 472, + 251, + 487 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/eba0482f1bb8098a1b8c9b9c14287373cd76f6f90ab9fc78d72b2f4cfafcf91d.jpg", + "text": "$$\n\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\sqrt { \\frac { T _ { a } } { T _ { b } H _ { a } | H _ { b } | } } \\exp \\left( { \\frac { s \\Delta L } { T _ { a } } + \\frac { ( 1 - s ) \\Delta L } { T _ { b } } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 487, + 683, + 530 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In the case of pure SGN, $\\begin{array} { r } { T _ { a } = \\frac { \\eta } { 2 B } H _ { a } } \\end{array}$ and $\\begin{array} { r } { T _ { b } = - \\frac { \\eta } { 2 B } H _ { b } } \\end{array}$ gives ", + "bbox": [ + 173, + 539, + 584, + 556 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/66676de4430a4fe7dee6c2eb0e787aea222c5216e54a1f703ffc598b9229d485.jpg", + "text": "$$\n\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\frac { 1 } { | H _ { b } | } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } ( \\frac { s } { H _ { a } } + \\frac { ( 1 - s ) } { | H _ { b } | } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 559, + 666, + 593 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We generalize the proof above into the high-dimensional SGD diffusion. ", + "bbox": [ + 173, + 603, + 648, + 618 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Step 1: ", + "bbox": [ + 173, + 625, + 222, + 638 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d62a8550510a939c026e05c51d3c0cb8d9894edd3c62617193b4d68f38b8679d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( \\theta \\in V _ { a } ) } \\\\ & { = \\displaystyle \\int _ { \\theta \\in V _ { a } } P ( \\theta ) d V } \\\\ & { = P ( a ) \\int _ { \\theta \\in V _ { a } } \\exp \\left[ - \\frac { 1 } { 2 } ( \\theta - a ) ^ { \\top } ( D _ { a } ^ { - \\frac { 1 } { 2 } } H _ { a } D _ { a } ^ { - \\frac { 1 } { 2 } } ) ( \\theta - a ) \\right] d V } \\\\ & { = P ( a ) \\frac { ( 2 \\pi ) ^ { \\frac { n } { 2 } } } { \\operatorname* { d e t } ( D _ { a } ^ { - 1 } H _ { a } ) ^ { \\frac { 1 } { 2 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 640, + 700, + 771 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Step 2: Based on the formula of the one-dimensional probability current and flux, we obtain the high-dimensional flux escaping through Col b: ", + "bbox": [ + 173, + 777, + 823, + 808 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7003e3a1fb7bb89246db1f27f280de569f32eeed6ca9bf74c1bf299498547d63.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { S _ { b } } J \\cdot d S } \\\\ & { = J _ { 1 d } \\int _ { S _ { b } } \\exp \\left[ - \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } [ D _ { b } ^ { - \\frac { 1 } { 2 } } H _ { b } D _ { b } ^ { - \\frac { 1 } { 2 } } ] ^ { \\perp e } ( \\theta - b ) \\right] d S } \\\\ & { = J _ { 1 d } \\frac { ( 2 \\pi ) ^ { \\frac { n - 1 } { 2 } } } { ( \\prod _ { i \\neq e } ( D _ { b i } ^ { - 1 } H _ { b i } ) ) ^ { \\frac { 1 } { 2 } } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 810, + 691, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $[ \\cdot ] ^ { \\perp e }$ indicates the directions perpendicular to the escape direction $e$ . So we have ", + "bbox": [ + 173, + 102, + 743, + 119 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c32f81dfe72beac8dbe890c674c7c9c15273beb55d6494fbab3bc2c420ecf544.jpg", + "text": "$$\n\\gamma = { \\frac { 1 } { 2 \\pi } } { \\sqrt { \\frac { \\operatorname* { d e t } ( H _ { a } D _ { a } ^ { - 1 } ) } { - \\operatorname* { d e t } ( H _ { b } D _ { b } ^ { - 1 } ) } } } | H _ { b e } | \\exp \\left( - { \\frac { s \\Delta L } { T _ { a } } } - { \\frac { ( 1 - s ) \\Delta L } { T _ { b } } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 292, + 123, + 704, + 166 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "$T _ { a }$ and $T _ { b }$ are the eigenvalues of $H _ { a } ^ { - 1 } D _ { a }$ and $H _ { b } ^ { - 1 } D _ { b }$ corresponding to the escape direction. We know $\\begin{array} { r } { D _ { a } \\ = \\ \\frac { \\eta } { 2 B } \\mathbf { \\bar { { H } } } _ { a } } \\end{array}$ and $\\begin{array} { r } { D _ { b } ~ = ~ \\frac { \\eta } { 2 B } [ H _ { b } ] ^ { + } } \\end{array}$ . As $D$ must be positive semidefinite, we replace $H _ { b } \\ = \\ U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , H _ { b e } ) U _ { b }$ by its positive semidefinite analog $[ H _ { b } ] ^ { + } =$ $U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , | H _ { b e } | ) U _ { b }$ . Thus, we have ", + "bbox": [ + 173, + 170, + 826, + 234 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/116c047fddc9a5b54efad6278202602a010b6e07b36f48fec0e8864eca7ceae0.jpg", + "text": "$$\n\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\frac { 1 } { \\left| H _ { b e } \\right| } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } \\left( \\frac { s } { H _ { a e } } + \\frac { \\left( 1 - s \\right) } { \\left| H _ { b e } \\right| } \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 239, + 679, + 275 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.3 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 309, + 400, + 323 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. A stationary distribution must have a balanced probability flux between valleys. So the probability flux of each valley must be equivalent, ", + "bbox": [ + 173, + 334, + 825, + 363 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0d6154c64d3f28a5bdb3bef0601e88275d31f0a99e0f7fb390785924d7d51d1a.jpg", + "text": "$$\nP ( \\theta \\in V _ { 1 } ) \\gamma _ { 1 2 } = P ( \\theta \\in V _ { 2 } ) \\gamma _ { 2 1 }\n$$", + "text_format": "latex", + "bbox": [ + 392, + 367, + 604, + 386 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As $\\tau = \\gamma ^ { - 1 }$ , it leads to $P ( \\theta \\in V _ { v } ) \\propto \\tau _ { v }$ . We normalize the total probability to 1, then we obtain the result. □ ", + "bbox": [ + 173, + 391, + 825, + 421 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 176, + 440, + 331, + 457 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily near saddle points. It means that $\\begin{array} { r } { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = - \\nabla \\cdot J ( \\theta , t ) \\approx 0 } \\end{array}$ holds near minima $a _ { 1 }$ and $a _ { 2 }$ , but not necessarily holds near saddle point $b$ . Quasi-Equilibrium Assumption is actually weaker but more useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt et al., 2017). Under Assumption 2, the probability density $P$ can behave like a stationary distribution only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about density motion can be obtained under the stationary assumption. ", + "bbox": [ + 173, + 470, + 825, + 599 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Low Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jastrz˛ebski et al., 2017), and is always justified when $\\frac { \\eta } { B }$ is small. Under Assumption 3, the probability densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor approximation, Assumption 1, even more reasonable in SGD diffusion. ", + "bbox": [ + 174, + 606, + 825, + 676 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Here, we try to provide a more intuitive explanation about Low Temperature Assumption in the domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The temperature can be interpreted as a real number $D$ . In SGD, we have the temperature as $\\begin{array} { r } { D = \\frac { \\eta } { 2 B } H } \\end{array}$ In statistical physics, if $\\scriptstyle { \\frac { \\Delta L } { D } }$ is large, then we call it Low Temperature Approximation. Note that $\\scriptstyle { \\frac { \\Delta L } { D } }$ appears insides an exponential function in the theoretical analysis. People usually believe that, numerically, $\\begin{array} { r } { \\frac { \\Delta L } { D } > 6 } \\end{array}$ can make a good approximation, for a similar reason of the 6-sigma rule in statistics. In the final training phase of deep networks, a common setting is $\\eta = 0 . 0 1$ and $B = 1 2 8$ . $\\textstyle { \\frac { \\Delta L } { H } } > 2 . 3 \\times \\mathbf { \\dot { 1 } } 0 ^ { - 4 }$ ly apply Assumption 3 to th. Empirically, the condition $\\begin{array} { r } { \\frac { \\Delta L } { H } > 2 . 3 \\times 1 0 ^ { - 4 } } \\end{array}$ h satisfy the very mild conditionholds well in SGD dynamics. It also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain barrier heights. ", + "bbox": [ + 173, + 683, + 826, + 844 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C THE STOCHASTIC GRADIENT NOISE ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 863, + 602, + 881 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet with $B = 5 0$ on CIFAR-10. We also note that the SGN on ResNet seems less Gaussian than SGN on ", + "bbox": [ + 173, + 895, + 823, + 922 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/6304cb3730f61ed48a4694c4774da457ddc5e526dfda5ae7dfe6a42b50aee709.jpg", + "image_caption": [ + "Figure 7: The Gradient Noise Analysis. The histogram of the norm of the gradient noises computed with ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009). " + ], + "image_footnote": [], + "bbox": [ + 235, + 162, + 738, + 537 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/c79c12433ca079002710133471885496c0dcf563a1d82028bab379f639344eb0.jpg", + "image_caption": [ + "Figure 8: The plot of the SGN covariance and the Hessian by training fully-connected network on MNIST. We display all elements $H _ { ( i , j ) } \\in [ - 0 . 0 3 , 0 . 0 3 ]$ of the Hessian matrix and the corresponding elements in gradient noise covariance matrix in the original coordinates. " + ], + "image_footnote": [], + "bbox": [ + 238, + 671, + 759, + 821 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/8d5acc2b749c95f09c4f662adbdfc17fc43a0921f1406e52bdae5c7bae9ae0f4.jpg", + "image_caption": [ + "Figure 9: The plot of the SGN covariance and the Hessian by training fully-connected network on Avila. We display all elements $H _ { ( i , j ) } \\in [ 1 e - 4 , 0 . 5 ]$ of the Hessian matrix and the corresponding elements in gradient noise covariance matrix in the space spanned by the eigenvectors of Hessians. " + ], + "image_footnote": [], + "bbox": [ + 240, + 122, + 741, + 267 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "fully-connected networks with the same batch size. Panigrahi et al. (2019) presented more results on the Gaussianity of SGN under various conditions. ", + "bbox": [ + 171, + 353, + 823, + 382 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By Figure 8, we validate $\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}$ in the original coordinates on MNIST. By Figure 9, we also validate $\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}$ on another dataset, Avila, in the space spanned by the eigenvectors of Hessian. The relationcan still be observed in these two cases. $\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}$ ", + "bbox": [ + 173, + 387, + 825, + 436 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Data Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on MNIST. We leave the preprocessing of Avila in D. Model: Fully-connected networks. ", + "bbox": [ + 173, + 443, + 825, + 472 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D MAIN EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 492, + 385, + 508 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Figure 10, 11, and 12 respectively validate that the exponential relation of the escape rate with the Hessian, the batch size and the learning rate. ", + "bbox": [ + 174, + 523, + 823, + 553 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D.1 EXPERIMENTAL SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 569, + 401, + 583 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Datasets: a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis. ", + "bbox": [ + 174, + 594, + 823, + 623 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Data Precessing: We perform per-pixel zero-mean and unit-variance normalization on input data. For simplicity, we also transform multi-class problems into binary-class problems by grouping labels, although this is unnecessary. ", + "bbox": [ + 174, + 630, + 823, + 672 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Model: Two-layer fully-connected networks with one hidden layer and 10 neurons per hidden layer. ", + "bbox": [ + 174, + 679, + 820, + 694 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Initializations: To ensure the initialized models are near minima, we first pretrain models with 200-1000 epochs to fit each data set as well as possible. We set the pretrained models’ parameters as the initialized $\\theta _ { t = 0 }$ . ", + "bbox": [ + 174, + 700, + 820, + 742 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Valleys’ Boundary: In principle, any small neighborhood around $\\theta _ { t = 0 }$ can be regarded as the inside of the start valleys. In our experiments, we set each dimension’s distance from $\\theta _ { t = 0 }$ should be less than 0.05, namely $| \\Delta \\theta _ { i } | \\le 0 . 0 5$ for each dimension $i$ . If we rescale the landscape by a factor $k$ , the neighborhood will also be rescaled by $k$ . Although we don’t know which loss valleys exist inside the neighborhood, we know the landscape of the neighborhood is invariant in each simulation. ", + "bbox": [ + 173, + 750, + 825, + 819 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Hyperparameters: In Figure 10: (a) $\\eta = 0 . 0 0 1 , B = 1$ , (b) $\\eta = 0 . 0 1 5 , B = 1 .$ , (c) $\\eta = 0 . 0 0 5 , B =$ 1, (d) $\\eta = 0 . 0 0 0 5 , B = 1$ . In Figure 11: (a) $\\eta = 0 . 0 2$ , (b) $\\eta = 0 . 6$ , (c) $\\eta = 0 . 1 8$ , (d) $\\eta = 0 . 0 1$ . In Figure 12: (a) $B = 1$ , (b) $B = 1$ , (c) $B = 1$ , (d) $B = 1$ . In Figure 13: (a) $\\eta = 0 . 0 0 0 2 , B = 1 0 0$ , (b) $\\eta = 0 . 0 0 1 , B = 1 0 0$ , (c) $\\eta = 0 . 0 0 0 2 , B = 1 0 0$ , (d) $\\eta = 0 . 0 0 0 1 , B = 1 0 0$ . In Figure 14: (a) $\\eta = 0 . 0 0 0 2$ , $B = 1 0 0 , D = 0 . 0 0 0 2$ , (b) $\\eta = 0 . 0 0 1 , B = 1 0 0 , D = 0 . 0 0 0 1$ , (c) $\\eta = 0 . 0 0 0 2 , B =$ $1 0 0 , D = 0 . 0 0 0 5$ , (d) $\\eta = 0 . 0 0 0 1 , B = 1 0 0 , D = 0 . 0 0 0 3$ . We note that the hyperparameters need be tuned for each initialized pretrained models, due to the stochastic property of deep learning. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/e611bd5ee1c1627c6eebbe465cd800ad1aa2a077166549a38354ad6ff51c8c74.jpg", + "image_caption": [ + "Figure 10: The escape rate exponentially depends on the “path Hessians” in the dynamics of SGD. $- \\log ( \\gamma )$ is linear with $\\textstyle { \\frac { 1 } { k } }$ . The “path Hessians” indicates the eigenvalues of Hessians corresponding to the escape directions. " + ], + "image_footnote": [], + "bbox": [ + 254, + 319, + 740, + 670 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/cc24938c36d0e50a1712f00debfae83e0ef961a9118b02da5c6b68dace8eb7b0.jpg", + "image_caption": [ + "Figure 11: The escape rate exponentially depends on the batch size in the dynamics of SGD. $- \\log ( \\gamma )$ is linear with $B$ . " + ], + "image_footnote": [], + "bbox": [ + 254, + 325, + 740, + 679 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/95d8cee65da16f84618d2949f3e4ec958b0cee119bbb90092d89784746e231aa.jpg", + "image_caption": [ + "Figure 12: The escape rate exponentially depends on the learning rate in the dynamics of SGD. $- \\log ( \\gamma )$ is linear with $\\frac { 1 } { \\eta }$ . The estimated escape rate has incorporated $\\eta$ as the time unit. " + ], + "image_footnote": [], + "bbox": [ + 251, + 324, + 740, + 675 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/32b28be96d16487646694dc86dedef57fef4d0cb3660ae3df14106d0d9ee0727.jpg", + "image_caption": [ + "Figure 13: The relation of the escape rate and the isotropic diffusion coefficient D. The escape formula that − $\\log ( \\gamma )$ is linear with $\\dot { \\frac { 1 } { D } }$ is validated. " + ], + "image_footnote": [], + "bbox": [ + 254, + 125, + 740, + 477 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "According to our experience, we can always find the hyperparameters to discover the quantitative relations as long as the pretrained model fits the data set well enough. The fined-tuned requirement can be avoided in Section E, because the models in Section E are artificially initialized. ", + "bbox": [ + 174, + 553, + 823, + 595 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Observation: we observe the number of iterations from the initialized position to the terminated position. We repeat experiments 100 times to estimate the escape rate $\\gamma$ and the mean escape time $\\tau$ . As the escape time is a random variable obeying an exponential distribution, $t \\sim E x p o n e n t i a l ( \\gamma )$ , the estimated escape rate can be written as ", + "bbox": [ + 174, + 602, + 826, + 659 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/a6c69eaa46cca8f5a5f30367cf330f3d8d639e665954ccde6db76406ba6a7e7a.jpg", + "text": "$$\n\\hat { \\gamma } = \\frac { 1 0 0 - 2 } { \\sum _ { i = 1 } ^ { 1 0 0 } t _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 450, + 670, + 547, + 707 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "The $9 5 \\%$ confidence interval of this estimator is ", + "bbox": [ + 174, + 718, + 488, + 733 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/296a92b12a07eb16d57f27ee87acdd41c9a8a3e9d36ad618bf979849d7c6a5d6.jpg", + "text": "$$\n\\hat { \\gamma } ( 1 - \\frac { 1 . 9 6 } { \\sqrt { 1 0 0 } } ) \\leq \\hat { \\gamma } \\leq \\hat { \\gamma } ( 1 + \\frac { 1 . 9 6 } { \\sqrt { 1 0 0 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 743, + 622, + 777 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "D.2 EXPERIMENTS ON SGLD ", + "text_level": 1, + "bbox": [ + 174, + 797, + 393, + 813 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Experimental Results: Figure 13 shows a highly precise exponential relation of the escape rate and the diffusion coefficient in the figure. Figure 14 shows a proportional relation of the escape rate and the Hessian determinant in the figure. Overall, the empirical results support the density diffusion theory in the dynamics of white noise. In experiments on SGLD, we carefully adjust the injected gradient noise scale in experiment to ensure that $D$ is significantly smaller than the loss barrier’ height and large enough to dominate SGN scale. If $D$ is too large, learning dynamics will be reduced to Free Brownian Motion. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/83ef704670a42a8d7638e9a3a2075a495a934d06c96343b2f88bb6398afe7b0c.jpg", + "image_caption": [ + "Figure 14: The relation of the escape rate and the Hessian determinant in the dynamics of white noise.The escape formula that $\\gamma$ is linear with $k$ is validated. " + ], + "image_footnote": [], + "bbox": [ + 250, + 325, + 736, + 678 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E EXPERIMENTS ON MORE MODELS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 495, + 118 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We supply experiments of training three models on artificial Gaussian datasets. In these experiments, we can analytically know the locations of the minima, Hessians and loss barriers, as each input feature is Gaussian noise. ", + "bbox": [ + 174, + 133, + 825, + 175 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "E.1 EXPERIMENTS SETTINGS ", + "text_level": 1, + "bbox": [ + 174, + 191, + 392, + 207 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Data Set: We generate 50000 Gaussian samples and random two-class labels as the training data set, $\\{ ( x ^ { ( i ) } , y ^ { ( i ) } ) | x ^ { ( \\bar { i } ) } \\sim \\mathcal { N } ( 0 , I ) , y ^ { ( i ) } \\in \\{ 0 , 1 \\} , i \\stackrel { \\cdot } { \\in } \\{ 1 , 2 , \\cdot \\cdot , 5 0 0 0 0 \\} \\}$ ", + "bbox": [ + 171, + 218, + 823, + 250 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Hyperparameters: In Figure 15: (a) $\\eta = 0 . 0 0 0 1 , B = 1 0 0$ , (b) $\\eta = 0 . 0 0 1 , B = 1 0 0$ , (c) $\\eta =$ 0.0003, $B = 1 0 0$ . In Figure 16: (a) $\\eta = 0 . 0 0 0 1 , B = 5 0 , D = 0 . 2$ , (b) $\\eta = 0 . 0 0 1 , B = 5 0 , D =$ 0.0005, (c) $\\eta = 0 . 0 0 0 3 , B = 1 , D = 0 . 0 0 0 3$ . In Figure 17: (a) $\\eta = 0 . 0 0 6 , B = 5 0$ , (b) $\\eta =$ 0.05, $, B = 5 0$ , (c) $\\eta = 0 . 0 0 5 , B = 1$ . In Figure 18: (a) $\\eta = 0 . 0 0 6$ , (b) $\\eta = 0 . 0 6$ , (c) $\\eta = 0 . 1$ . In Figure 19: (a) $B = 1$ , (b) $B = 1$ , (c) $B = 1$ . We note that the hyperparameters are recommended and needn’t be fine tuned again. The artificially initialized parameters avoids the stochastic property of the initial states. ", + "bbox": [ + 173, + 253, + 825, + 353 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Experiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex optimization, written as ", + "bbox": [ + 173, + 359, + 823, + 388 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/598af83511621d4f023c24c4739d7ba92c27423cac94fb900677543fd19c00fe.jpg", + "text": "$$\nf ( \\theta ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } ( \\theta _ { i } ^ { 4 } - 1 6 \\theta _ { i } ^ { 2 } + 5 \\theta _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 392, + 607, + 434 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as training data. ", + "bbox": [ + 173, + 439, + 823, + 468 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/65ad8b59461d7d4c73eccd54ecbf3361f263788d4b864f6b2fed2aade26f59c7.jpg", + "text": "$$\nL ( \\theta ) = f ( \\theta - x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 473, + 558, + 491 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where data samples $x \\sim \\mathcal { N } ( 0 , I )$ . The one-dimensional Styblinski-Tang Function has one global minimum located at $a = - 2 . 9 0 3 5 3 4$ , one local minimum located at $d$ , and one saddle point $b =$ 0.156731 as the boundary separating Valley $a _ { 1 }$ and Valley $a _ { 2 }$ . For a $\\mathbf { n }$ -dimensional Styblinski-Tang Function, we initialize parameters $\\theta _ { t = 0 } = \\textstyle { \\frac { 1 } { \\sqrt { k } } } ( - 2 . 9 0 3 5 3 4 , \\cdot \\cdot \\cdot , - 2 . 9 0 3 5 3 4 )$ , and set the valley’s boundary as $\\begin{array} { r } { \\theta _ { i } < \\frac { 1 } { \\sqrt { k } } 0 . 1 5 6 7 3 1 } \\end{array}$ , where $i$ is the dimension index. We record the number of iterations required to escape from the valley to the outside of valley. The setting 1 does not need labels. ", + "bbox": [ + 173, + 496, + 825, + 589 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Experiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initialization: $\\theta _ { t = 0 } = ( 0 , \\cdot \\cdot \\cdot , 0 )$ . Valley Boundary: $- 0 . 1 < \\theta _ { i } < 0 . 1$ . Due to the randomness of training data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin valley group, and study the mean escape time from the group of valleys. ", + "bbox": [ + 174, + 594, + 825, + 665 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Experiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization: $\\theta _ { t = 0 } = ( 0 . 1 , \\cdot \\cdot \\cdot , 0 . 1 )$ with a small Gaussian noise $\\epsilon = ( 0 , 0 . 0 1 I )$ . Valley Boundary: $0 . 0 5 < \\theta _ { i } < 0 . 1 5$ . To prevent the gradient disappearance problem of deep learning, we move the starting point from the origin. For symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value. Due to the complex loss landscape of deep networks, we can hardly know the exact information about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether or not the propositions can hold with very-small-batch gradient noise in practice. ", + "bbox": [ + 173, + 671, + 825, + 811 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "E.2 EXPERIMENTS RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 828, + 385, + 842 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. Figure 16 shows the relation of the escape rate and the Hessian determinant in the dynamics of white noise. Figure 17 shows the relation of the escape rate and the second order directional derivative in the dynamics of SGD. Figure 18 shows the relation of the escape rate and the batch size in the dynamics of SGD. Figure 19 shows the relation of the escape rate and the learning rate in the dynamics of SGD. ", + "bbox": [ + 173, + 853, + 826, + 924 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/a3af91f70dd76a14cfbf68c4af33655d6324c430f4d0a0f672fcb25cb4546d66.jpg", + "image_caption": [ + "Figure 15: The relation of the escape rate and the diffusion coefficient D in the dynamics of SGLD. The escape formula that $- \\log ( \\gamma )$ is linear with $\\textstyle { \\frac { 1 } { D } }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. " + ], + "image_footnote": [], + "bbox": [ + 191, + 147, + 808, + 285 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/9bdfc4ba7eacbef971e2839103509719b827e0264c02b872459d91774f4fc065.jpg", + "image_caption": [ + "Figure 16: The relation of the escape rate and the Hessian determinants in the dynamics of SGLD. The escape formula that $\\gamma$ is linear with $k$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. " + ], + "image_footnote": [], + "bbox": [ + 179, + 422, + 805, + 560 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/72037af24de8ea4733e366a2c5e96d4046e24eab734ab0312a61c39283d69b88.jpg", + "image_caption": [ + "Figure 17: The escape rate exponentially depends on the sharpness in the dynamics of SGD. The escape formula that $- \\log ( \\gamma )$ is linear with $\\frac { 1 } { k }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. " + ], + "image_footnote": [], + "bbox": [ + 189, + 698, + 807, + 837 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/6ee2fb941d89c488a872f7dd33c40187fc2f3f42d856655122e3c48cc20c441a.jpg", + "image_caption": [ + "Figure 18: The escape rate exponentially depends on the batch size in the dynamics of SGD. The escape formula that $\\bar { - } \\log ( \\gamma )$ is linear with $B$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. " + ], + "image_footnote": [], + "bbox": [ + 189, + 210, + 808, + 353 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/f24a88bb49321f7d8fee99556f21f5bdc68661b8c4d02b911e13f9652bb6df38.jpg", + "image_caption": [ + "Figure 19: The escape rate exponentially depends on the learning rate in the dynamics of SGD. The escape formula that $- \\log ( \\gamma )$ is linear with $\\frac { \\mathbf { i } } { \\eta }$ is validated in the setting of Styblinski-Tang Function, Logistic Regression and MLP. 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SGD is known to find a flat minimum that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 305, + 469, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 469, + 319 + ], + "score": 1.0, + "content": "often generalizes well. However, it is mathematically unclear how deep learning", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 316, + 469, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 316, + 469, + 329 + ], + "score": 1.0, + "content": "can select a flat minimum among so many minima. 