A mean curvature flow arising in adversarial training

TL;DR

This paper links adversarial training in binary classification to a geometric evolution of the decision boundary, showing it approximates a weighted mean curvature flow and suggesting a geometric interpretation of adversarial robustness.

Contribution

It introduces a novel geometric perspective on adversarial training by connecting it to mean curvature flow and provides a rigorous mathematical analysis of this connection.

Findings

Adversarial training approximates a weighted mean curvature flow.

The proposed scheme is monotone and consistent as the adversarial budget vanishes.

The analysis introduces new tools for nonlocal total variation subdifferential.

Abstract

We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.