Direct-Search for a Class of Stochastic Min-Max Problems

TL;DR

This paper introduces a novel derivative-free direct-search algorithm for stochastic min-max problems, proving its convergence under mild conditions, especially when the max-player satisfies the PL condition and the min-player is nonconvex.

Contribution

The work presents the first convergence analysis of a direct-search method for stochastic min-max problems with nonconvex and PL conditions.

Findings

Proves convergence of the proposed algorithm under mild assumptions.

Handles stochastic settings with dynamically estimated oracle accuracy.

Addresses nonconvex min-player and PL max-player scenarios.

Abstract

Recent applications in machine learning have renewed the interest of the community in min-max optimization problems. While gradient-based optimization methods are widely used to solve such problems, there are however many scenarios where these techniques are not well-suited, or even not applicable when the gradient is not accessible. We investigate the use of direct-search methods that belong to a class of derivative-free techniques that only access the objective function through an oracle. In this work, we design a novel algorithm in the context of min-max saddle point games where one sequentially updates the min and the max player. We prove convergence of this algorithm under mild assumptions, where the objective of the max-player satisfies the Polyak-\L{}ojasiewicz (PL) condition, while the min-player is characterized by a nonconvex objective. Our method only assumes dynamically adjusted accurate estimates of the oracle with a fixed probability. To the best of our knowledge, our analysis is the first one to address the convergence of a direct-search method for min-max objectives in a stochastic setting.