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To the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 349, + 470, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 470, + 361 + ], + "score": 1.0, + "content": "best of our knowledge, we are the first to theoretically and empirically prove that,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 359, + 470, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 470, + 372 + ], + "score": 1.0, + "content": "benefited from the Hessian-dependent covariance of stochastic gradient noise, SGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 371, + 470, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 470, + 383 + ], + "score": 1.0, + "content": "favors flat minima exponentially more than sharp minima, while Gradient Descent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 383, + 469, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 383, + 469, + 394 + ], + "score": 1.0, + "content": "(GD) with injected white noise favors flat minima only polynomially more than", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 393, + 469, + 406 + ], + "spans": [ + { + "bbox": [ + 141, + 393, + 469, + 406 + ], + "score": 1.0, + "content": "sharp minima. We also reveal that either a small learning rate or large-batch training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 405, + 469, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 469, + 416 + ], + "score": 1.0, + "content": "requires exponentially many iterations to escape from minima in terms of the ratio", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 415, + 470, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 415, + 470, + 427 + ], + "score": 1.0, + "content": "of the batch size and learning rate. 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Due to the over-parametrization and the highly complex loss landscape of deep", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 538, + 507, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 507, + 553 + ], + "score": 1.0, + "content": "1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 507, + 585 + ], + "score": 1.0, + "content": "study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 507, + 596 + ], + "score": 1.0, + "content": "2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 617, + 150, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 150, + 628 + ], + "score": 1.0, + "content": "minimum.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "The diffusion theory is an important theoretical tool to understand how deep learning dynamics", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "works. It helps us model the diffusion process of probability densities of parameters instead of model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "(SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrz˛ebski", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "a Beyesian perspective on generalization of SGD. Wu et al. 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SGD is known to find a flat minimum that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 305, + 469, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 469, + 319 + ], + "score": 1.0, + "content": "often generalizes well. However, it is mathematically unclear how deep learning", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 316, + 469, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 316, + 469, + 329 + ], + "score": 1.0, + "content": "can select a flat minimum among so many minima. To answer the question", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 327, + 469, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 469, + 339 + ], + "score": 1.0, + "content": "quantitatively, we develop a density diffusion theory to reveal how minima selection", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 338, + 469, + 350 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 469, + 350 + ], + "score": 1.0, + "content": "quantitatively depends on the minima sharpness and the hyperparameters. To the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 349, + 470, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 470, + 361 + ], + "score": 1.0, + "content": "best of our knowledge, we are the first to theoretically and empirically prove that,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 359, + 470, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 470, + 372 + ], + "score": 1.0, + "content": "benefited from the Hessian-dependent covariance of stochastic gradient noise, SGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 371, + 470, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 470, + 383 + ], + "score": 1.0, + "content": "favors flat minima exponentially more than sharp minima, while Gradient Descent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 383, + 469, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 383, + 469, + 394 + ], + "score": 1.0, + "content": "(GD) with injected white noise favors flat minima only polynomially more than", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 393, + 469, + 406 + ], + "spans": [ + { + "bbox": [ + 141, + 393, + 469, + 406 + ], + "score": 1.0, + "content": "sharp minima. We also reveal that either a small learning rate or large-batch training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 405, + 469, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 469, + 416 + ], + "score": 1.0, + "content": "requires exponentially many iterations to escape from minima in terms of the ratio", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 415, + 470, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 415, + 470, + 427 + ], + "score": 1.0, + "content": "of the batch size and learning rate. Thus, large-batch training cannot search flat", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 426, + 351, + 438 + ], + "spans": [ + { + "bbox": [ + 141, + 426, + 351, + 438 + ], + "score": 1.0, + "content": "minima efficiently in a realistic computational time.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5, + "bbox_fs": [ + 141, + 284, + 470, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 459, + 206, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 208, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 208, + 475 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "application areas. Due to the over-parametrization and the highly complex loss landscape of deep", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 538, + 507, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 507, + 553 + ], + "score": 1.0, + "content": "1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 507, + 585 + ], + "score": 1.0, + "content": "study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 507, + 596 + ], + "score": 1.0, + "content": "2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 617, + 150, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 150, + 628 + ], + "score": 1.0, + "content": "minimum.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 484, + 507, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "The diffusion theory is an important theoretical tool to understand how deep learning dynamics", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "works. It helps us model the diffusion process of probability densities of parameters instead of model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "(SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrz˛ebski", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "a Beyesian perspective on generalization of SGD. Wu et al. (2018) studied the escape problems of", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "SGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "a related proposition that describes the mean escape time in terms of a free energy that depends on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 466, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 466, + 161 + ], + "score": 1.0, + "content": "continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "However, the related papers mainly analyzed the diffusion process under parameter-independent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 164, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 164, + 243 + ], + "score": 1.0, + "content": "contributions:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 132, + 252, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "• The proposed theory formulates the fundamental roles of gradient noise, batch size, the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 262, + 347, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 347, + 275 + ], + "score": 1.0, + "content": "learning rate, and the Hessian in minima selection.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 137, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 137, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "The SGN covariance is approximately proportional to the Hessian and inverse to batch size.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 140, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "Either a small learning rate or large-batch training requires exponentially many iterations to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 304, + 395, + 316 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 395, + 316 + ], + "score": 1.0, + "content": "escape minima in terms of ratio of batch size and learning rate.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 131, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 131, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "• To the best of our knowledge, we are the first to theoretically and empirically reveal that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 329, + 398, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 398, + 343 + ], + "score": 1.0, + "content": "SGD favors flat minima exponentially more than sharp minima.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 356, + 401, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 402, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 402, + 372 + ], + "score": 1.0, + "content": "2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 390, + 507, + 409 + ], + "spans": [ + { + "bbox": [ + 103, + 390, + 221, + 409 + ], + "score": 1.0, + "content": "denote the data samples as", + "type": "text" + }, + { + "bbox": [ + 221, + 393, + 256, + 406 + ], + "score": 0.92, + "content": "\\{ x _ { j } \\} _ { j = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 390, + 365, + 409 + ], + "score": 1.0, + "content": ", the model parameters as", + "type": "text" + }, + { + "bbox": [ + 366, + 393, + 372, + 403 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 390, + 507, + 409 + ], + "score": 1.0, + "content": "and the loss function over data", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 142, + 418 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 142, + 407, + 149, + 415 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 405, + 162, + 418 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 162, + 405, + 192, + 417 + ], + "score": 0.93, + "content": "L ( \\theta , x )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 405, + 384, + 418 + ], + "score": 1.0, + "content": ". For simplicity, we denote the training loss as", + "type": "text" + }, + { + "bbox": [ + 385, + 405, + 405, + 417 + ], + "score": 0.92, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 405, + 506, + 418 + ], + "score": 1.0, + "content": ". Following Mandt et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 269, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 269, + 429 + ], + "score": 1.0, + "content": "(2017), we may write SGD dynamics as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 432, + 414, + 460 + ], + "lines": [ + { + "bbox": [ + 196, + 432, + 414, + 460 + ], + "spans": [ + { + "bbox": [ + 196, + 432, + 414, + 460 + ], + "score": 0.93, + "content": "\\theta _ { t + 1 } = \\theta _ { t } - \\eta \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } = \\theta _ { t } - \\eta \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } + \\eta C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } ,", + "type": "interline_equation", + "image_path": "6ef5022f684f592cdaef889ee88891f75529c47375e329ab2cb8554faec9688b.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 432, + 414, + 446.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 196, + 446.0, + 414, + 460.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 467, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 134, + 481 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 466, + 155, + 480 + ], + "score": 0.92, + "content": "\\hat { L } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 466, + 279, + 481 + ], + "score": 1.0, + "content": "is the loss of one minibatch,", + "type": "text" + }, + { + "bbox": [ + 279, + 467, + 337, + 479 + ], + "score": 0.92, + "content": "\\zeta _ { t } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 466, + 360, + 481 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 361, + 468, + 382, + 480 + ], + "score": 0.92, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "represents the gradient noise", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 423, + 491 + ], + "score": 1.0, + "content": "covariance matrix. The classic approach is to model SGN by Gaussian noise,", + "type": "text" + }, + { + "bbox": [ + 423, + 478, + 471, + 491 + ], + "score": 0.93, + "content": "{ \\mathcal { N } } ( 0 , { \\overline { { C } } } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "(Mandt", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 489, + 346, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 346, + 502 + ], + "score": 1.0, + "content": "et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Stochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 516, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 141, + 537 + ], + "score": 1.0, + "content": "training,", + "type": "text" + }, + { + "bbox": [ + 142, + 518, + 254, + 535 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } = \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } - \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 516, + 505, + 536 + ], + "score": 1.0, + "content": ", which is the difference between gradient descent and stochastic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "gradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "mean of many infinite-variance random variables converges to a stable distribution, while the mean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 392, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 392, + 578 + ], + "score": 1.0, + "content": "practice, we believe the Gaussian approximation of SGN is reasonable.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "Simsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "distribution looks closer to a stable distribution than a Gaussian distribution. However, this research", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 164, + 651 + ], + "score": 1.0, + "content": "“SGN” across", + "type": "text" + }, + { + "bbox": [ + 165, + 640, + 172, + 648 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 637, + 345, + 651 + ], + "score": 1.0, + "content": "model parameters and regarded “SGN\" as", + "type": "text" + }, + { + "bbox": [ + 345, + 640, + 353, + 648 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "samples drawn from a single-variant", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 426, + 684 + ], + "score": 1.0, + "content": "Gaussian noise. In our paper, SGN computed over different minibatches obeys a", + "type": "text" + }, + { + "bbox": [ + 426, + 673, + 433, + 681 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "-variant Gaussian", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 367, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 367, + 695 + ], + "score": 1.0, + "content": "distribution, which can be parameter-dependent and anisotropic.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "SGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "a related proposition that describes the mean escape time in terms of a free energy that depends on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 466, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 466, + 161 + ], + "score": 1.0, + "content": "continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "However, the related papers mainly analyzed the diffusion process under parameter-independent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 164, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 164, + 243 + ], + "score": 1.0, + "content": "contributions:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 165, + 506, + 243 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 252, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "• The proposed theory formulates the fundamental roles of gradient noise, batch size, the", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 262, + 347, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 347, + 275 + ], + "score": 1.0, + "content": "learning rate, and the Hessian in minima selection.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 137, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "The SGN covariance is approximately proportional to the Hessian and inverse to batch size.", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 140, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "Either a small learning rate or large-batch training requires exponentially many iterations to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 304, + 395, + 316 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 395, + 316 + ], + "score": 1.0, + "content": "escape minima in terms of ratio of batch size and learning rate.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 131, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "• To the best of our knowledge, we are the first to theoretically and empirically reveal that", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 329, + 398, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 398, + 343 + ], + "score": 1.0, + "content": "SGD favors flat minima exponentially more than sharp minima.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + } + ], + "index": 17, + "bbox_fs": [ + 131, + 251, + 506, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 356, + 401, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 402, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 402, + 372 + ], + "score": 1.0, + "content": "2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 390, + 507, + 409 + ], + "spans": [ + { + "bbox": [ + 103, + 390, + 221, + 409 + ], + "score": 1.0, + "content": "denote the data samples as", + "type": "text" + }, + { + "bbox": [ + 221, + 393, + 256, + 406 + ], + "score": 0.92, + "content": "\\{ x _ { j } \\} _ { j = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 390, + 365, + 409 + ], + "score": 1.0, + "content": ", the model parameters as", + "type": "text" + }, + { + "bbox": [ + 366, + 393, + 372, + 403 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 390, + 507, + 409 + ], + "score": 1.0, + "content": "and the loss function over data", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 142, + 418 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 142, + 407, + 149, + 415 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 405, + 162, + 418 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 162, + 405, + 192, + 417 + ], + "score": 0.93, + "content": "L ( \\theta , x )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 405, + 384, + 418 + ], + "score": 1.0, + "content": ". For simplicity, we denote the training loss as", + "type": "text" + }, + { + "bbox": [ + 385, + 405, + 405, + 417 + ], + "score": 0.92, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 405, + 506, + 418 + ], + "score": 1.0, + "content": ". Following Mandt et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 269, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 269, + 429 + ], + "score": 1.0, + "content": "(2017), we may write SGD dynamics as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 103, + 381, + 507, + 429 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 432, + 414, + 460 + ], + "lines": [ + { + "bbox": [ + 196, + 432, + 414, + 460 + ], + "spans": [ + { + "bbox": [ + 196, + 432, + 414, + 460 + ], + "score": 0.93, + "content": "\\theta _ { t + 1 } = \\theta _ { t } - \\eta \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } = \\theta _ { t } - \\eta \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } + \\eta C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } ,", + "type": "interline_equation", + "image_path": "6ef5022f684f592cdaef889ee88891f75529c47375e329ab2cb8554faec9688b.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 432, + 414, + 446.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 196, + 446.0, + 414, + 460.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 467, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 134, + 481 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 466, + 155, + 480 + ], + "score": 0.92, + "content": "\\hat { L } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 466, + 279, + 481 + ], + "score": 1.0, + "content": "is the loss of one minibatch,", + "type": "text" + }, + { + "bbox": [ + 279, + 467, + 337, + 479 + ], + "score": 0.92, + "content": "\\zeta _ { t } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 466, + 360, + 481 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 361, + 468, + 382, + 480 + ], + "score": 0.92, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "represents the gradient noise", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 423, + 491 + ], + "score": 1.0, + "content": "covariance matrix. The classic approach is to model SGN by Gaussian noise,", + "type": "text" + }, + { + "bbox": [ + 423, + 478, + 471, + 491 + ], + "score": 0.93, + "content": "{ \\mathcal { N } } ( 0 , { \\overline { { C } } } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "(Mandt", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 489, + 346, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 346, + 502 + ], + "score": 1.0, + "content": "et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 466, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Stochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 516, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 141, + 537 + ], + "score": 1.0, + "content": "training,", + "type": "text" + }, + { + "bbox": [ + 142, + 518, + 254, + 535 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C ( \\theta _ { t } ) ^ { \\frac { 1 } { 2 } } \\zeta _ { t } = \\frac { \\partial L ( \\theta _ { t } ) } { \\partial \\theta _ { t } } - \\frac { \\partial \\hat { L } ( \\theta _ { t } ) } { \\partial \\theta _ { t } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 516, + 505, + 536 + ], + "score": 1.0, + "content": ", which is the difference between gradient descent and stochastic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "gradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "mean of many infinite-variance random variables converges to a stable distribution, while the mean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 392, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 392, + 578 + ], + "score": 1.0, + "content": "practice, we believe the Gaussian approximation of SGN is reasonable.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 505, + 505, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "Simsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "distribution looks closer to a stable distribution than a Gaussian distribution. However, this research", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 164, + 651 + ], + "score": 1.0, + "content": "“SGN” across", + "type": "text" + }, + { + "bbox": [ + 165, + 640, + 172, + 648 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 637, + 345, + 651 + ], + "score": 1.0, + "content": "model parameters and regarded “SGN\" as", + "type": "text" + }, + { + "bbox": [ + 345, + 640, + 353, + 648 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "samples drawn from a single-variant", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 426, + 684 + ], + "score": 1.0, + "content": "Gaussian noise. In our paper, SGN computed over different minibatches obeys a", + "type": "text" + }, + { + "bbox": [ + 426, + 673, + 433, + 681 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "-variant Gaussian", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 367, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 367, + 695 + ], + "score": 1.0, + "content": "distribution, which can be parameter-dependent and anisotropic.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 582, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "score": 1.0, + "content": "the contradicted observations are from the different formulations of gradient noise. Simsekli et al.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 466, + 329 + ], + "score": 1.0, + "content": "(2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 466, + 318, + 474, + 326 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 474, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "-variant", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 442, + 339 + ], + "score": 1.0, + "content": "distribution. Our empirical analysis in Figure 1 holds well at least when the batch size", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 442, + 328, + 451, + 337 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 451, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "is larger than", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "16, which is common in practice. Similar empirical evidence can be observed for training ResNet18", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 349, + 419, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 419, + 362 + ], + "score": 1.0, + "content": "(He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 92, + 495, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 92, + 495, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 92, + 495, + 186 + ], + "spans": [ + { + "bbox": [ + 110, + 92, + 495, + 186 + ], + "score": 0.96, + "type": "image", + "image_path": "9f25e21da6eaf1f4dcc5fb4e55195174a32550f9f088693c21380f6b59439613.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 92, + 495, + 123.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 123.33333333333333, + 495, + 154.