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A mean curvature flow arising in adversarial training

Leon Bungert, Tim Laux, Kerrek Stinson

TL;DR

The paper develops a rigorous link between adversarial training for binary classification and geometric evolution of the decision boundary by recasting adversarial regularization as a nonlocal perimeter problem and analyzing a minimizing movements scheme. It establishes monotonicity, a selection principle, and consistency with a weighted mean curvature flow as the nonlocal scale $\varepsilon$ vanishes, proving convergence of time-parametrized boundary evolutions to $V = -\frac{1}{\varrho}\mathrm{div}(\varrho \nu_A)$ (equivalently $V = H_A - \nabla\varrho \cdot \nu_A$) on $\partial A$, starting from a smooth Bayes classifier. The work advances the mathematical understanding of adversarial robustness by connecting it to boundary-length minimization and geometric flows, and develops a detailed analysis of the nonlocal total variation $\mathrm{TV}_\varepsilon$ and its subdifferential. These results offer a rigorous, geometry-driven interpretation of how adversarial training concentrates on locally minimizing the decision boundary length, with a framework for studying robustness through a well-posed, convergent evolution process.

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.

A mean curvature flow arising in adversarial training

TL;DR

The paper develops a rigorous link between adversarial training for binary classification and geometric evolution of the decision boundary by recasting adversarial regularization as a nonlocal perimeter problem and analyzing a minimizing movements scheme. It establishes monotonicity, a selection principle, and consistency with a weighted mean curvature flow as the nonlocal scale vanishes, proving convergence of time-parametrized boundary evolutions to (equivalently ) on , starting from a smooth Bayes classifier. The work advances the mathematical understanding of adversarial robustness by connecting it to boundary-length minimization and geometric flows, and develops a detailed analysis of the nonlocal total variation and its subdifferential. These results offer a rigorous, geometry-driven interpretation of how adversarial training concentrates on locally minimizing the decision boundary length, with a framework for studying robustness through a well-posed, convergent evolution process.

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.
Paper Structure (10 sections, 18 theorems, 136 equations)

This paper contains 10 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. From adversarial training to mean curvature flow
  3. The minimizing movements scheme
  4. Main result
  5. Properties of the total variation
  6. Convergence of the adversarial training scheme
  7. Well-posedness
  8. Selection property
  9. Monotonicity
  10. Consistency

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^N$ be a bounded and convex domain. Suppose that in eq:selection the Bayes classifier $A_0 \subset\subset \Omega$ has $C^2$-boundary and that $t\mapsto A(t)$ is a parameterized curve evolving by the weighted mean curvature flow with normal velocity eq:normal_velocity up t

Theorems & Definitions (45)

  • Remark 2.1: The distance function
  • Theorem 1: Main theorem
  • Remark 2.2: Smooth Bayes classifiers
  • Remark 2.3: Convexity
  • Remark 2.4: The first singular time
  • Remark 2.5: Generalized solutions of mean curvature flow
  • Remark 2.6: Boundary conditions
  • Lemma 3.1: Coarea formula bungert2023geometry
  • Lemma 3.2
  • Remark 3.3
  • ...and 35 more