Solving Non-Convex Non-Concave Min-Max Games Under Polyak-{\L}ojasiewicz Condition

TL;DR

This paper addresses solving non-convex non-concave min-max games where one player's objective satisfies the Polyak-{ extL}ojasiewicz condition, proposing an efficient gradient-based method with convergence guarantees.

Contribution

It introduces a simple multi-step gradient descent-ascent algorithm that finds stationary points in non-convex non-concave min-max games under the PL condition, with proven convergence rates.

Findings

Algorithm converges to an $ ext{ extepsilon}$-stationary point in $ ilde{O}( ext{ extepsilon}^{-2})$ iterations.

Applicable to non-convex non-concave settings with PL condition on one player's objective.

Extends understanding of min-max game solutions beyond convex-concave regimes.

Abstract

In this short note, we consider the problem of solving a min-max zero-sum game. This problem has been extensively studied in the convex-concave regime where the global solution can be computed efficiently. Recently, there have also been developments for finding the first order stationary points of the game when one of the player's objective is concave or (weakly) concave. This work focuses on the non-convex non-concave regime where the objective of one of the players satisfies Polyak-{\L}ojasiewicz (PL) Condition. For such a game, we show that a simple multi-step gradient descent-ascent algorithm finds an $\varepsilon$--first order stationary point of the problem in $\widetilde{\mathcal{O}}(\varepsilon^{-2})$ iterations.