On the Application of Danskin's Theorem to Derivative-Free Minimax Optimization

TL;DR

This paper leverages Danskin's theorem to develop a gradient-free optimization method for black-box minimax problems, demonstrating competitive performance and effectiveness in high-dimensional and real-world scenarios.

Contribution

It introduces a novel approach using Evolution Strategies as stochastic estimators for descent directions in black-box minimax optimization, extending Danskin's theorem to gradient-free settings.

Findings

Performance comparable to coevolutionary methods

Favorable results in high-dimensional problems

Effective on a real-world application

Abstract

Motivated by Danskin's theorem, gradient-based methods have been applied with empirical success to solve minimax problems that involve non-convex outer minimization and non-concave inner maximization. On the other hand, recent work has demonstrated that Evolution Strategies (ES) algorithms are stochastic gradient approximators that seek robust solutions. In this paper, we address black-box (gradient-free) minimax problems that have long been tackled in a coevolutionary setup. To this end and guaranteed by Danskin's theorem, we employ ES as a stochastic estimator for the descent direction. The proposed approach is validated on a collection of black-box minimax problems. Based on our experiments, our method's performance is comparable with its coevolutionary counterparts and favorable for high-dimensional problems. Its efficacy is demonstrated on a real-world application.