Duality for the Adversarial Total Variation

2026-04-20

AI Summary

This work formulates the binary classification problem in adversarial training as a regularized risk minimization problem involving nonlocal total variation. By introducing nonlocal gradient and divergence operators, and leveraging duality theory from functional analysis together with variational methods on metric spaces, the authors establish a dual representation and an integration-by-parts formula for the nonlocal total variation. Building upon this foundation, they provide the first complete characterization of the subdifferential of this functional in both the space of continuous functions and the space of bounded measurable functions. This rigorous analytical framework offers a solid theoretical foundation for understanding and analyzing adversarial robustness in machine learning models.

Technology Category

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

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Adversarial training of binary classifiers can be reformulated as regularized risk minimization involving a nonlocal total variation. Building on this perspective, we establish a characterization of the subdifferential of this total variation using duality techniques. To achieve this, we derive a dual representation of the nonlocal total variation and a related integration of parts formula, involving a nonlocal gradient and divergence. We provide such duality statements both in the space of continuous functions vanishing at infinity on proper metric spaces and for the space of essentially bounded functions on Euclidean domains. Furthermore, under some additional conditions we provide characterizations of the subdifferential in these settings.

Problem

• adversarial total variation
• subdifferential
• duality
• nonlocal gradient
• regularized risk minimization

Innovation

• nonlocal total variation
• adversarial training
• duality
• subdifferential characterization
• nonlocal gradient