Improved Estimation of Concentration Under $\ell_p$-Norm Distance Metrics Using Half Spaces

1 INTRODUCTION . Despite achieving exceptional performance in benign settings , modern machine learning models have been shown to be highly vulnerable to inputs , known as adversarial examples , crafted with targeted but imperceptible perturbations ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) . This discovery has prompted a wave of research studies to propose defense mechanisms , including heuristic approaches ( Papernot et al. , 2016 ; Mądry et al. , 2018 ; Zhang et al. , 2019 ) and certifiable methods ( Wong & Kolter , 2018 ; Gowal et al. , 2019 ; Cohen et al. , 2019 ) . Unfortunately , none of these methods can successfully produce adversarially-robust models , even for classification tasks on toy datasets such as CIFAR-10 . To explain the prevalence of adversarial examples , a line of theoretical works ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Shafahi et al. , 2019 ; Dohmatob , 2019 ; Bhagoji et al. , 2019 ) have proven upper bounds on the maximum achievable adversarial robustness by imposing different assumptions on the underlying metric probability space . In particular , Mahloujifar et al . ( 2019a ) generalized the previous results showing that adversarial examples are inevitable as long as the input distributions are concentrated with respect to the perturbation metric . Thus , the question of whether or not natural image distributions are concentrated is highly relevant , as if they are it would rule out any possibility of there being adversarially robust image classifiers . Recently , Mahloujifar et al . ( 2019b ) proposed an empirical method to measure the concentration of an arbitrary distribution using data samples , then employed it to estimate a lower bound on intrinsic robustness ( see Definition 2.2 for its formal definition ) for several image benchmarks . By demonstrating the gap between the estimated bounds of intrinsic robustness and the robustness performance achieved by the best current models , they further concluded concentration of measure is not the sole reason behind the adversarial vulnerability of existing classifiers for benchmark image distributions . However , due to the heuristic nature of the proposed algorithm , it remains elusive whether the estimates it produces can serve as useful approximations of the underlying intrinsic robustness limits , thus hindering understanding of how much of the actual adversarial risk can be explained by the concentration of measure phenomenon . In this work , we address this issue by first characterizing the optimum of the actual concentration problem for general Gaussian spaces , then using our theoretical insights to develop an alternative algorithm for measuring concentration empirically that significantly improves both the accuracy and efficiency of estimates of intrinsic robustness . While we do not demonstrate a specific classifier which achieves this robustness upper bound , our results rule out inherent image distribution concentration as the reason for our current inability to find adversarially robust models . Contributions . We generalize the Gaussian Isoperimetric Inequality to non-spherical Gaussian distributions and ` p-norm distance metrics with p ≥ 2 ( including ` ∞ ) ( Theorem 3.3 ) . Motivated by the optimal concentration results for special Gaussian spaces ( Remark 3.4 ) , we develop a samplebased algorithm to estimate the concentration of measure using half spaces that works for arbitrary distribution and any ` p-norm distance ( Section 4 ) . Compared with prior approaches , we empirically demonstrate the significant increase in efficacy of our method under ` ∞-norm distance metric ( Section 6 ) . Not only does the proposed method converge to its limit with an order of magnitude fewer data ( Section 6.2 ) , it also finds a much tighter lower bound of intrinsic robustness for both simulated datasets whose underlying concentration function is analytically derivable and various benchmark image datasets ( Section 6.1 ) . In particular , we improve the best current estimated lower bound of intrinsic robustness from approximately 82 % to above 93 % for CIFAR-10 under ` ∞-norm bounded perturbations with = 8/255 . These tighter concentration estimates produced by our algorithm provide strong evidence that concentration of measure should not be considered as the main cause of adversarial vulnerability , at least for the image benchmarks evaluated in our experiments . Related Work . Several prior works