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 154.66666666666666, + 495, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 197, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Figure 1: The Stochastic Gradient Noise Analysis. The histogram of the norm of the gradient noises", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "computed with the three-layer fully-connected network on MNIST (LeCun, 1998). (a) and (c): the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 218, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 507, + 232 + ], + "score": 1.0, + "content": "histograms of the norms of two kinds of gradient noise: (a) “SGN” is computed over parameters,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "which is actually stochastic gradient rather than SGN; (c) SGN is computed over minibatches. (b)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "and (d): the histograms of the norms of (scaled) Gaussian noise and Lévy noise. Based on (a) and (b),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "Simsekli et al. (2019) argued that gradient noise across parameters is heavy-tailed Lévy noise. Based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 482, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 482, + 276 + ], + "score": 1.0, + "content": "on (c) and (d), we show that SGN without the isotropic restriction is approximately Gaussian.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "score": 1.0, + "content": "the contradicted observations are from the different formulations of gradient noise. Simsekli et al.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 466, + 329 + ], + "score": 1.0, + "content": "(2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a", + "type": "text" + }, + { + "bbox": [ + 466, + 318, + 474, + 326 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "-variant", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 442, + 339 + ], + "score": 1.0, + "content": "distribution. Our empirical analysis in Figure 1 holds well at least when the batch size", + "type": "text" + }, + { + "bbox": [ + 442, + 328, + 451, + 337 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "is larger than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "16, which is common in practice. Similar empirical evidence can be observed for training ResNet18", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 349, + 419, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 419, + 362 + ], + "score": 1.0, + "content": "(He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 504, + 378 + ], + "score": 1.0, + "content": "Panigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "is best described as Gaussian at-least in the early phases of training. Comparing our results with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 388, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 507, + 401 + ], + "score": 1.0, + "content": "Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 409, + 482, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 482, + 423 + ], + "score": 1.0, + "content": "matters. Because SGN on some layers/networks is more Gaussian than other layers/networks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "The isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameter-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "dependent and anisotropic SGN brings a large improvement over existing parameter-independent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 461, + 494 + ], + "score": 1.0, + "content": "parameter-independent anisotropic heavy-tailed noise, when the batch size is too small", + "type": "text" + }, + { + "bbox": [ + 462, + 482, + 491, + 493 + ], + "score": 0.83, + "content": "( B \\sim 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 356, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 356, + 505 + ], + "score": 1.0, + "content": "apply Central Limit Theorem. 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In standard", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 405, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 372, + 733 + ], + "score": 1.0, + "content": "SGLD, the injected gradient noise is fixed and isotropic Gaussian,", + "type": "text" + }, + { + "bbox": [ + 372, + 721, + 400, + 730 + ], + "score": 0.89, + "content": "D = I", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 720, + 405, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 92, + 495, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 92, + 495, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 92, + 495, + 186 + ], + "spans": [ + { + "bbox": [ + 110, + 92, + 495, + 186 + ], + "score": 0.96, + "type": "image", + "image_path": "9f25e21da6eaf1f4dcc5fb4e55195174a32550f9f088693c21380f6b59439613.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 92, + 495, + 123.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 123.33333333333333, + 495, + 154.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 154.66666666666666, + 495, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 197, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Figure 1: The Stochastic Gradient Noise Analysis. The histogram of the norm of the gradient noises", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "computed with the three-layer fully-connected network on MNIST (LeCun, 1998). (a) and (c): the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 218, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 507, + 232 + ], + "score": 1.0, + "content": "histograms of the norms of two kinds of gradient noise: (a) “SGN” is computed over parameters,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "which is actually stochastic gradient rather than SGN; (c) SGN is computed over minibatches. (b)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "and (d): the histograms of the norms of (scaled) Gaussian noise and Lévy noise. Based on (a) and (b),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "Simsekli et al. (2019) argued that gradient noise across parameters is heavy-tailed Lévy noise. Based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 482, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 482, + 276 + ], + "score": 1.0, + "content": "on (c) and (d), we show that SGN without the isotropic restriction is approximately Gaussian.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 361 + ], + "lines": [], + "index": 12, + "bbox_fs": [ + 105, + 305, + 506, + 362 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 504, + 378 + ], + "score": 1.0, + "content": "Panigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "is best described as Gaussian at-least in the early phases of training. Comparing our results with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 388, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 507, + 401 + ], + "score": 1.0, + "content": "Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 409, + 482, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 482, + 423 + ], + "score": 1.0, + "content": "matters. Because SGN on some layers/networks is more Gaussian than other layers/networks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 366, + 507, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "The isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameter-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "dependent and anisotropic SGN brings a large improvement over existing parameter-independent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 461, + 494 + ], + "score": 1.0, + "content": "parameter-independent anisotropic heavy-tailed noise, when the batch size is too small", + "type": "text" + }, + { + "bbox": [ + 462, + 482, + 491, + 493 + ], + "score": 0.83, + "content": "( B \\sim 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 356, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 356, + 505 + ], + "score": 1.0, + "content": "apply Central Limit Theorem. We will leave it as future work.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 427, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 237, + 522 + ], + "score": 1.0, + "content": "SGD Dynamics. Let us replace", + "type": "text" + }, + { + "bbox": [ + 237, + 512, + 244, + 522 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 509, + 258, + 522 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 258, + 510, + 268, + 520 + ], + "score": 0.77, + "content": "d t", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "as unit time. Then the continuous-time dynamics of SGD", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 520, + 273, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 273, + 533 + ], + "score": 1.0, + "content": "(Coffey & Kalmykov, 2012) is written as", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 509, + 505, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 547, + 376, + 573 + ], + "lines": [ + { + "bbox": [ + 234, + 547, + 376, + 573 + ], + "spans": [ + { + "bbox": [ + 234, + 547, + 376, + 573 + ], + "score": 0.94, + "content": "d \\theta = - \\frac { \\partial L ( \\theta ) } { \\partial \\theta } d t + [ 2 D ( \\theta ) ] ^ { \\frac { 1 } { 2 } } d W _ { t } ,", + "type": "interline_equation", + "image_path": "9941a935271e458ce4253fd77fc922fa780d0d1282c06cc73e97eabe4d867726.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 234, + 547, + 376, + 573 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 506, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 133, + 600 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 588, + 207, + 600 + ], + "score": 0.92, + "content": "d W _ { t } \\sim { \\mathcal { N } } ( 0 , I d t )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 588, + 225, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 587, + 288, + 601 + ], + "score": 0.94, + "content": "\\begin{array} { r } { D ( \\theta ) = \\frac { \\eta } { 2 } C ( \\theta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 588, + 425, + 600 + ], + "score": 1.0, + "content": ". 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The pret", + "type": "text" + }, + { + "bbox": [ + 247, + 189, + 303, + 205 + ], + "score": 0.94, + "content": "\\begin{array} { r } { C ( \\theta ) = \\frac { H ( \\theta ) } { B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 190, + 506, + 217 + ], + "score": 1.0, + "content": "by using three-layer fully-connected network ons are usually near critical points, while randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 391, + 228 + ], + "score": 1.0, + "content": "Initialized Models are far from critical points. We display all elements", + "type": "text" + }, + { + "bbox": [ + 392, + 214, + 478, + 227 + ], + "score": 0.92, + "content": "H _ { ( i , j ) } \\in [ 1 e - 4 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 213, + 506, + 228 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 304, + 239 + ], + "score": 1.0, + "content": "Hessian matrix and the corresponding elements", + "type": "text" + }, + { + "bbox": [ + 305, + 225, + 328, + 238 + ], + "score": 0.92, + "content": "C _ { ( i , j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "of gradient noise covariance matrix in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "space spanned by the eigenvectors of Hessian. Another supplementary experiment on Avila Dataset", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 103, + 245, + 507, + 262 + ], + "spans": [ + { + "bbox": [ + 103, + 245, + 308, + 262 + ], + "score": 1.0, + "content": "(De Stefano et al., 2018) in Appendix C reports", + "type": "text" + }, + { + "bbox": [ + 309, + 247, + 384, + 261 + ], + "score": 0.92, + "content": "\\hat { C } _ { a v i l a } \\approx 1 . 0 0 4 \\frac { H } { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 245, + 507, + 262 + ], + "score": 1.0, + "content": ". The small difference factor", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "between the empirical result and the ideal Equation is mainly because the pretrained network is not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 271, + 248, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 248, + 282 + ], + "score": 1.0, + "content": "perfectly located at a critical point.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 504, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 348, + 315 + ], + "score": 1.0, + "content": "The next question is how to formulate the SGN covariance", + "type": "text" + }, + { + "bbox": [ + 349, + 302, + 370, + 315 + ], + "score": 0.91, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "for SGD? Based on Smith & Le", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 314, + 294, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 294, + 325 + ], + "score": 1.0, + "content": "(2018), we can express the SGN covariance as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 329, + 507, + 369 + ], + "lines": [ + { + "bbox": [ + 111, + 329, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 111, + 329, + 507, + 369 + ], + "score": 0.92, + "content": "\\boldsymbol { \\Sigma } ( \\theta ) = \\frac { 1 } { B } \\left[ \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } - \\boldsymbol { \\nabla } L ( \\theta ) \\boldsymbol { \\nabla } L ( \\theta ) ^ { \\top } \\right] \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } .", + "type": "interline_equation", + "image_path": "b9d79fb860e473b985959cf631bc674c905731483c5fe79f22f112928253e4b8.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 111, + 329, + 507, + 342.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 342.3333333333333, + 507, + 355.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 111, + 355.66666666666663, + 507, + 368.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "The approximation is true near critical points, due to the fact that the gradient noise variance dominates", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "the gradient mean near critical points. We know the observed fisher information matrix satisfies", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 406, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 407, + 174, + 419 + ], + "score": 0.92, + "content": "\\operatorname { F I M } ( \\theta ) \\approx H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 406, + 507, + 420 + ], + "score": 1.0, + "content": "near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrz˛ebski et al.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 418, + 252, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 252, + 431 + ], + "score": 1.0, + "content": "(2017); Zhu et al. (2019), we obtain", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 433, + 439, + 468 + ], + "lines": [ + { + "bbox": [ + 171, + 433, + 439, + 468 + ], + "spans": [ + { + "bbox": [ + 171, + 433, + 439, + 468 + ], + "score": 0.95, + "content": "C ( \\theta ) \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\nabla L ( \\theta , x _ { j } ) \\nabla L ( \\theta , x _ { j } ) ^ { \\top } = \\frac { 1 } { B } \\mathrm { F I M } ( \\theta ) \\approx \\frac { 1 } { B } H ( \\theta ) ,", + "type": "interline_equation", + "image_path": "484eebf70737b8e39684e093ac75f159d42ec466e794097965f95b4e72b7c815.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 171, + 433, + 439, + 444.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 171, + 444.6666666666667, + 439, + 456.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 171, + 456.33333333333337, + 439, + 468.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 216, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 216, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 216, + 487 + ], + "score": 1.0, + "content": "which approximately gives", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 488, + 364, + 511 + ], + "lines": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "spans": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "score": 0.93, + "content": "D ( \\theta ) = \\frac { \\eta } { 2 } C ( \\theta ) = \\frac { \\eta } { 2 B } H ( \\theta )", + "type": "interline_equation", + "image_path": "bca581a28a061cac16042fb3570f808fa0d6498893fcb26a8494b5ab75188d0a.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 104, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 326, + 528 + ], + "score": 1.0, + "content": "near minima. It indicates that the SGN covariance", + "type": "text" + }, + { + "bbox": [ + 326, + 515, + 348, + 528 + ], + "score": 0.92, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "is approximately proportional to the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 143, + 540 + ], + "score": 1.0, + "content": "Hessian", + "type": "text" + }, + { + "bbox": [ + 143, + 527, + 165, + 538 + ], + "score": 0.88, + "content": "H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 525, + 295, + 540 + ], + "score": 1.0, + "content": "and inverse to the batch size", + "type": "text" + }, + { + "bbox": [ + 295, + 527, + 304, + 537 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 525, + 505, + 540 + ], + "score": 1.0, + "content": ". Obviously, we can generalize Equation 7 by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 538, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 234, + 553 + ], + "score": 0.93, + "content": "\\begin{array} { r } { D ( \\theta ) ~ = ~ \\frac { \\eta C ( \\theta ) } { 2 } ~ = ~ \\frac { \\eta } { 2 B } [ H ( \\theta ) ] ^ { + } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 539, + 493, + 553 + ], + "score": 1.0, + "content": "near critical points, when there exist negative eigenvalues in", + "type": "text" + }, + { + "bbox": [ + 493, + 540, + 504, + 550 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 238, + 565 + ], + "score": 1.0, + "content": "along some directions. We use", + "type": "text" + }, + { + "bbox": [ + 239, + 552, + 254, + 565 + ], + "score": 0.9, + "content": "[ \\cdot ] ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 552, + 507, + 565 + ], + "score": 1.0, + "content": "to denote the positive semidefinite transformation of a sym-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 323, + 576 + ], + "score": 1.0, + "content": "metric matrix: if we have the eigendecomposation", + "type": "text" + }, + { + "bbox": [ + 323, + 563, + 479, + 576 + ], + "score": 0.89, + "content": "H = U \\mathrm { d i a g } ( H _ { 1 } , \\cdot \\cdot \\cdot , H _ { n - 1 } , H _ { n } ) U ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 563, + 505, + 576 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 574, + 291, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 288, + 587 + ], + "score": 0.89, + "content": "[ H ] ^ { + } = U \\mathrm { d i a g } ( | H _ { 1 } | , \\cdots , | H _ { n - 1 } | , | H _ { n } | ) U ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 574, + 291, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a follow-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "up paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "the relation still approximately holds for even the randomly network, which is far from critical points.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "The correlation is especially high along the flat directions with small-magnitude eigenvalues of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 645, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 660 + ], + "score": 1.0, + "content": "Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 449, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 449, + 670 + ], + "score": 1.0, + "content": "noise approximation all failed to capture this core relation in deep learning dynamics.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 257, + 697 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 259, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 259, + 699 + ], + "score": 1.0, + "content": "3 SGD DIFFUSION THEORY", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 278, + 733 + ], + "score": 1.0, + "content": "assume there are two valleys, Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 278, + 722, + 289, + 732 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 352, + 733 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 352, + 722, + 363, + 732 + ], + "score": 0.84, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", seen in Figure 3. Also Col b is the", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 92, + 494, + 177 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 92, + 494, + 177 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 92, + 494, + 177 + ], + "spans": [ + { + "bbox": [ + 113, + 92, + 494, + 177 + ], + "score": 0.966, + "type": "image", + "image_path": "084198cb7082c38ce670ef6ab05aa9b80fc9eb6c2d5851d7717d8d63c4a5ddfd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 92, + 494, + 120.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 120.33333333333333, + 494, + 148.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 148.66666666666666, + 494, + 177.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 189, + 506, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 189, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 247, + 217 + ], + "score": 1.0, + "content": "Figure 2: We empirically verified MNIST (LeCun, 1998). The pret", + "type": "text" + }, + { + "bbox": [ + 247, + 189, + 303, + 205 + ], + "score": 0.94, + "content": "\\begin{array} { r } { C ( \\theta ) = \\frac { H ( \\theta ) } { B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 190, + 506, + 217 + ], + "score": 1.0, + "content": "by using three-layer fully-connected network ons are usually near critical points, while randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 391, + 228 + ], + "score": 1.0, + "content": "Initialized Models are far from critical points. We display all elements", + "type": "text" + }, + { + "bbox": [ + 392, + 214, + 478, + 227 + ], + "score": 0.92, + "content": "H _ { ( i , j ) } \\in [ 1 e - 4 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 213, + 506, + 228 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 304, + 239 + ], + "score": 1.0, + "content": "Hessian matrix and the corresponding elements", + "type": "text" + }, + { + "bbox": [ + 305, + 225, + 328, + 238 + ], + "score": 0.92, + "content": "C _ { ( i , j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "of gradient noise covariance matrix in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "space spanned by the eigenvectors of Hessian. Another supplementary experiment on Avila Dataset", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 103, + 245, + 507, + 262 + ], + "spans": [ + { + "bbox": [ + 103, + 245, + 308, + 262 + ], + "score": 1.0, + "content": "(De Stefano et al., 2018) in Appendix C reports", + "type": "text" + }, + { + "bbox": [ + 309, + 247, + 384, + 261 + ], + "score": 0.92, + "content": "\\hat { C } _ { a v i l a } \\approx 1 . 0 0 4 \\frac { H } { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 245, + 507, + 262 + ], + "score": 1.0, + "content": ". The small difference factor", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "between the empirical result and the ideal Equation is mainly because the pretrained network is not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 271, + 248, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 248, + 282 + ], + "score": 1.0, + "content": "perfectly located at a critical point.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 504, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 348, + 315 + ], + "score": 1.0, + "content": "The next question is how to formulate the SGN covariance", + "type": "text" + }, + { + "bbox": [ + 349, + 302, + 370, + 315 + ], + "score": 0.91, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "for SGD? Based on Smith & Le", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 314, + 294, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 294, + 325 + ], + "score": 1.0, + "content": "(2018), we can express the SGN covariance as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 302, + 505, + 325 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 329, + 507, + 369 + ], + "lines": [ + { + "bbox": [ + 111, + 329, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 111, + 329, + 507, + 369 + ], + "score": 0.92, + "content": "\\boldsymbol { \\Sigma } ( \\theta ) = \\frac { 1 } { B } \\left[ \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } - \\boldsymbol { \\nabla } L ( \\theta ) \\boldsymbol { \\nabla } L ( \\theta ) ^ { \\top } \\right] \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) \\boldsymbol { \\nabla } L ( \\theta , x _ { j } ) ^ { \\top } .", + "type": "interline_equation", + "image_path": "b9d79fb860e473b985959cf631bc674c905731483c5fe79f22f112928253e4b8.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 111, + 329, + 507, + 342.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 342.3333333333333, + 507, + 355.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 111, + 355.66666666666663, + 507, + 368.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "The approximation is true near critical points, due to the fact that the gradient noise variance dominates", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "the gradient mean near critical points. We know the observed fisher information matrix satisfies", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 406, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 407, + 174, + 419 + ], + "score": 0.92, + "content": "\\operatorname { F I M } ( \\theta ) \\approx H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 406, + 507, + 420 + ], + "score": 1.0, + "content": "near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrz˛ebski et al.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 418, + 252, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 252, + 431 + ], + "score": 1.0, + "content": "(2017); Zhu et al. (2019), we obtain", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 385, + 507, + 431 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 433, + 439, + 468 + ], + "lines": [ + { + "bbox": [ + 171, + 433, + 439, + 468 + ], + "spans": [ + { + "bbox": [ + 171, + 433, + 439, + 468 + ], + "score": 0.95, + "content": "C ( \\theta ) \\approx \\frac { 1 } { B m } \\sum _ { j = 1 } ^ { m } \\nabla L ( \\theta , x _ { j } ) \\nabla L ( \\theta , x _ { j } ) ^ { \\top } = \\frac { 1 } { B } \\mathrm { F I M } ( \\theta ) \\approx \\frac { 1 } { B } H ( \\theta ) ,", + "type": "interline_equation", + "image_path": "484eebf70737b8e39684e093ac75f159d42ec466e794097965f95b4e72b7c815.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 171, + 433, + 439, + 444.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 171, + 444.6666666666667, + 439, + 456.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 171, + 456.33333333333337, + 439, + 468.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 216, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 216, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 216, + 487 + ], + "score": 1.0, + "content": "which approximately gives", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 471, + 216, + 487 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 488, + 364, + 511 + ], + "lines": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "spans": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "score": 0.93, + "content": "D ( \\theta ) = \\frac { \\eta } { 2 } C ( \\theta ) = \\frac { \\eta } { 2 B } H ( \\theta )", + "type": "interline_equation", + "image_path": "bca581a28a061cac16042fb3570f808fa0d6498893fcb26a8494b5ab75188d0a.