On the Existence of the Adversarial Bayes Classifier (Extended Version)

TL;DR

This paper investigates the conditions under which a Bayes optimal classifier exists in the context of adversarial robustness, providing theoretical guarantees and extending previous results to all norms.

Contribution

It offers general sufficient conditions for the existence of adversarial Bayes classifiers and extends prior results to include all norms, correcting earlier errors.

Findings

Provided conditions guarantee the existence of adversarial Bayes classifiers.

Extended results to all norms, including non-strictly convex ones.

Corrected previous errors in the theoretical framework.

Abstract

Adversarial robustness is a critical property in a variety of modern machine learning applications. While it has been the subject of several recent theoretical studies, many important questions related to adversarial robustness are still open. In this work, we study a fundamental question regarding Bayes optimality for adversarial robustness. We provide general sufficient conditions under which the existence of a Bayes optimal classifier can be guaranteed for adversarial robustness. Our results can provide a useful tool for a subsequent study of surrogate losses in adversarial robustness and their consistency properties. This manuscript is the extended and corrected version of the paper \emph{On the Existence of the Adversarial Bayes Classifier} published in NeurIPS 2021. There were two errors in theorem statements in the original paper -- one in the definition of pseudo-certifiable robustness and the other in the measurability of $A^\e$ for arbitrary metric spaces. In this version we correct the errors. Furthermore, the results of the original paper did not apply to some non-strictly convex norms and here we extend our results to all possible norms.