Lower Bounds on Adversarial Robustness from Optimal Transport

TL;DR

This paper uses optimal transport theory to establish fundamental lower bounds on adversarial robustness in classification, providing insights into the limits of classifier performance under adversarial perturbations.

Contribution

It introduces a novel framework linking optimal transport costs to minimum achievable adversarial loss, offering new bounds and analysis for robust classification.

Findings

Optimal transport cost bounds the minimum adversarial loss.

Linear classifiers are optimal under certain conditions.

Sample complexity results extend previous bounds.

Abstract

While progress has been made in understanding the robustness of machine learning classifiers to test-time adversaries (evasion attacks), fundamental questions remain unresolved. In this paper, we use optimal transport to characterize the minimum possible loss in an adversarial classification scenario. In this setting, an adversary receives a random labeled example from one of two classes, perturbs the example subject to a neighborhood constraint, and presents the modified example to the classifier. We define an appropriate cost function such that the minimum transportation cost between the distributions of the two classes determines the minimum $0-1$ loss for any classifier. When the classifier comes from a restricted hypothesis class, the optimal transportation cost provides a lower bound. We apply our framework to the case of Gaussian data with norm-bounded adversaries and explicitly show matching bounds for the classification and transport problems as well as the optimality of linear classifiers. We also characterize the sample complexity of learning in this setting, deriving and extending previously known results as a special case. Finally, we use our framework to study the gap between the optimal classification performance possible and that currently achieved by state-of-the-art robustly trained neural networks for datasets of interest, namely, MNIST, Fashion MNIST and CIFAR-10.