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 247, + 488, + 364, + 511 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 104, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 326, + 528 + ], + "score": 1.0, + "content": "near minima. It indicates that the SGN covariance", + "type": "text" + }, + { + "bbox": [ + 326, + 515, + 348, + 528 + ], + "score": 0.92, + "content": "C ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "is approximately proportional to the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 143, + 540 + ], + "score": 1.0, + "content": "Hessian", + "type": "text" + }, + { + "bbox": [ + 143, + 527, + 165, + 538 + ], + "score": 0.88, + "content": "H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 525, + 295, + 540 + ], + "score": 1.0, + "content": "and inverse to the batch size", + "type": "text" + }, + { + "bbox": [ + 295, + 527, + 304, + 537 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 525, + 505, + 540 + ], + "score": 1.0, + "content": ". Obviously, we can generalize Equation 7 by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 538, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 234, + 553 + ], + "score": 0.93, + "content": "\\begin{array} { r } { D ( \\theta ) ~ = ~ \\frac { \\eta C ( \\theta ) } { 2 } ~ = ~ \\frac { \\eta } { 2 B } [ H ( \\theta ) ] ^ { + } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 539, + 493, + 553 + ], + "score": 1.0, + "content": "near critical points, when there exist negative eigenvalues in", + "type": "text" + }, + { + "bbox": [ + 493, + 540, + 504, + 550 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 238, + 565 + ], + "score": 1.0, + "content": "along some directions. We use", + "type": "text" + }, + { + "bbox": [ + 239, + 552, + 254, + 565 + ], + "score": 0.9, + "content": "[ \\cdot ] ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 552, + 507, + 565 + ], + "score": 1.0, + "content": "to denote the positive semidefinite transformation of a sym-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 323, + 576 + ], + "score": 1.0, + "content": "metric matrix: if we have the eigendecomposation", + "type": "text" + }, + { + "bbox": [ + 323, + 563, + 479, + 576 + ], + "score": 0.89, + "content": "H = U \\mathrm { d i a g } ( H _ { 1 } , \\cdot \\cdot \\cdot , H _ { n - 1 } , H _ { n } ) U ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 563, + 505, + 576 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 574, + 291, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 288, + 587 + ], + "score": 0.89, + "content": "[ H ] ^ { + } = U \\mathrm { d i a g } ( | H _ { 1 } | , \\cdots , | H _ { n - 1 } | , | H _ { n } | ) U ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 574, + 291, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 515, + 507, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a follow-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "up paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "the relation still approximately holds for even the randomly network, which is far from critical points.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "The correlation is especially high along the flat directions with small-magnitude eigenvalues of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 645, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 660 + ], + "score": 1.0, + "content": "Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 449, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 449, + 670 + ], + "score": 1.0, + "content": "noise approximation all failed to capture this core relation in deep learning dynamics.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 591, + 506, + 670 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 257, + 697 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 259, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 259, + 699 + ], + "score": 1.0, + "content": "3 SGD DIFFUSION THEORY", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 278, + 733 + ], + "score": 1.0, + "content": "assume there are two valleys, Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 278, + 722, + 289, + 732 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 352, + 733 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 352, + 722, + 363, + 732 + ], + "score": 0.84, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", seen in Figure 3. Also Col b is the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "boundary between two valleys. What is the mean escape time for a particle governed by Equation 2", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 223, + 281 + ], + "score": 1.0, + "content": "to escape from Sharp Valley", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 223, + 271, + 235, + 280 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 235, + 269, + 293, + 281 + ], + "score": 1.0, + "content": "to Flat Valley", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 293, + 270, + 304, + 280 + ], + "score": 0.72, + "content": "a _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 305, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "? The mean escape time is widely used in related", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 280, + 442, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 442, + 292 + ], + "score": 1.0, + "content": "statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019).", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 188, + 96, + 422, + 202 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 188, + 96, + 422, + 202 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 96, + 422, + 202 + ], + "spans": [ + { + "bbox": [ + 188, + 96, + 422, + 202 + ], + "score": 0.959, + "type": "image", + "image_path": "73689560624558dfdf080463c0fd2eec0e7953c645f73e038906c281f99c7fba.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 188, + 96, + 422, + 111.14285714285714 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 188, + 111.14285714285714, + 422, + 126.28571428571428 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 188, + 126.28571428571428, + 422, + 141.42857142857142 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 188, + 141.42857142857142, + 422, + 156.57142857142856 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 188, + 156.57142857142856, + 422, + 171.7142857142857 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 188, + 171.7142857142857, + 422, + 186.85714285714283 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 188, + 186.85714285714283, + 422, + 201.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 214, + 503, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 249, + 227 + ], + "score": 1.0, + "content": "Figure 3: Kramers Escape Problem.", + "type": "text" + }, + { + "bbox": [ + 250, + 217, + 261, + 226 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 214, + 278, + 227 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 279, + 217, + 290, + 226 + ], + "score": 0.86, + "content": " { \\boldsymbol { a } } _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 214, + 447, + 227 + ], + "score": 1.0, + "content": "are minima of two neighboring valleys.", + "type": "text" + }, + { + "bbox": [ + 448, + 215, + 453, + 225 + ], + "score": 0.6, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "is the saddle", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 225, + 361, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 238, + 240 + ], + "score": 1.0, + "content": "point separating the two valleys.", + "type": "text" + }, + { + "bbox": [ + 238, + 228, + 244, + 236 + ], + "score": 0.61, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 225, + 345, + 240 + ], + "score": 1.0, + "content": "locates outside of Valley", + "type": "text" + }, + { + "bbox": [ + 345, + 228, + 356, + 237 + ], + "score": 0.83, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 225, + 361, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 5.25 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "boundary between two valleys. What is the mean escape time for a particle governed by Equation 2", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 223, + 281 + ], + "score": 1.0, + "content": "to escape from Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 223, + 271, + 235, + 280 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 269, + 293, + 281 + ], + "score": 1.0, + "content": "to Flat Valley", + "type": "text" + }, + { + "bbox": [ + 293, + 270, + 304, + 280 + ], + "score": 0.72, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "? The mean escape time is widely used in related", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 280, + 442, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 442, + 292 + ], + "score": 1.0, + "content": "statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "Gauss’s Divergence Theorem (Arfken & Weber, 1999; Lipschutz et al., 2009) states that the surface", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "integral of a vector field over a closed surface, which is called the flux through the surface, is equal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "to the volume integral of the divergence over the region inside the surface. We respectively denote", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 205, + 343 + ], + "score": 1.0, + "content": "the mean escape time as", + "type": "text" + }, + { + "bbox": [ + 205, + 331, + 212, + 339 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 329, + 288, + 343 + ], + "score": 1.0, + "content": ", the escape rate as", + "type": "text" + }, + { + "bbox": [ + 288, + 331, + 295, + 341 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 329, + 419, + 343 + ], + "score": 1.0, + "content": ", and the probability current as", + "type": "text" + }, + { + "bbox": [ + 419, + 330, + 426, + 339 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 505, + 343 + ], + "score": 1.0, + "content": ". 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We have", + "type": "text" + }, + { + "bbox": [ + 252, + 476, + 277, + 487 + ], + "score": 0.91, + "content": "j = J", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 476, + 434, + 489 + ], + "score": 1.0, + "content": "in the case of one-dimensional escape.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 507, + 506 + ], + "score": 1.0, + "content": "Classical Assumptions. We state three classical assumptions first for the density diffusion theory.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "score": 1.0, + "content": "Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 524, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "score": 1.0, + "content": "and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "(Jastrz˛ebski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 568, + 150, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 150, + 582 + ], + "score": 1.0, + "content": "Problems.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 503, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "Assumption 1 (The Second Order Taylor Approximation). The loss function around critical points", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 595, + 249, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 596, + 117, + 605 + ], + "score": 0.84, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 595, + 249, + 609 + ], + "score": 1.0, + "content": "can be approximately written as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 612, + 431, + 636 + ], + "lines": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "spans": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "score": 0.93, + "content": "L ( \\theta ) = L ( \\theta ^ { \\star } ) + g ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) + \\frac { 1 } { 2 } ( \\theta - \\theta ^ { \\star } ) ^ { \\top } H ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) .", + "type": "interline_equation", + "image_path": "51182f3a03e686be47ce14ef11732d6ba605ccb76a1cc4c2d7b85c9425862cd8.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "score": 1.0, + "content": "Assumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 656, + 501, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 501, + 668 + ], + "score": 1.0, + "content": "Assumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "We will dive into these two assumptions deeper than previous papers for SGD dynamics. 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We respectively denote", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 205, + 343 + ], + "score": 1.0, + "content": "the mean escape time as", + "type": "text" + }, + { + "bbox": [ + 205, + 331, + 212, + 339 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 329, + 288, + 343 + ], + "score": 1.0, + "content": ", the escape rate as", + "type": "text" + }, + { + "bbox": [ + 288, + 331, + 295, + 341 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 329, + 419, + 343 + ], + "score": 1.0, + "content": ", and the probability current as", + "type": "text" + }, + { + "bbox": [ + 419, + 330, + 426, + 339 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 505, + 343 + ], + "score": 1.0, + "content": ". We apply Gauss’s", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 340, + 365, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 365, + 354 + ], + "score": 1.0, + "content": "Divergence Theorem to the Fokker-Planck Equation resulting in", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 297, + 505, + 354 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 357, + 447, + 382 + ], + "lines": [ + { + "bbox": [ + 164, + 357, + 447, + 382 + ], + "spans": [ + { + "bbox": [ + 164, + 357, + 447, + 382 + ], + "score": 0.93, + "content": "\\nabla \\cdot \\left[ P ( \\theta , t ) \\nabla L ( \\theta ) \\right] + \\nabla \\cdot \\nabla D ( \\theta ) P ( \\theta , t ) = \\frac { \\partial P ( \\theta , t ) } { \\partial t } = - \\nabla \\cdot J ( \\theta , t ) .", + "type": "interline_equation", + "image_path": "81ae893b80fe266236bb47f161db168952995549551e12a490ee4ee02a12db9d.jpg" + } + ] + } + ], + "index": 17, + 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+ "content": ". We have", + "type": "text" + }, + { + "bbox": [ + 252, + 476, + 277, + 487 + ], + "score": 0.91, + "content": "j = J", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 476, + 434, + 489 + ], + "score": 1.0, + "content": "in the case of one-dimensional escape.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 437, + 505, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 507, + 506 + ], + "score": 1.0, + "content": "Classical Assumptions. We state three classical assumptions first for the density diffusion theory.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "score": 1.0, + "content": "Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 524, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "score": 1.0, + "content": "and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "(Jastrz˛ebski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 568, + 150, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 150, + 582 + ], + "score": 1.0, + "content": "Problems.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 492, + 507, + 582 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 503, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "Assumption 1 (The Second Order Taylor Approximation). The loss function around critical points", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 595, + 249, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 596, + 117, + 605 + ], + "score": 0.84, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 595, + 249, + 609 + ], + "score": 1.0, + "content": "can be approximately written as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 584, + 505, + 609 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 612, + 431, + 636 + ], + "lines": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "spans": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "score": 0.93, + "content": "L ( \\theta ) = L ( \\theta ^ { \\star } ) + g ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) + \\frac { 1 } { 2 } ( \\theta - \\theta ^ { \\star } ) ^ { \\top } H ( \\theta ^ { \\star } ) ( \\theta - \\theta ^ { \\star } ) .", + "type": "interline_equation", + "image_path": "51182f3a03e686be47ce14ef11732d6ba605ccb76a1cc4c2d7b85c9425862cd8.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 179, + 612, + 431, + 636 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "list", + "bbox": [ + 106, + 640, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "score": 1.0, + "content": "Assumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima.", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 656, + 501, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 501, + 668 + ], + "score": 1.0, + "content": "Assumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature).", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 640, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "We will dive into these two assumptions deeper than previous papers for SGD dynamics. Assumptions", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "2 and 3 both mean that our diffusion theory can better describe the escape processes that cost more", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "iterations. As this class of “slow” escape processes takes main computational time compared with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "“fast” escape processes, this class of “slow” escape process is more interesting for training of deep", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "neural networks. Our empirical analysis in Section 4 supports that the escape processes in the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "score": 1.0, + "content": "wide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 482, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 482, + 106 + ], + "score": 1.0, + "content": "Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 676, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "score": 1.0, + "content": "wide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 482, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 482, + 106 + ], + "score": 1.0, + "content": "Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "Escape paths. We generalize the concept of critical points into critical paths as the path where 1)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "the gradient perpendicular to the path direction must be zero, and 2) the second order directional", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "for escaping must be critical paths. The most possible escape direction at one point must be the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 177 + ], + "score": 1.0, + "content": "far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 390, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 390, + 221 + ], + "score": 1.0, + "content": "which are strictly flat, is ignorable according to our following analysis.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 506, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 507, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 250, + 239 + ], + "score": 1.0, + "content": "The boundary between Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 250, + 228, + 262, + 237 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 226, + 324, + 239 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 324, + 228, + 335, + 237 + ], + "score": 0.84, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 226, + 410, + 239 + ], + "score": 1.0, + "content": "is the saddle point", + "type": "text" + }, + { + "bbox": [ + 410, + 227, + 415, + 236 + ], + "score": 0.44, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 226, + 480, + 239 + ], + "score": 1.0, + "content": ". The Hessian at", + "type": "text" + }, + { + "bbox": [ + 480, + 227, + 486, + 236 + ], + "score": 0.6, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 226, + 489, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 489, + 226, + 502, + 237 + ], + "score": 0.79, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 226, + 507, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "must have only one negative eigenvalue and the corresponding eigenvector is the escape direction.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 480, + 261 + ], + "score": 1.0, + "content": "Without losing generality, we first assume that there is only one most possible path through", + "type": "text" + }, + { + "bbox": [ + 481, + 248, + 504, + 258 + ], + "score": 0.46, + "content": "\\operatorname { C o l } b", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 320, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 229, + 273 + ], + "score": 1.0, + "content": "existing between Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 230, + 261, + 240, + 270 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 258, + 304, + 273 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 304, + 261, + 315, + 270 + ], + "score": 0.83, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 258, + 320, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "SGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 377, + 299 + ], + "score": 1.0, + "content": "are interested in SGLD, when the injected noise dominates SGN as", + "type": "text" + }, + { + "bbox": [ + 377, + 287, + 404, + 298 + ], + "score": 0.92, + "content": "\\eta 0", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 287, + 506, + 299 + ], + "score": 1.0, + "content": "in final epochs. Because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 321 + ], + "score": 1.0, + "content": "usually simplified as Gradient Descent with injected white noise, whose behavior is identical to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "score": 1.0, + "content": "Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 274, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 274, + 354 + ], + "score": 1.0, + "content": "mixture of injected white noise and SGN.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 345, + 369 + ], + "score": 1.0, + "content": "Theorem 3.1 (SGLD Escapes Minima). The loss function", + "type": "text" + }, + { + "bbox": [ + 346, + 356, + 366, + 368 + ], + "score": 0.9, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 355, + 411, + 369 + ], + "score": 1.0, + "content": "is of class", + "type": "text" + }, + { + "bbox": [ + 411, + 356, + 424, + 366 + ], + "score": 0.88, + "content": "C ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 355, + 443, + 369 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 443, + 358, + 450, + 366 + ], + "score": 0.64, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "-dimensional.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 192, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 192, + 402 + ], + "score": 1.0, + "content": "outside of Valley a is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 399, + 399, + 432 + ], + "lines": [ + { + "bbox": [ + 211, + 399, + 399, + 432 + ], + "spans": [ + { + "bbox": [ + 211, + 399, + 399, + 432 + ], + "score": 0.95, + "content": "\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\sqrt { \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\frac { \\Delta L } { D } \\right) .", + "type": "interline_equation", + "image_path": "425a3e09bd66d9bf92cb9a14d67ccbb4cc6bbbd38570a307c9f37b27183c2f35.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 399, + 399, + 415.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 211, + 415.5, + 399, + 432.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 169, + 446 + ], + "score": 1.0, + "content": "We denote that", + "type": "text" + }, + { + "bbox": [ + 169, + 435, + 183, + 445 + ], + "score": 0.88, + "content": "H _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 434, + 203, + 446 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 435, + 216, + 445 + ], + "score": 0.87, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "are the Hessians of the loss function at the minimum a and the saddle", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 130, + 457 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 131, + 446, + 135, + 455 + ], + "score": 0.5, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 445, + 140, + 457 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 140, + 446, + 220, + 457 + ], + "score": 0.88, + "content": "\\Delta L = L ( b ) - L ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 445, + 477, + 457 + ], + "score": 1.0, + "content": "is the loss barrier height, e indicates the escape direction, and", + "type": "text" + }, + { + "bbox": [ + 478, + 446, + 494, + 456 + ], + "score": 0.87, + "content": "H _ { b e }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 226, + 468 + ], + "score": 1.0, + "content": "the eigenvalue of the Hessian", + "type": "text" + }, + { + "bbox": [ + 226, + 457, + 239, + 467 + ], + "score": 0.86, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 456, + 494, + 468 + ], + "score": 1.0, + "content": "corresponding to the escape direction. The diffusion coefficient", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 504, + 466 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 467, + 218, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 218, + 479 + ], + "score": 1.0, + "content": "is usually set to 1 in SGLD.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "SGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 509, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 281, + 527 + ], + "score": 1.0, + "content": "is far from the Gibs-Boltzmann distribution,", + "type": "text" + }, + { + "bbox": [ + 281, + 509, + 381, + 529 + ], + "score": 0.94, + "content": "\\begin{array} { r } { P ( \\theta ) = \\frac { 1 } { Z } \\exp \\left( - \\frac { L ( \\theta ) } { D } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 511, + 505, + 526 + ], + "score": 1.0, + "content": ". These different characteristics", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "make SGD diffusion behave differently from known physical dynamical systems and much less", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 321, + 573 + ], + "score": 1.0, + "content": "and injected white noise, as long as the eigenvectors of", + "type": "text" + }, + { + "bbox": [ + 322, + 560, + 343, + 572 + ], + "score": 0.91, + "content": "D ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "are closely aligned with the eigenvectors", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 145, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 117, + 584 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 571, + 140, + 583 + ], + "score": 0.91, + "content": "H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 570, + 145, + 584 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 506, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 343, + 599 + ], + "score": 1.0, + "content": "Theorem 3.2 (SGD Escapes Minima). The loss function", + "type": "text" + }, + { + "bbox": [ + 343, + 586, + 363, + 598 + ], + "score": 0.91, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 585, + 410, + 599 + ], + "score": 1.0, + "content": "is of class", + "type": "text" + }, + { + "bbox": [ + 410, + 585, + 423, + 596 + ], + "score": 0.88, + "content": "C ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 585, + 443, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 588, + 450, + 596 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 585, + 507, + 599 + ], + "score": 1.0, + "content": "-dimensional.