have sought to empirically estimate lower bounds on intrinsic robustness using data samples . The pioneering work of Gilmer et al . ( 2018 ) introduced the connection between adversarial examples and the concentration phenomenon for uniform n-spheres , then proposed a simple heuristic to find a half space that expands slowly under Euclidean distance for the MNIST dataset . Our work can be seen as a strict generalization of Gilmer et al . ( 2018 ) ’ s , which applies to arbitrary ` p-norm distance metrics ( including ` ∞ ) . By characterizing the optimal transport cost between conditional distributions , Bhagoji et al . ( 2019 ) estimated a lower bound on the best possible adversarial robustness for several image datasets . However , when applied to adversaries beyond ` 2 , such as ` ∞ , the lower bound produced by their method is not informative ( that is , it is close to zero ) . The most relevant previous work is Mahloujifar et al . ( 2019b ) , which proposed a general method for measuring concentration using special collections of subsets . Although the optimal value of the considered empirical concentration problem is proven to asymptotically converge to the actual concentration , there is no guarantee that the proposed searching algorithm for solving the empirical problem finds the optimum . Our approach follows the framework introduced by Mahloujifar et al . ( 2019b ) ’ s , but considers a different collection of subsets for the empirical concentration problem . This not only results in optimality for theoretical Gaussian distributions , but also significantly improves the estimation performance for typical image benchmarks . Another line of work attempts to provide estimates of intrinsic robustness upper bounds based on generative assumptions . In order to justify the theoretically-derived impossibility results , Fawzi et al . ( 2018 ) estimated the smoothness parameters of the state-of-the-art generative models on CIFAR-10 and SVHN datasets , which yield approximated upper bounds on adversarial robustness for any classifiers . Zhang et al . ( 2020 ) generalized their results to non-smoothed data manifolds , such as datasets that can be captured by a conditional generative model . However , these methods only work for simulated generative distributions , which may deviate from the actual distributions they are intended to understand . Notation . For any n ∈ Z+ , denote by [ n ] the set { 1 , 2 , . . . , n } . Lowercase boldface letters denote vectors and uppercase boldface letters represent matrices . For any vector x and p ∈ [ 1 , ∞ ) , let xj , ‖x‖p and ‖x‖∞ be the j-th element , the ` p-norm and the ` ∞-norm of x . For any matrix A , B is said to be a square root of A if A = BB , and the induced matrix p-norm of A is defined as ‖A‖p = supx6=0 { ‖Ax‖p/‖x‖p } . Denote by N ( θ , Σ ) the Gaussian distribution with mean θ and covariance matrix Σ . Let γn be the probability measure of N ( 0 , In ) , where In denotes the identity matrix . Let Φ ( · ) be the cumulative distribution function of N ( 0 , 1 ) and Φ−1 ( · ) be its inverse . For any set A , let pow ( A ) and 1A ( · ) be all measurable subsets and the indicator function of A . Let ( X , µ , ∆ ) be a metric probability space , where ∆ : X × X → R≥0 denotes a distance metric on X . Define the empirical measure with respect to a sample set { xi } i∈ [ m ] as µ̂m ( A ) = 1m ∑ i∈ [ m ] 1A ( xi ) , ∀A ∈ pow ( X ) . Let B ( x , , ∆ ) = { x′ ∈ X : ∆ ( x′ , x ) ≤ } be the ball around x with radius . Define the -expansion of A as A ( ∆ ) = { x ∈ X : ∃ x′ ∈ B ( x , , ∆ ) ∩ A } . 2 PRELIMINARIES . In this section , we introduce the problem of measuring concentration and its connection to adversarial robustness . Consider a metric probability space of instances ( X , µ , ∆ ) . Given parameters ≥ 0 and α > 0 , the concentration of measure problem1 can be cast as the following optimization problem : minimize E∈Pow ( X ) µ ( E ( ∆ ) ) subject to µ ( E ) ≥ α . ( 2.1 ) We focus on the case where ∆ is some ` p-norm distance metric ( including ` ∞ ) in this work . Concentration of measure has been shown to be closely related to adversarial examples ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Mahloujifar et al. , 2019a ) . In particular , one can prove that for a given robust learning problem , if the input distribution is concentrated with respect to the perturbation metric , no adversarially robust model exists . The concentration parameter ( which corresponds to the optimal value of optimization problem ( 2.1 ) ) determines an inherent