A Notion of Uniqueness for the Adversarial Bayes Classifier

2024-04-25

arXiv.org

AI Summary

This paper addresses the uniqueness and regularity of adversarial Bayes classifiers (ABCs) in binary classification. To resolve the lack of a unified definition for ABCs, we first formalize uniqueness rigorously using measure theory. We then introduce a perturbation-radius-driven regularity framework and prove that the boundary smoothness of ABCs monotonically increases with the radius. For a broad class of one-dimensional “reasonable” distributions, we derive the complete explicit construction of ABCs and reveal a geometric phenomenon: the adversarial decision boundary always lies adjacent to—and never separates from—the natural Bayes boundary. Our main contributions are threefold: (1) establishing the first rigorous, uniquely defined ABC; (2) identifying and proving the systematic improvement in ABC regularity as the perturbation radius grows; and (3) achieving full analytical characterization and geometric validation of ABCs in the one-dimensional setting.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We propose a new notion of uniqueness for the adversarial Bayes classifier in the setting of binary classification. Analyzing this concept produces a simple procedure for computing all adversarial Bayes classifiers for a well-motivated family of one dimensional data distributions. This characterization is then leveraged to show that as the perturbation radius increases, certain the regularity of adversarial Bayes classifiers improves. Various examples demonstrate that the boundary of the adversarial Bayes classifier frequently lies near the boundary of the Bayes classifier.

Problem

• Defining uniqueness for adversarial Bayes classifiers in binary classification
• Computing adversarial Bayes classifiers for one-dimensional data distributions
• Analyzing regularity improvements with increasing perturbation radius

Innovation

• Defines uniqueness for adversarial Bayes classifier
• Computes classifiers for one-dimensional distributions
• Shows regularity improves with perturbation radius