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 418, + 609 + ], + "score": 1.0, + "content": "Only one most possible path exists between Valley a and the outside of Valley", + "type": "text" + }, + { + "bbox": [ + 418, + 599, + 424, + 607 + ], + "score": 0.43, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 597, + 487, + 609 + ], + "score": 1.0, + "content": ". 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Each escape path contributes to the total escape rate. Multiple paths combined", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "together have a total escape rate. If there are multiple parallel from the start valley to the end valley,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "we can compute the total escape rate easily based on the following computation rule. The computation", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 507, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "Escape paths. We generalize the concept of critical points into critical paths as the path where 1)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "the gradient perpendicular to the path direction must be zero, and 2) the second order directional", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "for escaping must be critical paths. The most possible escape direction at one point must be the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 177 + ], + "score": 1.0, + "content": "far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 390, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 390, + 221 + ], + "score": 1.0, + "content": "which are strictly flat, is ignorable according to our following analysis.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 111, + 506, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 506, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 507, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 250, + 239 + ], + "score": 1.0, + "content": "The boundary between Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 250, + 228, + 262, + 237 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 226, + 324, + 239 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 324, + 228, + 335, + 237 + ], + "score": 0.84, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 226, + 410, + 239 + ], + "score": 1.0, + "content": "is the saddle point", + "type": "text" + }, + { + "bbox": [ + 410, + 227, + 415, + 236 + ], + "score": 0.44, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 226, + 480, + 239 + ], + "score": 1.0, + "content": ". The Hessian at", + "type": "text" + }, + { + "bbox": [ + 480, + 227, + 486, + 236 + ], + "score": 0.6, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 226, + 489, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 489, + 226, + 502, + 237 + ], + "score": 0.79, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 226, + 507, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "must have only one negative eigenvalue and the corresponding eigenvector is the escape direction.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 480, + 261 + ], + "score": 1.0, + "content": "Without losing generality, we first assume that there is only one most possible path through", + "type": "text" + }, + { + "bbox": [ + 481, + 248, + 504, + 258 + ], + "score": 0.46, + "content": "\\operatorname { C o l } b", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 320, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 229, + 273 + ], + "score": 1.0, + "content": "existing between Sharp Valley", + "type": "text" + }, + { + "bbox": [ + 230, + 261, + 240, + 270 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 258, + 304, + 273 + ], + "score": 1.0, + "content": "and Flat Valley", + "type": "text" + }, + { + "bbox": [ + 304, + 261, + 315, + 270 + ], + "score": 0.83, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 258, + 320, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 226, + 507, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "SGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 377, + 299 + ], + "score": 1.0, + "content": "are interested in SGLD, when the injected noise dominates SGN as", + "type": "text" + }, + { + "bbox": [ + 377, + 287, + 404, + 298 + ], + "score": 0.92, + "content": "\\eta 0", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 287, + 506, + 299 + ], + "score": 1.0, + "content": "in final epochs. Because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 321 + ], + "score": 1.0, + "content": "usually simplified as Gradient Descent with injected white noise, whose behavior is identical to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "score": 1.0, + "content": "Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 274, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 274, + 354 + ], + "score": 1.0, + "content": "mixture of injected white noise and SGN.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 275, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 345, + 369 + ], + "score": 1.0, + "content": "Theorem 3.1 (SGLD Escapes Minima). The loss function", + "type": "text" + }, + { + "bbox": [ + 346, + 356, + 366, + 368 + ], + "score": 0.9, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 355, + 411, + 369 + ], + "score": 1.0, + "content": "is of class", + "type": "text" + }, + { + "bbox": [ + 411, + 356, + 424, + 366 + ], + "score": 0.88, + "content": "C ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 355, + 443, + 369 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 443, + 358, + 450, + 366 + ], + "score": 0.64, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "-dimensional.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 192, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 192, + 402 + ], + "score": 1.0, + "content": "outside of Valley a is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 355, + 506, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 399, + 399, + 432 + ], + "lines": [ + { + "bbox": [ + 211, + 399, + 399, + 432 + ], + "spans": [ + { + "bbox": [ + 211, + 399, + 399, + 432 + ], + "score": 0.95, + "content": "\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\sqrt { \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\frac { \\Delta L } { D } \\right) .", + "type": "interline_equation", + "image_path": "425a3e09bd66d9bf92cb9a14d67ccbb4cc6bbbd38570a307c9f37b27183c2f35.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 399, + 399, + 415.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 211, + 415.5, + 399, + 432.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 169, + 446 + ], + "score": 1.0, + "content": "We denote that", + "type": "text" + }, + { + "bbox": [ + 169, + 435, + 183, + 445 + ], + "score": 0.88, + "content": "H _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 434, + 203, + 446 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 435, + 216, + 445 + ], + "score": 0.87, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "are the Hessians of the loss function at the minimum a and the saddle", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 130, + 457 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 131, + 446, + 135, + 455 + ], + "score": 0.5, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 445, + 140, + 457 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 140, + 446, + 220, + 457 + ], + "score": 0.88, + "content": "\\Delta L = L ( b ) - L ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 445, + 477, + 457 + ], + "score": 1.0, + "content": "is the loss barrier height, e indicates the escape direction, and", + "type": "text" + }, + { + "bbox": [ + 478, + 446, + 494, + 456 + ], + "score": 0.87, + "content": "H _ { b e }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 226, + 468 + ], + "score": 1.0, + "content": "the eigenvalue of the Hessian", + "type": "text" + }, + { + "bbox": [ + 226, + 457, + 239, + 467 + ], + "score": 0.86, + "content": "H _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 456, + 494, + 468 + ], + "score": 1.0, + "content": "corresponding to the escape direction. The diffusion coefficient", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 504, + 466 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 467, + 218, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 218, + 479 + ], + "score": 1.0, + "content": "is usually set to 1 in SGLD.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 434, + 506, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "SGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 509, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 281, + 527 + ], + "score": 1.0, + "content": "is far from the Gibs-Boltzmann distribution,", + "type": "text" + }, + { + "bbox": [ + 281, + 509, + 381, + 529 + ], + "score": 0.94, + "content": "\\begin{array} { r } { P ( \\theta ) = \\frac { 1 } { Z } \\exp \\left( - \\frac { L ( \\theta ) } { D } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 511, + 505, + 526 + ], + "score": 1.0, + "content": ". These different characteristics", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "make SGD diffusion behave differently from known physical dynamical systems and much less", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 321, + 573 + ], + "score": 1.0, + "content": "and injected white noise, as long as the eigenvectors of", + "type": "text" + }, + { + "bbox": [ + 322, + 560, + 343, + 572 + ], + "score": 0.91, + "content": "D ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "are closely aligned with the eigenvectors", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 145, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 117, + 584 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 571, + 140, + 583 + ], + "score": 0.91, + "content": "H ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 570, + 145, + 584 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 487, + 505, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 506, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 343, + 599 + ], + "score": 1.0, + "content": "Theorem 3.2 (SGD Escapes Minima). The loss function", + "type": "text" + }, + { + "bbox": [ + 343, + 586, + 363, + 598 + ], + "score": 0.91, + "content": "L ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 585, + 410, + 599 + ], + "score": 1.0, + "content": "is of class", + "type": "text" + }, + { + "bbox": [ + 410, + 585, + 423, + 596 + ], + "score": 0.88, + "content": "C ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 585, + 443, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 588, + 450, + 596 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 585, + 507, + 599 + ], + "score": 1.0, + "content": "-dimensional.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 418, + 609 + ], + "score": 1.0, + "content": "Only one most possible path exists between Valley a and the outside of Valley", + "type": "text" + }, + { + "bbox": [ + 418, + 599, + 424, + 607 + ], + "score": 0.43, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 597, + 487, + 609 + ], + "score": 1.0, + "content": ". If Assumption", + "type": "text" + }, + { + "bbox": [ + 487, + 599, + 492, + 606 + ], + "score": 0.37, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 597, + 506, + 609 + ], + "score": 1.0, + "content": ", 2,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 289, + 621 + ], + "score": 1.0, + "content": "and 3 hold, and the dynamics is governed by", + "type": "text" + }, + { + "bbox": [ + 290, + 608, + 311, + 618 + ], + "score": 0.33, + "content": "S G D", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 607, + 506, + 621 + ], + "score": 1.0, + "content": ", then the mean escape time from Valley a to the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 619, + 192, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 192, + 631 + ], + "score": 1.0, + "content": "outside of Valley a is", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 585, + 507, + 631 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 635, + 404, + 662 + ], + "lines": [ + { + "bbox": [ + 205, + 635, + 404, + 662 + ], + "spans": [ + { + "bbox": [ + 205, + 635, + 404, + 662 + ], + "score": 0.94, + "content": "\\tau = 2 \\pi \\frac { 1 } { | H _ { b e } | } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } \\left( \\frac { s } { H _ { a e } } + \\frac { ( 1 - s ) } { | H _ { b e } | } \\right) \\right] ,", + "type": "interline_equation", + "image_path": "ec8a128a9aa153b1c19ff5fa460949aff35a421d555e99c40ce59e45380110bc.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 635, + 404, + 648.5 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 205, + 648.5, + 404, + 662.0 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 503, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 133, + 680 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 668, + 173, + 679 + ], + "score": 0.91, + "content": "s \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 667, + 317, + 680 + ], + "score": 1.0, + "content": "is a path-dependent parameter, and", + "type": "text" + }, + { + "bbox": [ + 318, + 668, + 335, + 678 + ], + "score": 0.89, + "content": "H _ { a e }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 667, + 354, + 680 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 668, + 371, + 678 + ], + "score": 0.9, + "content": "H _ { b e }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "are, respectively, the eigenvalues", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 495, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 326, + 691 + ], + "score": 1.0, + "content": "of the Hessians at the minimum a and the saddle point", + "type": "text" + }, + { + "bbox": [ + 326, + 680, + 331, + 688 + ], + "score": 0.66, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 677, + 495, + 691 + ], + "score": 1.0, + "content": "corresponding to the escape direction e.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 667, + 505, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Multiple-path escape. Each escape path contributes to the total escape rate. Multiple paths combined", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "together have a total escape rate. If there are multiple parallel from the start valley to the end valley,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "we can compute the total escape rate easily based on the following computation rule. The computation", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "rule is based on the fact that the probability flux integrals are additive. We can easily generalize the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 281, + 503, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 496, + 296 + ], + "score": 1.0, + "content": "mean escape time analysis into the cases that there are multiple parallel escape paths indexed by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 497, + 284, + 503, + 294 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "dimensionality directly, higher dimensionality may increase the number of escape paths and loss", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 327, + 306, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 306, + 338 + ], + "score": 1.0, + "content": "valleys, and change the spectrum of the Hessians.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 50, + "bbox_fs": [ + 106, + 699, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 121, + 92, + 488, + 204 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 121, + 92, + 488, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 92, + 488, + 204 + ], + "spans": [ + { + "bbox": [ + 121, + 92, + 488, + 204 + ], + "score": 0.966, + "type": "image", + "image_path": "40850afd82cda33fe28c5aa92bf1019fad755c726e83948318f7a64db61eaf3d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 121, + 92, + 488, + 129.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 121, + 129.33333333333334, + 488, + 166.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 121, + 166.66666666666669, + 488, + 204.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 216, + 506, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Figure 4: The mean escape time analysis of SGD by using Styblinski-Tang Function. The Pearson", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "Correlation is higher than 0.99. Left Column: Sharpness. Middle Column: Batch Size. Right Column:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 168, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 168, + 252 + ], + "score": 1.0, + "content": "Learning Rate.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "rule is based on the fact that the probability flux integrals are additive. We can easily generalize the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 281, + 503, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 496, + 296 + ], + "score": 1.0, + "content": "mean escape time analysis into the cases that there are multiple parallel escape paths indexed by", + "type": "text" + }, + { + "bbox": [ + 497, + 284, + 503, + 294 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "dimensionality directly, higher dimensionality may increase the number of escape paths and loss", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 327, + 306, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 306, + 338 + ], + "score": 1.0, + "content": "valleys, and change the spectrum of the Hessians.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 105, + 340, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 104, + 335, + 503, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 439, + 357 + ], + "score": 1.0, + "content": "Rule 1. If there are multiple MPPs between the start valley and the end valley, then", + "type": "text" + }, + { + "bbox": [ + 439, + 340, + 503, + 355 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\gamma _ { t o t a l } = \\sum _ { p } \\gamma _ { p } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 362, + 502, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "Thus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 374, + 207, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 207, + 387 + ], + "score": 1.0, + "content": "analyze the escape rates.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 507, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 507, + 403 + ], + "score": 1.0, + "content": "Minima selection. Now, we may formulate the probability of minima selection as Proposition 1.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "the landscape contain many good modes and bad modes. SGD transits from one mode to another", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "mode during training. The mean escape time of one mode corresponds to the number of iterations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "which SGD spends on this mode during training, which is naturally proportional to the probability of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 444, + 242, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 242, + 459 + ], + "score": 1.0, + "content": "selecting this mode after training.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "Proposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 450, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 450, + 484 + ], + "score": 1.0, + "content": "Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 488, + 350, + 513 + ], + "lines": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "spans": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "score": 0.94, + "content": "P ( \\theta \\in V _ { a } ) = \\frac { \\tau _ { a } } { \\sum _ { v } \\tau _ { v } } ,", + "type": "interline_equation", + "image_path": "ed10cf147b3b58efa64c529c4d2fec7e55a1a844d052c80b1fed9db1d4901ac2.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 503, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 245, + 532 + ], + "score": 1.0, + "content": "where v is the index of valleys, and", + "type": "text" + }, + { + "bbox": [ + 245, + 519, + 255, + 529 + ], + "score": 0.86, + "content": "\\tau _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 515, + 496, + 532 + ], + "score": 1.0, + "content": "is the mean escape time from Valley v to the outside of Valley", + "type": "text" + }, + { + "bbox": [ + 497, + 520, + 503, + 528 + ], + "score": 0.41, + "content": "v", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 545, + 240, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 241, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 241, + 560 + ], + "score": 1.0, + "content": "4 EMPIRICAL ANALYSIS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "In this section, we try to directly validate the escape formulas on real-world datasets. Each escape", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 592, + 443, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 443, + 605 + ], + "score": 1.0, + "content": "times under various gradient noise scales, batch sizes, learning rates, and sharpness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "How to compare the escape rates under the same settings with various minima sharpness? Our√", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 263, + 635 + ], + "score": 1.0, + "content": "method is to multiply a rescaling factor", + "type": "text" + }, + { + "bbox": [ + 263, + 621, + 278, + 633 + ], + "score": 0.91, + "content": "\\sqrt { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "to each parameter, and the Hessian will be proportionally√", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 185, + 647 + ], + "score": 1.0, + "content": "rescaled by a factor", + "type": "text" + }, + { + "bbox": [ + 186, + 635, + 192, + 644 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 634, + 232, + 647 + ], + "score": 1.0, + "content": ". If we let", + "type": "text" + }, + { + "bbox": [ + 232, + 633, + 365, + 646 + ], + "score": 0.93, + "content": "L ( \\theta ) = f ( \\theta ) L ( \\theta ) = f ( { \\sqrt { k } } \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 634, + 388, + 647 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 388, + 633, + 506, + 646 + ], + "score": 0.92, + "content": "H ( \\theta ) = \\nabla ^ { 2 } f ( \\theta ) \\to H ( \\theta ) =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 144, + 657 + ], + "score": 0.92, + "content": "k \\nabla ^ { 2 } f ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 644, + 222, + 658 + ], + "score": 1.0, + "content": ". Thus, we can use", + "type": "text" + }, + { + "bbox": [ + 222, + 646, + 229, + 655 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "to indicate the minima sharpness. The theoretical relations of SGD", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 278, + 669 + ], + "score": 1.0, + "content": "we try to validate can be formulated as:", + "type": "text" + }, + { + "bbox": [ + 279, + 656, + 370, + 669 + ], + "score": 0.86, + "content": "( 1 ) - \\log ( \\gamma ) = \\bar { \\mathcal { O } } ( \\textstyle { \\frac { 1 } { k } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 657, + 378, + 669 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 379, + 656, + 468, + 669 + ], + "score": 0.82, + "content": "2 ) - \\log ( \\gamma ) = \\mathcal { O } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 657, + 506, + 669 + ], + "score": 1.0, + "content": ", and (3)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 185, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 180, + 684 + ], + "score": 0.91, + "content": "- \\log ( \\gamma ) = \\mathcal { O } ( \\textstyle { \\frac { 1 } { \\eta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 666, + 185, + 685 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The mean escape time analysis of SGD. Styblinski-Tang Function, which has multiple minima and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "saddle points, is a common test function for nonconvex optimization. We conduct an intuitional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "10-dimensional experiment, where the simulations start from a given minimum and terminate when", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "reaching the boundary of the loss valley. 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The Pearson", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "Correlation is higher than 0.99. Left Column: Sharpness. Middle Column: Batch Size. Right Column:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 168, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 168, + 252 + ], + "score": 1.0, + "content": "Learning Rate.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 337 + ], + "lines": [], + "index": 8.5, + "bbox_fs": [ + 104, + 270, + 506, + 338 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 105, + 340, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 104, + 335, + 503, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 439, + 357 + ], + "score": 1.0, + "content": "Rule 1. If there are multiple MPPs between the start valley and the end valley, then", + "type": "text" + }, + { + "bbox": [ + 439, + 340, + 503, + 355 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\gamma _ { t o t a l } = \\sum _ { p } \\gamma _ { p } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 335, + 503, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 362, + 502, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "Thus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 374, + 207, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 207, + 387 + ], + "score": 1.0, + "content": "analyze the escape rates.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 361, + 505, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 507, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 507, + 403 + ], + "score": 1.0, + "content": "Minima selection. Now, we may formulate the probability of minima selection as Proposition 1.