A Scenario Approach to the Robustness of Nonconvex-Nonconcave Minimax Problems

TL;DR

This paper explores the probabilistic robustness of nonconvex-nonconcave minimax problems using the scenario approach, providing guarantees for stationary points and global minimax points under various conditions.

Contribution

It introduces a probabilistic robustness guarantee for stationary points in convex strategy sets and a relaxed bound for global minimax points in nonconvex sets, advancing theoretical understanding.

Findings

Established a robustness guarantee for $oldsymbol{ ext{ε}}$-stationary points.

Proved the monotonicity of the stationary residual with scenario count.

Provided a relaxed probabilistic bound for global minimax points.

Abstract

This paper investigates probabilistic robustness of nonconvex-nonconcave minimax problems via the scenario approach. Specifically, under convex strategy sets for all players, inspired by recent advances in scenario optimization, we first establish a probabilistic robustness guarantee for an $\varepsilon$-stationary point, overcoming the dependence on the non-degeneracy assumption by proving the monotonicity of the stationary residual in the number of scenarios. Furthermore, in the presence of nonconvex strategy sets, we reveal the fundamental difficulty of obtaining a tight theoretical bound based on this recent framework. Consequently, we establish a relaxed, yet rigorously valid, probabilistic bound for a global minimax point. A numerical experiment corroborates our theoretical findings.