Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning

TL;DR

This paper proves that a nonlocal Minkowski-type perimeter model, used to describe adversarial training effects in binary classification, converges to a local anisotropic perimeter, revealing insights into classification stability and asymptotics.

Contribution

It establishes Gamma-convergence of a nonlocal perimeter to a local anisotropic perimeter under minimal regularity assumptions, connecting adversarial training to sharp interface limits.

Findings

Gamma-convergence of nonlocal to local perimeter established

Results apply to total variations and graph discretizations

Insights into adversarial training stability and asymptotics

Abstract

In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowski type to a local anisotropic perimeter. The nonlocal model describes the regularizing effect of adversarial training in binary classifications. The energy essentially depends on the interaction between two distributions modelling likelihoods for the associated classes. We overcome typical strict regularity assumptions for the distributions by only assuming that they have bounded $BV$ densities. In the natural topology coming from compactness, we prove Gamma-convergence to a weighted perimeter with weight determined by an anisotropic function of the two densities. Despite being local, this sharp interface limit reflects classification stability with respect to adversarial perturbations. We further apply our results to deduce Gamma-convergence of the associated total variations, to study the asymptotics of adversarial training, and to prove Gamma-convergence of graph discretizations for the nonlocal perimeter.