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "the landscape contain many good modes and bad modes. SGD transits from one mode to another", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "mode during training. The mean escape time of one mode corresponds to the number of iterations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "which SGD spends on this mode during training, which is naturally proportional to the probability of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 444, + 242, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 242, + 459 + ], + "score": 1.0, + "content": "selecting this mode after training.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 390, + 507, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "Proposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 450, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 450, + 484 + ], + "score": 1.0, + "content": "Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 460, + 506, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 488, + 350, + 513 + ], + "lines": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "spans": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "score": 0.94, + "content": "P ( \\theta \\in V _ { a } ) = \\frac { \\tau _ { a } } { \\sum _ { v } \\tau _ { v } } ,", + "type": "interline_equation", + "image_path": "ed10cf147b3b58efa64c529c4d2fec7e55a1a844d052c80b1fed9db1d4901ac2.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 261, + 488, + 350, + 513 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 503, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 245, + 532 + ], + "score": 1.0, + "content": "where v is the index of valleys, and", + "type": "text" + }, + { + "bbox": [ + 245, + 519, + 255, + 529 + ], + "score": 0.86, + "content": "\\tau _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 515, + 496, + 532 + ], + "score": 1.0, + "content": "is the mean escape time from Valley v to the outside of Valley", + "type": "text" + }, + { + "bbox": [ + 497, + 520, + 503, + 528 + ], + "score": 0.41, + "content": "v", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 515, + 503, + 532 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 545, + 240, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 241, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 241, + 560 + ], + "score": 1.0, + "content": "4 EMPIRICAL ANALYSIS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "In this section, we try to directly validate the escape formulas on real-world datasets. Each escape", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 592, + 443, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 443, + 605 + ], + "score": 1.0, + "content": "times under various gradient noise scales, batch sizes, learning rates, and sharpness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 569, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "How to compare the escape rates under the same settings with various minima sharpness? 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We try to validate", + "type": "text" + }, + { + "bbox": [ + 355, + 687, + 396, + 699 + ], + "score": 0.93, + "content": "\\gamma = \\mathcal { O } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 686, + 414, + 702 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 687, + 490, + 701 + ], + "score": 0.9, + "content": "- \\log ( \\gamma ) = \\mathcal { O } ( \\frac { 1 } { D } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "SGLD (dominated by injected Gaussian noise). Figure 6 shows that SGLD only favors flat minima", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "polynomially more than sharp minima as Theorem 3.1 indicates. Figure 6 also verifies that the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 721, + 396, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 396, + 733 + ], + "score": 1.0, + "content": "injected gradient noise scale exponentially affects flat minima selection.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 686, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 190, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "SGD favors flat minima exponentially more than sharp minima. We can discover a few inter-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "esting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 433, + 141 + ], + "score": 1.0, + "content": "depends on the eigenvalue of the Hessian at minima along the escape direction,", + "type": "text" + }, + { + "bbox": [ + 433, + 129, + 451, + 140 + ], + "score": 0.9, + "content": "H _ { a e }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 128, + 506, + 141 + ], + "score": 1.0, + "content": ". Thus, SGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "comes from the exponential relation of the mean escape time and the minima sharpness. The measure", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 162, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 506, + 173 + ], + "score": 1.0, + "content": "of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness”", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "score": 1.0, + "content": "by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 178, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 178, + 217 + ], + "score": 1.0, + "content": "more than Adam.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "The ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "score": 1.0, + "content": "why large-batch training can easily get trapped near sharp minima, and increasing the learning rate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 265, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 265, + 506, + 280 + ], + "score": 1.0, + "content": "is large-batch training expects exponentially longer time to escape minima. Note that, as the mean", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "escape time in the theorems is equivalent to the product of the learning rate and the number of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "batch size and the learning rate. The practical computational time in large-batch training is usually", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "too short to search many enough flat minima. We conjecture that exponentially increasing training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 321, + 482, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 482, + 334 + ], + "score": 1.0, + "content": "iterations may be helpful for large batch training, while this is often too expensive in practice.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "Low dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of over-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "parametrized deep networks are close to zero, while only a small number of eigenvalues are large", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "score": 1.0, + "content": "(Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 472, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 472, + 393 + ], + "score": 1.0, + "content": "that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "As the escape rate exponentially depends the corresponding eigenvalues, a small number of large", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 507, + 422 + ], + "score": 1.0, + "content": "eigenvalues means that the process of minima selection mainly happens in the relatively low di-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "mensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 442, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 507, + 456 + ], + "score": 1.0, + "content": "hardly depends on those “meaningless” dimensions with small second order directional derivatives.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "This novel characteristic of SGD significantly reduces the explorable parameter space around one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 464, + 300, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 300, + 478 + ], + "score": 1.0, + "content": "minimum into a much lower dimensional space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "High-order effects. As we have applied the second-order Taylor approximation near critical points,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "paper. However, we also argue that the third-order effect is much smaller than the second-order effect", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "under the low temperature assumption in Kramers Escape Problems. We will leave the more refined", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 536, + 241, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 241, + 549 + ], + "score": 1.0, + "content": "high-order theory as future work.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 564, + 195, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 197, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 197, + 581 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "formulate the exponential relation of minima selection to the minima sharpness, the batch size, and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "not only helps us understand how SGD selects flat minima, but also will provide researchers a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 504, + 669 + ], + "score": 1.0, + "content": "powerful theoretical tool to analyze more learning behaviors and design better optimizers in future.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 218, + 696 + ], + "lines": [ + { + "bbox": [ + 107, + 685, + 220, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 685, + 220, + 699 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENT", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 761 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 190, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "SGD favors flat minima exponentially more than sharp minima. We can discover a few inter-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "esting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 433, + 141 + ], + "score": 1.0, + "content": "depends on the eigenvalue of the Hessian at minima along the escape direction,", + "type": "text" + }, + { + "bbox": [ + 433, + 129, + 451, + 140 + ], + "score": 0.9, + "content": "H _ { a e }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 128, + 506, + 141 + ], + "score": 1.0, + "content": ". Thus, SGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "comes from the exponential relation of the mean escape time and the minima sharpness. The measure", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 162, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 506, + 173 + ], + "score": 1.0, + "content": "of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness”", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "score": 1.0, + "content": "by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 178, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 178, + 217 + ], + "score": 1.0, + "content": "more than Adam.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 106, + 506, + 217 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "The ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "score": 1.0, + "content": "why large-batch training can easily get trapped near sharp minima, and increasing the learning rate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 265, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 265, + 506, + 280 + ], + "score": 1.0, + "content": "is large-batch training expects exponentially longer time to escape minima. Note that, as the mean", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "escape time in the theorems is equivalent to the product of the learning rate and the number of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "batch size and the learning rate. The practical computational time in large-batch training is usually", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "too short to search many enough flat minima. We conjecture that exponentially increasing training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 321, + 482, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 482, + 334 + ], + "score": 1.0, + "content": "iterations may be helpful for large batch training, while this is often too expensive in practice.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 221, + 506, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "Low dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of over-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "parametrized deep networks are close to zero, while only a small number of eigenvalues are large", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "score": 1.0, + "content": "(Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 472, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 472, + 393 + ], + "score": 1.0, + "content": "that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 337, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "As the escape rate exponentially depends the corresponding eigenvalues, a small number of large", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 507, + 422 + ], + "score": 1.0, + "content": "eigenvalues means that the process of minima selection mainly happens in the relatively low di-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "mensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 442, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 507, + 456 + ], + "score": 1.0, + "content": "hardly depends on those “meaningless” dimensions with small second order directional derivatives.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "This novel characteristic of SGD significantly reduces the explorable parameter space around one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 464, + 300, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 300, + 478 + ], + "score": 1.0, + "content": "minimum into a much lower dimensional space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 398, + 507, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "High-order effects. As we have applied the second-order Taylor approximation near critical points,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "paper. However, we also argue that the third-order effect is much smaller than the second-order effect", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "under the low temperature assumption in Kramers Escape Problems. We will leave the more refined", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 536, + 241, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 241, + 549 + ], + "score": 1.0, + "content": "high-order theory as future work.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 481, + 506, + 549 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 564, + 195, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 197, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 197, + 581 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "formulate the exponential relation of minima selection to the minima sharpness, the batch size, and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "not only helps us understand how SGD selects flat minima, but also will provide researchers a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 504, + 669 + ], + "score": 1.0, + "content": "powerful theoretical tool to analyze more learning behaviors and design better optimizers in future.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 590, + 506, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 218, + 696 + ], + "lines": [ + { + "bbox": [ + 107, + 685, + 220, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 685, + 220, + 699 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENT", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 106, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 112 + ], + "score": 1.0, + "content": "Alessandro Achille and Stefano Soatto. 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This proposition is a well known conclusion in statistical physics under Assumption 1, 2 and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "3. We still provide an intuitional proof here, and the following proof of SGD Diffusion will closely", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "relate to this proof. 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A hitting time analysis of stochastic gradient", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 115, + 146, + 435, + 158 + ], + "spans": [ + { + "bbox": [ + 115, + 146, + 435, + 158 + ], + "score": 1.0, + "content": "langevin dynamics. In Conference on Learning Theory, pp. 1980–2022, 2017b.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 106, + 135, + 505, + 158 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 503, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "Huan-Xiang Zhou. Rate theories for biologists. Quarterly reviews of biophysics, 43(2):219–293,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 176, + 143, + 190 + ], + "spans": [ + { + "bbox": [ + 114, + 176, + 143, + 190 + ], + "score": 1.0, + "content": "2010.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 164, + 505, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 506, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "Zhanxing Zhu, Jingfeng Wu, Bing Yu, Lei Wu, and Jinwen Ma. The anisotropic noise in stochastic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 208, + 507, + 221 + ], + "spans": [ + { + "bbox": [ + 115, + 208, + 507, + 221 + ], + "score": 1.0, + "content": "gradient descent: Its behavior of escaping from sharp minima and regularization effects. In ICML,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 114, + 219, + 207, + 230 + ], + "spans": [ + { + "bbox": [ + 114, + 219, + 207, + 230 + ], + "score": 1.0, + "content": "pp. 7654–7663, 2019.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 197, + 507, + 230 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 170, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 171, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 171, + 267 + ], + "score": 1.0, + "content": "A PROOFS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 277, + 238, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 239, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 239, + 290 + ], + "score": 1.0, + "content": "A.1 PROOF OF THEOREM 3.1", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "Proof. This proposition is a well known conclusion in statistical physics under Assumption 1, 2 and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "3. We still provide an intuitional proof here, and the following proof of SGD Diffusion will closely", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "relate to this proof. 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\\frac { \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ^ { + } ( \\theta - b ) } { T } \\right] d S } \\\\ & { = \\displaystyle J _ { b } \\frac { ( 2 \\pi T ) ^ { \\frac { n - 1 } { 2 } } } { ( \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } ) ^ { \\frac { 1 } { 2 } } } } \\end{array}", + "type": "interline_equation", + "image_path": "6d036e0291db65976eaa462fa7b09a7523085fe2d93ddea9ab3e7877252500ab.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 240, + 395, + 255.83333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 215, + 255.83333333333334, + 395, + 271.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 215, + 271.6666666666667, + 395, + 287.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 215, + 287.5, + 395, + 303.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 215, + 303.3333333333333, + 395, + 319.16666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 215, + 319.16666666666663, + 395, + 334.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 343, + 388, + 412 + ], + "lines": [ + { + "bbox": [ + 223, + 343, + 388, + 412 + ], + "spans": [ + { + "bbox": [ + 223, + 343, + 388, + 412 + ], + "score": 0.91, + "content": "\\begin{array} { c } { { \\tau = 2 \\pi \\sqrt { \\displaystyle \\frac { \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } } { \\operatorname * { d e t } ( H _ { a } ) | H _ { b e } | } } \\exp \\left( \\displaystyle \\frac { \\Delta L } { T } \\right) } } \\\\ { { = 2 \\pi \\sqrt { \\displaystyle \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\displaystyle \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\displaystyle \\frac { \\Delta L } { D } \\right) . } } \\end{array}", + "type": "interline_equation", + "image_path": "c1fff91c97fdb3a8ca6647cfd560bf9c72d6976c78eaa5818adf5bec6d53555c.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 343, + 388, + 360.25 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 223, + 360.25, + 388, + 377.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 223, + 377.5, + 388, + 394.75 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 223, + 394.75, + 388, + 412.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 239, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 240, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 240, + 447 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 3.2", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 506, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "Proof. 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\\frac { \\frac { 1 } { 2 } ( \\theta - b ) ^ { \\top } H _ { b } ^ { + } ( \\theta - b ) } { T } \\right] d S } \\\\ & { = \\displaystyle J _ { b } \\frac { ( 2 \\pi T ) ^ { \\frac { n - 1 } { 2 } } } { ( \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } ) ^ { \\frac { 1 } { 2 } } } } \\end{array}", + "type": "interline_equation", + "image_path": "6d036e0291db65976eaa462fa7b09a7523085fe2d93ddea9ab3e7877252500ab.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 240, + 395, + 255.83333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 215, + 255.83333333333334, + 395, + 271.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 215, + 271.6666666666667, + 395, + 287.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 215, + 287.5, + 395, + 303.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 215, + 303.3333333333333, + 395, + 319.16666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 215, + 319.16666666666663, + 395, + 334.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 343, + 388, + 412 + ], + "lines": [ + { + "bbox": [ + 223, + 343, + 388, + 412 + ], + "spans": [ + { + "bbox": [ + 223, + 343, + 388, + 412 + ], + "score": 0.91, + "content": "\\begin{array} { c } { { \\tau = 2 \\pi \\sqrt { \\displaystyle \\frac { \\prod _ { i = 1 } ^ { n - 1 } H _ { b i } } { \\operatorname * { d e t } ( H _ { a } ) | H _ { b e } | } } \\exp \\left( \\displaystyle \\frac { \\Delta L } { T } \\right) } } \\\\ { { = 2 \\pi \\sqrt { \\displaystyle \\frac { - \\operatorname * { d e t } ( H _ { b } ) } { \\operatorname * { d e t } ( H _ { a } ) } } \\displaystyle \\frac { 1 } { | H _ { b e } | } \\exp \\left( \\displaystyle \\frac { \\Delta L } { D } \\right) . } } \\end{array}", + "type": "interline_equation", + "image_path": "c1fff91c97fdb3a8ca6647cfd560bf9c72d6976c78eaa5818adf5bec6d53555c.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 343, + 388, + 360.25 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 223, + 360.25, + 388, + 377.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 223, + 377.5, + 388, + 394.75 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 223, + 394.75, + 388, + 412.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 239, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 240, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 240, + 447 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 3.2", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 506, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "Proof. 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So we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 98, + 431, + 132 + ], + "lines": [ + { + "bbox": [ + 179, + 98, + 431, + 132 + ], + "spans": [ + { + "bbox": [ + 179, + 98, + 431, + 132 + ], + "score": 0.94, + "content": "\\gamma = { \\frac { 1 } { 2 \\pi } } { \\sqrt { \\frac { \\operatorname* { d e t } ( H _ { a } D _ { a } ^ { - 1 } ) } { - \\operatorname* { d e t } ( H _ { b } D _ { b } ^ { - 1 } ) } } } | H _ { b e } | \\exp \\left( - { \\frac { s \\Delta L } { T _ { a } } } - { \\frac { ( 1 - s ) \\Delta L } { T _ { b } } } \\right)", + "type": "interline_equation", + "image_path": "c32f81dfe72beac8dbe890c674c7c9c15273beb55d6494fbab3bc2c420ecf544.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 179, + 98, + 431, + 109.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 179, + 109.33333333333333, + 431, + 120.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 179, + 120.66666666666666, + 431, + 132.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 135, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 118, + 148 + ], + "score": 0.86, + "content": "T _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 135, + 140, + 149 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 140, + 137, + 151, + 148 + ], + "score": 0.87, + "content": "T _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 135, + 251, + 149 + ], + "score": 1.0, + "content": "are the eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 252, + 136, + 285, + 148 + ], + "score": 0.91, + "content": "H _ { a } ^ { - 1 } D _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 135, + 307, + 149 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 136, + 340, + 149 + ], + "score": 0.93, + "content": "H _ { b } ^ { - 1 } D _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "corresponding to the escape direction.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 144, + 509, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 151, + 164 + ], + "score": 1.0, + "content": "We know", + "type": "text" + }, + { + "bbox": [ + 151, + 147, + 211, + 160 + ], + "score": 0.91, + "content": "\\begin{array} { r } { D _ { a } \\ = \\ \\frac { \\eta } { 2 B } \\mathbf { \\bar { { H } } } _ { a } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 144, + 234, + 164 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 148, + 305, + 161 + ], + "score": 0.89, + "content": "\\begin{array} { r } { D _ { b } ~ = ~ \\frac { \\eta } { 2 B } [ H _ { b } ] ^ { + } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 144, + 331, + 164 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 331, + 149, + 342, + 158 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 144, + 509, + 164 + ], + "score": 1.0, + "content": "must be positive semidefinite, we re-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 132, + 174 + ], + "score": 1.0, + "content": "place", + "type": "text" + }, + { + "bbox": [ + 133, + 161, + 312, + 173 + ], + "score": 0.9, + "content": "H _ { b } \\ = \\ U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , H _ { b e } ) U _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 159, + 464, + 174 + ], + "score": 1.0, + "content": "by its positive semidefinite analog", + "type": "text" + }, + { + "bbox": [ + 465, + 160, + 505, + 173 + ], + "score": 0.91, + "content": "[ H _ { b } ] ^ { + } =", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 325, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 259, + 187 + ], + "score": 0.89, + "content": "U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , | H _ { b e } | ) U _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 172, + 325, + 187 + ], + "score": 1.0, + "content": ". Thus, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 190, + 416, + 218 + ], + "lines": [ + { + "bbox": [ + 194, + 190, + 416, + 218 + ], + "spans": [ + { + "bbox": [ + 194, + 190, + 416, + 218 + ], + "score": 0.91, + "content": "\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\frac { 1 } { \\left| H _ { b e } \\right| } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } \\left( \\frac { s } { H _ { a e } } + \\frac { \\left( 1 - s \\right) } { \\left| H _ { b e } \\right| } \\right) \\right] .", + "type": "interline_equation", + "image_path": "116c047fddc9a5b54efad6278202602a010b6e07b36f48fec0e8864eca7ceae0.