Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a class of adversarial classification problems which admit a continuous solution

Guillaume Carlier, Maxime Sylvestre

TL;DR

We formulate robust binary classification under adversarial perturbations as a zero-sum game between a classifier with loss functions $l_i$ and an adversary incurring transport costs $c_i$. The paper proves the existence of a game value and a continuous (even Lipschitz) optimal classifier under mild regularity, using convex duality and Kantorovich-type arguments. It also develops a softmax regularization that links to entropic optimal transport, establishes $\,\,\\Gamma$-convergence to the original problem, and provides numerical demonstrations on a one-dimensional torus illustrating how the regularization parameter $\,\\\varepsilon$ and transport costs shape both the classifier and the adversarial strategy. These results offer a principled, computable framework for robust adversarial classification with transport-based perturbations and connect to modern OT techniques for scalable optimization.

Abstract

We consider a class of adversarial classification problems in the form of zero-sum games between a classifier and an adversary. The latter is able to corrupt data, at the expense of some optimal transport cost. We show that quite general assumptions on the loss functions of the classifier and the transport cost functions of the adversary ensure the existence of a Nash equilibrium with a continuous (or even Lipschitz) classifier's strategy. We also consider a softmax-like regularization of this problem and present numerical results for this regularization.

On a class of adversarial classification problems which admit a continuous solution

TL;DR

We formulate robust binary classification under adversarial perturbations as a zero-sum game between a classifier with loss functions and an adversary incurring transport costs . The paper proves the existence of a game value and a continuous (even Lipschitz) optimal classifier under mild regularity, using convex duality and Kantorovich-type arguments. It also develops a softmax regularization that links to entropic optimal transport, establishes -convergence to the original problem, and provides numerical demonstrations on a one-dimensional torus illustrating how the regularization parameter and transport costs shape both the classifier and the adversarial strategy. These results offer a principled, computable framework for robust adversarial classification with transport-based perturbations and connect to modern OT techniques for scalable optimization.

Abstract

We consider a class of adversarial classification problems in the form of zero-sum games between a classifier and an adversary. The latter is able to corrupt data, at the expense of some optimal transport cost. We show that quite general assumptions on the loss functions of the classifier and the transport cost functions of the adversary ensure the existence of a Nash equilibrium with a continuous (or even Lipschitz) classifier's strategy. We also consider a softmax-like regularization of this problem and present numerical results for this regularization.

Paper Structure

This paper contains 11 sections, 12 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and Assumptions
  3. Concise Formulations
  4. Duality and Attainment Results
  5. Duality
  6. Existence of Continuous Optimal Classifiers
  7. Primal-Dual Conditions and Consequences
  8. Softmax Regularization, Numerical Results
  9. Softmax Approximation
  10. Numerical Results
  11. Conclusions

Key Result

Lemma 3.1

Let $(\nu_1, \nu_{-1}) \in {\mathscr{P}}(X)^2$ let ${\overline \nu}:=\nu_1+\nu_{-1}$ and let $\alpha_i$ be the density of $\nu_i$ with respect to ${\overline \nu}$ (so that $\alpha_{-1}=1-\alpha_1$ and $\alpha_1 \in [0, 1]$, ${\overline \nu}$-almost everywhere), then where

Figures (3)

  • Figure 1: Impact of $\varepsilon$
  • Figure 2: Impact of the cost
  • Figure 3: Impact of the initial measures

Theorems & Definitions (29)

  • Lemma 3.1
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • Proposition 4.4
  • proof
  • Remark 4.5
  • ...and 19 more