Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning
This work addresses the theoretical foundation of adversarial machine learning for binary classification, providing a rigorous mathematical framework to analyze stability and regularization effects, though it is incremental in extending convergence results to less regular distributions.
The paper tackles the problem of understanding the regularizing effect of adversarial training in binary classification by proving Gamma-convergence of a nonlocal perimeter to a local anisotropic perimeter, with results applicable to distributions having bounded BV densities and leading to a weighted perimeter that reflects classification stability against adversarial perturbations.
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.