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 190, + 416, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 194, + 204.0, + 416, + 218.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 245, + 245, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 246, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 246, + 258 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 504, + 278 + ], + "score": 1.0, + "content": "Proof. A stationary distribution must have a balanced probability flux between valleys. So the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 308, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 308, + 290 + ], + "score": 1.0, + "content": "probability flux of each valley must be equivalent,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 291, + 370, + 306 + ], + "lines": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "spans": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "score": 0.92, + "content": "P ( \\theta \\in V _ { 1 } ) \\gamma _ { 1 2 } = P ( \\theta \\in V _ { 2 } ) \\gamma _ { 2 1 }", + "type": "interline_equation", + "image_path": "0d6154c64d3f28a5bdb3bef0601e88275d31f0a99e0f7fb390785924d7d51d1a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 119, + 324 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 119, + 311, + 155, + 323 + ], + "score": 0.92, + "content": "\\tau = \\gamma ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 309, + 200, + 324 + ], + "score": 1.0, + "content": ", it leads to", + "type": "text" + }, + { + "bbox": [ + 201, + 311, + 267, + 323 + ], + "score": 0.91, + "content": "P ( \\theta \\in V _ { v } ) \\propto \\tau _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 309, + 506, + 324 + ], + "score": 1.0, + "content": ". We normalize the total probability to 1, then we obtain the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 321, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 135, + 334 + ], + "score": 1.0, + "content": "result.", + "type": "text" + }, + { + "bbox": [ + 496, + 324, + 504, + 331 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 349, + 203, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 205, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 205, + 365 + ], + "score": 1.0, + "content": "B ASSUMPTIONS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 373, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 384, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 240, + 402 + ], + "score": 1.0, + "content": "near saddle points. It means that", + "type": "text" + }, + { + "bbox": [ + 240, + 385, + 352, + 401 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = - \\nabla \\cdot J ( \\theta , t ) \\approx 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 384, + 430, + 402 + ], + "score": 1.0, + "content": "holds near minima", + "type": "text" + }, + { + "bbox": [ + 430, + 389, + 441, + 398 + ], + "score": 0.84, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 384, + 459, + 402 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 389, + 470, + 398 + ], + "score": 0.83, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 384, + 507, + 402 + ], + "score": 1.0, + "content": ", but not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 250, + 410 + ], + "score": 1.0, + "content": "necessarily holds near saddle point", + "type": "text" + }, + { + "bbox": [ + 251, + 400, + 256, + 408 + ], + "score": 0.61, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 399, + 504, + 410 + ], + "score": 1.0, + "content": ". Quasi-Equilibrium Assumption is actually weaker but more", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 336, + 433 + ], + "score": 1.0, + "content": "et al., 2017). Under Assumption 2, the probability density", + "type": "text" + }, + { + "bbox": [ + 336, + 421, + 345, + 430 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "can behave like a stationary distribution", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "score": 1.0, + "content": "only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 365, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 365, + 476 + ], + "score": 1.0, + "content": "density motion can be obtained under the stationary assumption.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "Low Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jas-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 490, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 308, + 507 + ], + "score": 1.0, + "content": "trz˛ebski et al., 2017), and is always justified when", + "type": "text" + }, + { + "bbox": [ + 309, + 492, + 318, + 504 + ], + "score": 0.88, + "content": "\\frac { \\eta } { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 490, + 506, + 507 + ], + "score": 1.0, + "content": "is small. Under Assumption 3, the probability", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 393, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 393, + 537 + ], + "score": 1.0, + "content": "approximation, Assumption 1, even more reasonable in SGD diffusion.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 506, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Here, we try to provide a more intuitive explanation about Low Temperature Assumption in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 559, + 503, + 582 + ], + "spans": [ + { + "bbox": [ + 102, + 559, + 298, + 582 + ], + "score": 1.0, + "content": "temperature can be interpreted as a real number", + "type": "text" + }, + { + "bbox": [ + 298, + 564, + 308, + 573 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 559, + 458, + 582 + ], + "score": 1.0, + "content": ". In SGD, we have the temperature as", + "type": "text" + }, + { + "bbox": [ + 459, + 563, + 503, + 577 + ], + "score": 0.93, + "content": "\\begin{array} { r } { D = \\frac { \\eta } { 2 B } H } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 574, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 205, + 591 + ], + "score": 1.0, + "content": "In statistical physics, if", + "type": "text" + }, + { + "bbox": [ + 205, + 575, + 220, + 590 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { \\Delta L } { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 574, + 507, + 591 + ], + "score": 1.0, + "content": "is large, then we call it Low Temperature Approximation. Note that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 583, + 510, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 122, + 603 + ], + "score": 0.9, + "content": "\\scriptstyle { \\frac { \\Delta L } { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 583, + 510, + 609 + ], + "score": 1.0, + "content": "appears insides an exponential function in the theoretical analysis. People usually believe that,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 160, + 618 + ], + "score": 1.0, + "content": "numerically,", + "type": "text" + }, + { + "bbox": [ + 160, + 601, + 195, + 615 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\Delta L } { D } > 6 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 599, + 506, + 618 + ], + "score": 1.0, + "content": "can make a good approximation, for a similar reason of the 6-sigma rule in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 409, + 627 + ], + "score": 1.0, + "content": "statistics. In the final training phase of deep networks, a common setting is", + "type": "text" + }, + { + "bbox": [ + 410, + 614, + 447, + 626 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 614, + 465, + 627 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 466, + 614, + 503, + 624 + ], + "score": 0.9, + "content": "B = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 624, + 508, + 655 + ], + "spans": [ + { + "bbox": [ + 107, + 635, + 182, + 649 + ], + "score": 0.91, + "content": "\\textstyle { \\frac { \\Delta L } { H } } > 2 . 3 \\times \\mathbf { \\dot { 1 } } 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 624, + 296, + 655 + ], + "score": 1.0, + "content": "ly apply Assumption 3 to th. Empirically, the condition", + "type": "text" + }, + { + "bbox": [ + 297, + 635, + 371, + 648 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\Delta L } { H } > 2 . 3 \\times 1 0 ^ { - 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 624, + 508, + 655 + ], + "score": 1.0, + "content": "h satisfy the very mild conditionholds well in SGD dynamics. It", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 657, + 169, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 169, + 671 + ], + "score": 1.0, + "content": "barrier heights.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 684, + 369, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 683, + 370, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 370, + 699 + ], + "score": 1.0, + "content": "C THE STOCHASTIC GRADIENT NOISE ANALYSIS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 126, + 732 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 721, + 159, + 731 + ], + "score": 0.89, + "content": "B = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "on CIFAR-10. We also note that the SGN on ResNet seems less Gaussian than SGN on", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 221, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 496, + 222, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 496, + 222, + 504, + 232 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 455, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 457, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 133, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 82, + 153, + 95 + ], + "score": 0.92, + "content": "[ \\cdot ] ^ { \\perp e }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 81, + 398, + 96 + ], + "score": 1.0, + "content": "indicates the directions perpendicular to the escape direction", + "type": "text" + }, + { + "bbox": [ + 398, + 85, + 403, + 92 + ], + "score": 0.56, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 81, + 457, + 96 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 457, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 98, + 431, + 132 + ], + "lines": [ + { + "bbox": [ + 179, + 98, + 431, + 132 + ], + "spans": [ + { + "bbox": [ + 179, + 98, + 431, + 132 + ], + "score": 0.94, + "content": "\\gamma = { \\frac { 1 } { 2 \\pi } } { \\sqrt { \\frac { \\operatorname* { d e t } ( H _ { a } D _ { a } ^ { - 1 } ) } { - \\operatorname* { d e t } ( H _ { b } D _ { b } ^ { - 1 } ) } } } | H _ { b e } | \\exp \\left( - { \\frac { s \\Delta L } { T _ { a } } } - { \\frac { ( 1 - s ) \\Delta L } { T _ { b } } } \\right)", + "type": "interline_equation", + "image_path": "c32f81dfe72beac8dbe890c674c7c9c15273beb55d6494fbab3bc2c420ecf544.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 179, + 98, + 431, + 109.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 179, + 109.33333333333333, + 431, + 120.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 179, + 120.66666666666666, + 431, + 132.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 135, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 118, + 148 + ], + "score": 0.86, + "content": "T _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 135, + 140, + 149 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 140, + 137, + 151, + 148 + ], + "score": 0.87, + "content": "T _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 135, + 251, + 149 + ], + "score": 1.0, + "content": "are the eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 252, + 136, + 285, + 148 + ], + "score": 0.91, + "content": "H _ { a } ^ { - 1 } D _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 135, + 307, + 149 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 136, + 340, + 149 + ], + "score": 0.93, + "content": "H _ { b } ^ { - 1 } D _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "corresponding to the escape direction.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 144, + 509, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 151, + 164 + ], + "score": 1.0, + "content": "We know", + "type": "text" + }, + { + "bbox": [ + 151, + 147, + 211, + 160 + ], + "score": 0.91, + "content": "\\begin{array} { r } { D _ { a } \\ = \\ \\frac { \\eta } { 2 B } \\mathbf { \\bar { { H } } } _ { a } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 144, + 234, + 164 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 148, + 305, + 161 + ], + "score": 0.89, + "content": "\\begin{array} { r } { D _ { b } ~ = ~ \\frac { \\eta } { 2 B } [ H _ { b } ] ^ { + } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 144, + 331, + 164 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 331, + 149, + 342, + 158 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 144, + 509, + 164 + ], + "score": 1.0, + "content": "must be positive semidefinite, we re-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 132, + 174 + ], + "score": 1.0, + "content": "place", + "type": "text" + }, + { + "bbox": [ + 133, + 161, + 312, + 173 + ], + "score": 0.9, + "content": "H _ { b } \\ = \\ U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , H _ { b e } ) U _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 159, + 464, + 174 + ], + "score": 1.0, + "content": "by its positive semidefinite analog", + "type": "text" + }, + { + "bbox": [ + 465, + 160, + 505, + 173 + ], + "score": 0.91, + "content": "[ H _ { b } ] ^ { + } =", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 325, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 259, + 187 + ], + "score": 0.89, + "content": "U _ { b } ^ { \\top } d i a g ( H _ { b 1 } , \\cdot \\cdot \\cdot , H _ { b ( n - 1 ) } , | H _ { b e } | ) U _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 172, + 325, + 187 + ], + "score": 1.0, + "content": ". Thus, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 135, + 509, + 187 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 190, + 416, + 218 + ], + "lines": [ + { + "bbox": [ + 194, + 190, + 416, + 218 + ], + "spans": [ + { + "bbox": [ + 194, + 190, + 416, + 218 + ], + "score": 0.91, + "content": "\\tau = \\frac { 1 } { \\gamma } = 2 \\pi \\frac { 1 } { \\left| H _ { b e } \\right| } \\exp \\left[ \\frac { 2 B \\Delta L } { \\eta } \\left( \\frac { s } { H _ { a e } } + \\frac { \\left( 1 - s \\right) } { \\left| H _ { b e } \\right| } \\right) \\right] .", + "type": "interline_equation", + "image_path": "116c047fddc9a5b54efad6278202602a010b6e07b36f48fec0e8864eca7ceae0.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 190, + 416, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 194, + 204.0, + 416, + 218.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 245, + 245, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 246, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 246, + 258 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 504, + 278 + ], + "score": 1.0, + "content": "Proof. A stationary distribution must have a balanced probability flux between valleys. So the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 308, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 308, + 290 + ], + "score": 1.0, + "content": "probability flux of each valley must be equivalent,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 265, + 504, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 291, + 370, + 306 + ], + "lines": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "spans": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "score": 0.92, + "content": "P ( \\theta \\in V _ { 1 } ) \\gamma _ { 1 2 } = P ( \\theta \\in V _ { 2 } ) \\gamma _ { 2 1 }", + "type": "interline_equation", + "image_path": "0d6154c64d3f28a5bdb3bef0601e88275d31f0a99e0f7fb390785924d7d51d1a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 240, + 291, + 370, + 306 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 119, + 324 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 119, + 311, + 155, + 323 + ], + "score": 0.92, + "content": "\\tau = \\gamma ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 309, + 200, + 324 + ], + "score": 1.0, + "content": ", it leads to", + "type": "text" + }, + { + "bbox": [ + 201, + 311, + 267, + 323 + ], + "score": 0.91, + "content": "P ( \\theta \\in V _ { v } ) \\propto \\tau _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 309, + 506, + 324 + ], + "score": 1.0, + "content": ". We normalize the total probability to 1, then we obtain the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 321, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 135, + 334 + ], + "score": 1.0, + "content": "result.", + "type": "text" + }, + { + "bbox": [ + 496, + 324, + 504, + 331 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 309, + 506, + 334 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 349, + 203, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 205, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 205, + 365 + ], + "score": 1.0, + "content": "B ASSUMPTIONS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 373, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 384, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 240, + 402 + ], + "score": 1.0, + "content": "near saddle points. It means that", + "type": "text" + }, + { + "bbox": [ + 240, + 385, + 352, + 401 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial P ( \\theta , t ) } { \\partial t } = - \\nabla \\cdot J ( \\theta , t ) \\approx 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 384, + 430, + 402 + ], + "score": 1.0, + "content": "holds near minima", + "type": "text" + }, + { + "bbox": [ + 430, + 389, + 441, + 398 + ], + "score": 0.84, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 384, + 459, + 402 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 389, + 470, + 398 + ], + "score": 0.83, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 384, + 507, + 402 + ], + "score": 1.0, + "content": ", but not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 250, + 410 + ], + "score": 1.0, + "content": "necessarily holds near saddle point", + "type": "text" + }, + { + "bbox": [ + 251, + 400, + 256, + 408 + ], + "score": 0.61, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 399, + 504, + 410 + ], + "score": 1.0, + "content": ". Quasi-Equilibrium Assumption is actually weaker but more", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 336, + 433 + ], + "score": 1.0, + "content": "et al., 2017). Under Assumption 2, the probability density", + "type": "text" + }, + { + "bbox": [ + 336, + 421, + 345, + 430 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "can behave like a stationary distribution", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "score": 1.0, + "content": "only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 365, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 365, + 476 + ], + "score": 1.0, + "content": "density motion can be obtained under the stationary assumption.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 374, + 507, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "Low Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jas-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 490, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 308, + 507 + ], + "score": 1.0, + "content": "trz˛ebski et al., 2017), and is always justified when", + "type": "text" + }, + { + "bbox": [ + 309, + 492, + 318, + 504 + ], + "score": 0.88, + "content": "\\frac { \\eta } { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 490, + 506, + 507 + ], + "score": 1.0, + "content": "is small. Under Assumption 3, the probability", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 393, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 393, + 537 + ], + "score": 1.0, + "content": "approximation, Assumption 1, even more reasonable in SGD diffusion.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 481, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 506, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Here, we try to provide a more intuitive explanation about Low Temperature Assumption in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 559, + 503, + 582 + ], + "spans": [ + { + "bbox": [ + 102, + 559, + 298, + 582 + ], + "score": 1.0, + "content": "temperature can be interpreted as a real number", + "type": "text" + }, + { + "bbox": [ + 298, + 564, + 308, + 573 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 559, + 458, + 582 + ], + "score": 1.0, + "content": ". In SGD, we have the temperature as", + "type": "text" + }, + { + "bbox": [ + 459, + 563, + 503, + 577 + ], + "score": 0.93, + "content": "\\begin{array} { r } { D = \\frac { \\eta } { 2 B } H } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 574, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 205, + 591 + ], + "score": 1.0, + "content": "In statistical physics, if", + "type": "text" + }, + { + "bbox": [ + 205, + 575, + 220, + 590 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { \\Delta L } { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 574, + 507, + 591 + ], + "score": 1.0, + "content": "is large, then we call it Low Temperature Approximation. Note that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 583, + 510, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 122, + 603 + ], + "score": 0.9, + "content": "\\scriptstyle { \\frac { \\Delta L } { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 583, + 510, + 609 + ], + "score": 1.0, + "content": "appears insides an exponential function in the theoretical analysis. People usually believe that,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 160, + 618 + ], + "score": 1.0, + "content": "numerically,", + "type": "text" + }, + { + "bbox": [ + 160, + 601, + 195, + 615 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\Delta L } { D } > 6 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 599, + 506, + 618 + ], + "score": 1.0, + "content": "can make a good approximation, for a similar reason of the 6-sigma rule in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 409, + 627 + ], + "score": 1.0, + "content": "statistics. In the final training phase of deep networks, a common setting is", + "type": "text" + }, + { + "bbox": [ + 410, + 614, + 447, + 626 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 614, + 465, + 627 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 466, + 614, + 503, + 624 + ], + "score": 0.9, + "content": "B = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 624, + 508, + 655 + ], + "spans": [ + { + "bbox": [ + 107, + 635, + 182, + 649 + ], + "score": 0.91, + "content": "\\textstyle { \\frac { \\Delta L } { H } } > 2 . 3 \\times \\mathbf { \\dot { 1 } } 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 624, + 296, + 655 + ], + "score": 1.0, + "content": "ly apply Assumption 3 to th. Empirically, the condition", + "type": "text" + }, + { + "bbox": [ + 297, + 635, + 371, + 648 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\Delta L } { H } > 2 . 3 \\times 1 0 ^ { - 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 624, + 508, + 655 + ], + "score": 1.0, + "content": "h satisfy the very mild conditionholds well in SGD dynamics. It", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 657, + 169, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 169, + 671 + ], + "score": 1.0, + "content": "barrier heights.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 102, + 541, + 510, + 671 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 684, + 369, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 683, + 370, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 370, + 699 + ], + "score": 1.0, + "content": "C THE STOCHASTIC GRADIENT NOISE ANALYSIS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 126, + 732 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 721, + 159, + 731 + ], + "score": 0.89, + "content": "B = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "on CIFAR-10. 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The relationcan still be observed in these two cases.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 334, + 138, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 138, + 348 + ], + "score": 0.9, + "content": "\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "Data Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 451, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 451, + 375 + ], + "score": 1.0, + "content": "MNIST. We leave the preprocessing of Avila in D. 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We display all elements", + "type": "text" + }, + { + "bbox": [ + 234, + 238, + 321, + 251 + ], + "score": 0.93, + "content": "H _ { ( i , j ) } \\in [ 1 e - 4 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 236, + 505, + 253 + ], + "score": 1.0, + "content": "of the Hessian matrix and the corresponding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 248, + 502, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 502, + 261 + ], + "score": 1.0, + "content": "elements in gradient noise covariance matrix in the space spanned by the eigenvectors of Hessians.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 105, + 280, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "fully-connected networks with the same batch size. Panigrahi et al. (2019) presented more results on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 292, + 307, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 307, + 303 + ], + "score": 1.0, + "content": "the Gaussianity of SGN under various conditions.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 279, + 505, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 104, + 306, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 204, + 324 + ], + "score": 1.0, + "content": "By Figure 8, we validate", + "type": "text" + }, + { + "bbox": [ + 205, + 307, + 236, + 321 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 306, + 506, + 324 + ], + "score": 1.0, + "content": "in the original coordinates on MNIST. By Figure 9, we also validate", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 319, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 138, + 334 + ], + "score": 0.89, + "content": "\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 319, + 505, + 348 + ], + "score": 1.0, + "content": "on another dataset, Avila, in the space spanned by the eigenvectors of Hessian. The relationcan still be observed in these two cases.