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

Leon Bungert, Kerrek Stinson

TL;DR

The paper establishes a Γ-convergence result for a nonlocal Minkowski-type perimeter arising from adversarial training in binary classification, showing that as the nonlocal interaction vanishes, the energy converges to a local anisotropic perimeter weighted by a function β that depends on BV densities of the two classes. The Γ-limit retains anisotropy only through density discontinuities and hence captures the regularizing effect of adversarial perturbations in a sharp-interface limit. The authors prove compactness, the liminf and limsup inequalities, and demonstrate consequences for the total variation, the asymptotics of adversarial training, and graph-based discretizations via TL^p convergence. This provides a rigorous continuum framework connecting adversarial robustness to weighted minimal-surface-type energies, with implications for both theory and graph-based algorithms in ML contexts.

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.

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

TL;DR

The paper establishes a Γ-convergence result for a nonlocal Minkowski-type perimeter arising from adversarial training in binary classification, showing that as the nonlocal interaction vanishes, the energy converges to a local anisotropic perimeter weighted by a function β that depends on BV densities of the two classes. The Γ-limit retains anisotropy only through density discontinuities and hence captures the regularizing effect of adversarial perturbations in a sharp-interface limit. The authors prove compactness, the liminf and limsup inequalities, and demonstrate consequences for the total variation, the asymptotics of adversarial training, and graph-based discretizations via TL^p convergence. This provides a rigorous continuum framework connecting adversarial robustness to weighted minimal-surface-type energies, with implications for both theory and graph-based algorithms in ML contexts.

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 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.
Paper Structure (17 sections, 21 theorems, 186 equations)

This paper contains 17 sections, 21 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Notation
  4. Balls and cubes
  5. Measures and sets
  6. Measure-theoretic set quantities
  7. Functions of bounded variation
  8. Main results
  9. Auxiliary definitions and reformulations of the perimeter
  10. Gamma-convergence and compactness
  11. Compactness
  12. Liminf bound
  13. Limsup bound
  14. Applications
  15. Gamma-convergence of total variation
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\Omega\subset\mathbb{R}^d$ be an open and bounded Lipschitz domain, and let $\rho_0,\rho_1\in BV(\Omega)\cap L^\infty(\Omega)$ satisfy $\mathop{\mathrm{ess\,inf}}\limits_\Omega(\rho_0+\rho_1)>0$. Then for any sequence $(\varepsilon_k)_{k\in\mathbb{N}}$ with $\lim_{k\to\infty}\varepsilon_k=0$ an

Theorems & Definitions (40)

  • Theorem 2.1
  • Remark 2.2
  • Example 1
  • Theorem 2.3
  • Remark 2.4: Decomposition of the limit perimeter
  • Theorem 2.5: Conditional convergence of adversarial training
  • Example 2
  • Lemma 2.6
  • proof : Proof of \ref{['thm:gammaCompact']}
  • Lemma 3.1
  • ...and 30 more