upper bound on the maximum adversarial robustness that any model can achieve for the given problem . To explain the connection between concentration of measure and robust learning in a more formal way , we lay out the definition of adversarial risk that we work with . We draw this definition from several previous works , including Gilmer et al . ( 2018 ) ; Bubeck et al . ( 2019 ) ; Mahloujifar et al . ( 2019a ; b ) .2 Definition 2.1 ( Adversarial Risk ) . Let ( X , µ , ∆ ) be the input metric probability space . Assume f∗ is the underlying ground-truth classifier that gives labels to any input . Given classifier f and ≥ 0 , the adversarial risk of f with respect to -perturbations measured by ∆ is defined as : AdvRisk ( f ) = Pr x∼µ [ ∃ x′ ∈ B ( x , , ∆ ) s.t . f ( x′ ) 6= f∗ ( x′ ) ] . Correspondingly , we define the adversarial robustness of f as AdvRob ( f ) = 1−AdvRisk ( f ) . When = 0 , adversarial risk degenerates to standard risk . In other words , it holds for any f that AdvRisk0 ( f ) = Risk ( f ) : = Prx∼µ [ f ( x ) 6= f∗ ( x ) ] . We remark that this definition assumes the existence of an underlying ground-truth labeling function , which does not apply to the agnostic setting where inputs can have non-deterministic labels . Initially introduced in Mahloujifar et al . ( 2019b ) , intrinsic robustness captures the maximum adversarial robustness that can be achieved by any imperfect classifier for a robust classification problem . Definition 2.2 ( Intrinsic Robustness ) . Consider the same setting as in Definition 2.1 . For any α > 0 , let Fα = { f : Risk ( f ) ≥ α } be the set of imperfect classifiers whose risk is at least α . Then the intrinsic robustness of the given robust classification problem with respect to Fα is defined as : AdvRob ( Fα ) = 1− inf f∈Fα { AdvRisk ( f ) } = sup f∈Fα { AdvRob ( f ) } . It is worth noting that the value of AdvRob ( Fα ) is only determined by the underlying input data distribution , the perturbation set and the risk threshold parameter α , which is independent of the model class one would choose for learning . By relating the robustness of a classifier to the -expansion of its induced error region , the following lemma , proved in Mahloujifar et al . ( 2019a ) , establishes a fundamental connection between the concentration of measure and the intrinsic robustness one can hope for a robust classification problem . Lemma 2.3 . Consider the same setting as in Definition 2.2 . Let h ( ∆ ) µ ( α , ) be the concentration function that captures the optimal value of the concentration of measure problem ( 2.1 ) : h ( ∆ ) µ ( α , ) = inf { µ ( E ( ∆ ) ) : E ∈ pow ( X ) and µ ( E ) ≥ α } . Then , AdvRob ( Fα ) = 1− h ( ∆ ) µ ( α , ) holds for any α > 0 and ≥ 0 . 1The standard notion of concentration of measure ( Talagrand , 1995 ) corresponds to the case where α = 0.5 . 2Other related definitions , such as the one used in most empirical works for robustness evaluation , are equivalent to this , as long as small perturbations preserve the ground truth . See Diochnos et al . ( 2018 ) for a detailed comparison of these and other definitions of adversarial robustness . Lemma 2.3 suggests that one can characterize the intrinsic robustness limit for a robust classification problem by measuring the concentration of the input data with respect to the perturbation metric . In this paper , we aim to understand and empirically estimate the intrinsic robustness limit for typical robust classification tasks by measuring concentration . It is worth noting that solving the concentration problem ( 2.1 ) itself only shows the existence of an error region E whose -expansion has certain ( small ) measure . This further implies the possibility of existing an optimally robust classifier ( with risk at least α ) , whose robustness matches the intrinsic robustness limit AdvRob ( Fα ) . However , actually finding such optimal classifier using a learning algorithm might be a much more challenging task , which is beyond the scope of this work .

Summary: This paper considers the estimation of the concentration of measures, which possibly causes of the vulnerability of machine learning models to adversarial attacks. Towards such a goal, the authors first extend the Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary l_p norm. An algorithm that uses half-spaces as feasible set in the concentration of measure problem is then proposed. Empirical studies are conducted to show the efficiency as well as the efficacy of the proposed method when compared to [Mahloujifar et al. (2019b)].