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 334, + 138, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 138, + 348 + ], + "score": 0.9, + "content": "\\begin{array} { r } { C = \\frac { H } { B } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 306, + 506, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "Data Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 451, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 451, + 375 + ], + "score": 1.0, + "content": "MNIST. We leave the preprocessing of Avila in D. Model: Fully-connected networks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 351, + 505, + 375 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 390, + 236, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 237, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 237, + 404 + ], + "score": 1.0, + "content": "D MAIN EXPERIMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 504, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 504, + 427 + ], + "score": 1.0, + "content": "Figure 10, 11, and 12 respectively validate that the exponential relation of the escape rate with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 425, + 286, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 286, + 438 + ], + "score": 1.0, + "content": "Hessian, the batch size and the learning rate.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 414, + 504, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 451, + 246, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 248, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 248, + 464 + ], + "score": 1.0, + "content": "D.1 EXPERIMENTAL SETTINGS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 504, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "Datasets: a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 482, + 152, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 152, + 495 + ], + "score": 1.0, + "content": "Diagnosis.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 471, + 505, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "Data Precessing: We perform per-pixel zero-mean and unit-variance normalization on input data.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "For simplicity, we also transform multi-class problems into binary-class problems by grouping labels,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 520, + 223, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 223, + 535 + ], + "score": 1.0, + "content": "although this is unnecessary.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 499, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 502, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 504, + 551 + ], + "score": 1.0, + "content": "Model: Two-layer fully-connected networks with one hidden layer and 10 neurons per hidden layer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 537, + 504, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 502, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "score": 1.0, + "content": "Initializations: To ensure the initialized models are near minima, we first pretrain models with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "200-1000 epochs to fit each data set as well as possible. 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In our experiments, we set each dimension’s distance from", + "type": "text" + }, + { + "bbox": [ + 425, + 605, + 445, + 617 + ], + "score": 0.91, + "content": "\\theta _ { t = 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "should be less", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 181, + 628 + ], + "score": 1.0, + "content": "than 0.05, namely", + "type": "text" + }, + { + "bbox": [ + 181, + 616, + 235, + 628 + ], + "score": 0.92, + "content": "| \\Delta \\theta _ { i } | \\le 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 616, + 315, + 628 + ], + "score": 1.0, + "content": "for each dimension", + "type": "text" + }, + { + "bbox": [ + 315, + 617, + 320, + 626 + ], + "score": 0.46, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 616, + 479, + 628 + ], + "score": 1.0, + "content": ". 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The artificially initialized parameters avoids the stochastic property", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 185, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 185, + 280 + ], + "score": 1.0, + "content": "of the initial states.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Experiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex opti-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 296, + 189, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 189, + 309 + ], + "score": 1.0, + "content": "mization, written as", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 311, + 372, + 344 + ], + "lines": [ + { + "bbox": [ + 238, + 311, + 372, + 344 + ], + "spans": [ + { + "bbox": [ + 238, + 311, + 372, + 344 + ], + "score": 0.94, + "content": "f ( \\theta ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } ( \\theta _ { i } ^ { 4 } - 1 6 \\theta _ { i } ^ { 2 } + 5 \\theta _ { i } ) .", + "type": "interline_equation", + "image_path": "598af83511621d4f023c24c4739d7ba92c27423cac94fb900677543fd19c00fe.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 311, + 372, + 327.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 238, + 327.5, + 372, + 344.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 504, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "score": 1.0, + "content": "We use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 162, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 162, + 372 + ], + "score": 1.0, + "content": "training data.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 375, + 342, + 389 + ], + "lines": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "spans": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "score": 0.9, + "content": "L ( \\theta ) = f ( \\theta - x ) ,", + "type": "interline_equation", + "image_path": "65ad8b59461d7d4c73eccd54ecbf3361f263788d4b864f6b2fed2aade26f59c7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 189, + 406 + ], + "score": 1.0, + "content": "where data samples", + "type": "text" + }, + { + "bbox": [ + 189, + 394, + 242, + 406 + ], + "score": 0.92, + "content": "x \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 394, + 505, + 406 + ], + "score": 1.0, + "content": ". The one-dimensional Styblinski-Tang Function has one global", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 191, + 417 + ], + "score": 1.0, + "content": "minimum located at", + "type": "text" + }, + { + "bbox": [ + 192, + 406, + 259, + 416 + ], + "score": 0.86, + "content": "a = - 2 . 9 0 3 5 3 4", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 405, + 388, + 417 + ], + "score": 1.0, + "content": ", one local minimum located at", + "type": "text" + }, + { + "bbox": [ + 388, + 406, + 395, + 415 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 405, + 487, + 417 + ], + "score": 1.0, + "content": ", and one saddle point", + "type": "text" + }, + { + "bbox": [ + 487, + 405, + 505, + 416 + ], + "score": 0.85, + "content": "b =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 284, + 429 + ], + "score": 1.0, + "content": "0.156731 as the boundary separating Valley", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 295, + 427 + ], + "score": 0.82, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 415, + 340, + 429 + ], + "score": 1.0, + "content": "and Valley", + "type": "text" + }, + { + "bbox": [ + 341, + 417, + 352, + 427 + ], + "score": 0.81, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 415, + 379, + 429 + ], + "score": 1.0, + "content": ". For a", + "type": "text" + }, + { + "bbox": [ + 379, + 417, + 385, + 426 + ], + "score": 0.49, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "-dimensional Styblinski-Tang", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 424, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 103, + 424, + 249, + 444 + ], + "score": 1.0, + "content": "Function, we initialize parameters", + "type": "text" + }, + { + "bbox": [ + 249, + 427, + 419, + 442 + ], + "score": 0.91, + "content": "\\theta _ { t = 0 } = \\textstyle { \\frac { 1 } { \\sqrt { k } } } ( - 2 . 9 0 3 5 3 4 , \\cdot \\cdot \\cdot , - 2 . 9 0 3 5 3 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 424, + 507, + 444 + ], + "score": 1.0, + "content": ", and set the valley’s", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 102, + 437, + 509, + 462 + ], + "spans": [ + { + "bbox": [ + 102, + 437, + 158, + 462 + ], + "score": 1.0, + "content": "boundary as", + "type": "text" + }, + { + "bbox": [ + 158, + 441, + 230, + 457 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\theta _ { i } < \\frac { 1 } { \\sqrt { k } } 0 . 1 5 6 7 3 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 437, + 261, + 462 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 262, + 443, + 266, + 452 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 437, + 509, + 462 + ], + "score": 1.0, + "content": "is the dimension index. We record the number of iterations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 456, + 480, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 480, + 467 + ], + "score": 1.0, + "content": "required to escape from the valley to the outside of valley. The setting 1 does not need labels.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "Experiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initial-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 140, + 496 + ], + "score": 1.0, + "content": "ization:", + "type": "text" + }, + { + "bbox": [ + 140, + 483, + 214, + 495 + ], + "score": 0.92, + "content": "\\theta _ { t = 0 } = ( 0 , \\cdot \\cdot \\cdot , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 482, + 292, + 496 + ], + "score": 1.0, + "content": ". Valley Boundary:", + "type": "text" + }, + { + "bbox": [ + 293, + 483, + 361, + 494 + ], + "score": 0.92, + "content": "- 0 . 1 < \\theta _ { i } < 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Due to the randomness of training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 516, + 396, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 396, + 529 + ], + "score": 1.0, + "content": "valley group, and study the mean escape time from the group of valleys.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 546 + ], + "score": 1.0, + "content": "Experiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 543, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 415, + 556 + ], + "score": 1.0, + "content": "losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization:", + "type": "text" + }, + { + "bbox": [ + 415, + 543, + 504, + 555 + ], + "score": 0.9, + "content": "\\theta _ { t = 0 } = ( 0 . 1 , \\cdot \\cdot \\cdot , 0 . 1 )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 223, + 567 + ], + "score": 1.0, + "content": "with a small Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 224, + 555, + 283, + 567 + ], + "score": 0.92, + "content": "\\epsilon = ( 0 , 0 . 0 1 I )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 554, + 364, + 567 + ], + "score": 1.0, + "content": ". Valley Boundary:", + "type": "text" + }, + { + "bbox": [ + 365, + 555, + 438, + 565 + ], + "score": 0.91, + "content": "0 . 0 5 < \\theta _ { i } < 0 . 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 554, + 506, + 567 + ], + "score": 1.0, + "content": ". To prevent the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "gradient disappearance problem of deep learning, we move the starting point from the origin. For", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 576, + 507, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 507, + 590 + ], + "score": 1.0, + "content": "symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "Due to the complex loss landscape of deep networks, we can hardly know the exact information", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "score": 1.0, + "content": "about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 621, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 632 + ], + "score": 1.0, + "content": "the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 631, + 431, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 431, + 644 + ], + "score": 1.0, + "content": "or not the propositions can hold with very-small-batch gradient noise in practice.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 107, + 656, + 236, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 236, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 236, + 668 + ], + "score": 1.0, + "content": "E.2 EXPERIMENTS RESULTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. Figure 16", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "shows the relation of the escape rate and the Hessian determinant in the dynamics of white noise.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Figure 17 shows the relation of the escape rate and the second order directional derivative in the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "dynamics of SGD. Figure 18 shows the relation of the escape rate and the batch size in the dynamics", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "of SGD. Figure 19 shows the relation of the escape rate and the learning rate in the dynamics of SGD.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 303, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 304, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 304, + 96 + ], + "score": 1.0, + "content": "E EXPERIMENTS ON MORE MODELS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "We supply experiments of training three models on artificial Gaussian datasets. In these experiments,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "we can analytically know the locations of the minima, Hessians and loss barriers, as each input feature", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 181, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 181, + 140 + ], + "score": 1.0, + "content": "is Gaussian noise.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 105, + 506, + 140 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 240, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 240, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 240, + 165 + ], + "score": 1.0, + "content": "E.1 EXPERIMENTS SETTINGS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 173, + 504, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "Data Set: We generate 50000 Gaussian samples and random two-class labels as the training data set,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 184, + 372, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 372, + 198 + ], + "score": 0.82, + "content": "\\{ ( x ^ { ( i ) } , y ^ { ( i ) } ) | x ^ { ( \\bar { i } ) } \\sim \\mathcal { N } ( 0 , I ) , y ^ { ( i ) } \\in \\{ 0 , 1 \\} , i \\stackrel { \\cdot } { \\in } \\{ 1 , 2 , \\cdot \\cdot , 5 0 0 0 0 \\} \\}", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 172, + 506, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 263, + 215 + ], + "score": 1.0, + "content": "Hyperparameters: In Figure 15: (a)", + "type": "text" + }, + { + "bbox": [ + 263, + 203, + 358, + 213 + ], + "score": 0.85, + "content": "\\eta = 0 . 0 0 0 1 , B = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 201, + 377, + 215 + ], + "score": 1.0, + "content": ", (b)", + "type": "text" + }, + { + "bbox": [ + 377, + 203, + 466, + 213 + ], + "score": 0.61, + "content": "\\eta = 0 . 0 0 1 , B = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 201, + 485, + 215 + ], + "score": 1.0, + "content": ", (c)", + "type": "text" + }, + { + "bbox": [ + 486, + 204, + 505, + 214 + ], + "score": 0.84, + "content": "\\eta =", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 138, + 226 + ], + "score": 1.0, + "content": "0.0003,", + "type": "text" + }, + { + "bbox": [ + 139, + 213, + 177, + 224 + ], + "score": 0.88, + "content": "B = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 212, + 254, + 226 + ], + "score": 1.0, + "content": ". In Figure 16: (a)", + "type": "text" + }, + { + "bbox": [ + 255, + 214, + 380, + 225 + ], + "score": 0.79, + "content": "\\eta = 0 . 0 0 0 1 , B = 5 0 , D = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 212, + 399, + 226 + ], + "score": 1.0, + "content": ", (b)", + "type": "text" + }, + { + "bbox": [ + 399, + 213, + 504, + 225 + ], + "score": 0.82, + "content": "\\eta = 0 . 0 0 1 , B = 5 0 , D =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 223, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 155, + 238 + ], + "score": 1.0, + "content": "0.0005, (c)", + "type": "text" + }, + { + "bbox": [ + 155, + 225, + 297, + 235 + ], + "score": 0.85, + "content": "\\eta = 0 . 0 0 0 3 , B = 1 , D = 0 . 0 0 0 3", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 223, + 379, + 238 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 165, + 259 + ], + "score": 1.0, + "content": "Figure 19: (a)", + "type": "text" + }, + { + "bbox": [ + 166, + 247, + 194, + 257 + ], + "score": 0.88, + "content": "B = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 245, + 212, + 259 + ], + "score": 1.0, + "content": ", (b)", + "type": "text" + }, + { + "bbox": [ + 212, + 247, + 240, + 257 + ], + "score": 0.89, + "content": "B = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 245, + 258, + 259 + ], + "score": 1.0, + "content": ", (c)", + "type": "text" + }, + { + "bbox": [ + 258, + 246, + 286, + 257 + ], + "score": 0.88, + "content": "B = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 245, + 506, + 259 + ], + "score": 1.0, + "content": ". We note that the hyperparameters are recommended", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "and needn’t be fine tuned again. The artificially initialized parameters avoids the stochastic property", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 185, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 185, + 280 + ], + "score": 1.0, + "content": "of the initial states.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 201, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Experiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex opti-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 296, + 189, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 189, + 309 + ], + "score": 1.0, + "content": "mization, written as", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 285, + 505, + 309 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 311, + 372, + 344 + ], + "lines": [ + { + "bbox": [ + 238, + 311, + 372, + 344 + ], + "spans": [ + { + "bbox": [ + 238, + 311, + 372, + 344 + ], + "score": 0.94, + "content": "f ( \\theta ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } ( \\theta _ { i } ^ { 4 } - 1 6 \\theta _ { i } ^ { 2 } + 5 \\theta _ { i } ) .", + "type": "interline_equation", + "image_path": "598af83511621d4f023c24c4739d7ba92c27423cac94fb900677543fd19c00fe.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 311, + 372, + 327.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 238, + 327.5, + 372, + 344.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 504, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "score": 1.0, + "content": "We use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 162, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 162, + 372 + ], + "score": 1.0, + "content": "training data.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 346, + 506, + 372 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 375, + 342, + 389 + ], + "lines": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "spans": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "score": 0.9, + "content": "L ( \\theta ) = f ( \\theta - x ) ,", + "type": "interline_equation", + "image_path": "65ad8b59461d7d4c73eccd54ecbf3361f263788d4b864f6b2fed2aade26f59c7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 268, + 375, + 342, + 389 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 189, + 406 + ], + "score": 1.0, + "content": "where data samples", + "type": "text" + }, + { + "bbox": [ + 189, + 394, + 242, + 406 + ], + "score": 0.92, + "content": "x \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 394, + 505, + 406 + ], + "score": 1.0, + "content": ". The one-dimensional Styblinski-Tang Function has one global", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 191, + 417 + ], + "score": 1.0, + "content": "minimum located at", + "type": "text" + }, + { + "bbox": [ + 192, + 406, + 259, + 416 + ], + "score": 0.86, + "content": "a = - 2 . 9 0 3 5 3 4", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 405, + 388, + 417 + ], + "score": 1.0, + "content": ", one local minimum located at", + "type": "text" + }, + { + "bbox": [ + 388, + 406, + 395, + 415 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 405, + 487, + 417 + ], + "score": 1.0, + "content": ", and one saddle point", + "type": "text" + }, + { + "bbox": [ + 487, + 405, + 505, + 416 + ], + "score": 0.85, + "content": "b =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 284, + 429 + ], + "score": 1.0, + "content": "0.156731 as the boundary separating Valley", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 295, + 427 + ], + "score": 0.82, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 415, + 340, + 429 + ], + "score": 1.0, + "content": "and Valley", + "type": "text" + }, + { + "bbox": [ + 341, + 417, + 352, + 427 + ], + "score": 0.81, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 415, + 379, + 429 + ], + "score": 1.0, + "content": ". For a", + "type": "text" + }, + { + "bbox": [ + 379, + 417, + 385, + 426 + ], + "score": 0.49, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "-dimensional Styblinski-Tang", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 424, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 103, + 424, + 249, + 444 + ], + "score": 1.0, + "content": "Function, we initialize parameters", + "type": "text" + }, + { + "bbox": [ + 249, + 427, + 419, + 442 + ], + "score": 0.91, + "content": "\\theta _ { t = 0 } = \\textstyle { \\frac { 1 } { \\sqrt { k } } } ( - 2 . 9 0 3 5 3 4 , \\cdot \\cdot \\cdot , - 2 . 9 0 3 5 3 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 424, + 507, + 444 + ], + "score": 1.0, + "content": ", and set the valley’s", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 102, + 437, + 509, + 462 + ], + "spans": [ + { + "bbox": [ + 102, + 437, + 158, + 462 + ], + "score": 1.0, + "content": "boundary as", + "type": "text" + }, + { + "bbox": [ + 158, + 441, + 230, + 457 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\theta _ { i } < \\frac { 1 } { \\sqrt { k } } 0 . 1 5 6 7 3 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 437, + 261, + 462 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 262, + 443, + 266, + 452 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 437, + 509, + 462 + ], + "score": 1.0, + "content": "is the dimension index. We record the number of iterations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 456, + 480, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 480, + 467 + ], + "score": 1.0, + "content": "required to escape from the valley to the outside of valley. The setting 1 does not need labels.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 102, + 394, + 509, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "Experiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initial-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 140, + 496 + ], + "score": 1.0, + "content": "ization:", + "type": "text" + }, + { + "bbox": [ + 140, + 483, + 214, + 495 + ], + "score": 0.92, + "content": "\\theta _ { t = 0 } = ( 0 , \\cdot \\cdot \\cdot , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 482, + 292, + 496 + ], + "score": 1.0, + "content": ". Valley Boundary:", + "type": "text" + }, + { + "bbox": [ + 293, + 483, + 361, + 494 + ], + "score": 0.92, + "content": "- 0 . 1 < \\theta _ { i } < 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Due to the randomness of training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 516, + 396, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 396, + 529 + ], + "score": 1.0, + "content": "valley group, and study the mean escape time from the group of valleys.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 472, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 546 + ], + "score": 1.0, + "content": "Experiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 543, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 415, + 556 + ], + "score": 1.0, + "content": "losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization:", + "type": "text" + }, + { + "bbox": [ + 415, + 543, + 504, + 555 + ], + "score": 0.9, + "content": "\\theta _ { t = 0 } = ( 0 . 1 , \\cdot \\cdot \\cdot , 0 . 1 )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 223, + 567 + ], + "score": 1.0, + "content": "with a small Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 224, + 555, + 283, + 567 + ], + "score": 0.92, + "content": "\\epsilon = ( 0 , 0 . 0 1 I )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 554, + 364, + 567 + ], + "score": 1.0, + "content": ". Valley Boundary:", + "type": "text" + }, + { + "bbox": [ + 365, + 555, + 438, + 565 + ], + "score": 0.91, + "content": "0 . 0 5 < \\theta _ { i } < 0 . 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 554, + 506, + 567 + ], + "score": 1.0, + "content": ". To prevent the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "gradient disappearance problem of deep learning, we move the starting point from the origin. For", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 576, + 507, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 507, + 590 + ], + "score": 1.0, + "content": "symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "Due to the complex loss landscape of deep networks, we can hardly know the exact information", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "score": 1.0, + "content": "about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 621, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 632 + ], + "score": 1.0, + "content": "the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 631, + 431, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 431, + 644 + ], + "score": 1.0, + "content": "or not the propositions can hold with very-small-batch gradient noise in practice.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 530, + 507, + 644 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 656, + 236, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 236, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 236, + 668 + ], + "score": 1.0, + "content": "E.2 EXPERIMENTS RESULTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. 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