Local Convergence of Gradient Methods for Min-Max Games: Partial Curvature Generically Suffices

TL;DR

This paper demonstrates that gradient methods for two-player zero-sum games converge under broader conditions than previously known, specifically when the symmetric part of the Jacobian is nonzero, and explores how partial curvature influences convergence rates and algorithm performance.

Contribution

The paper establishes convergence of gradient methods under partial curvature conditions and analyzes how eigenvector positions affect convergence, extending understanding beyond the positive definite case.

Findings

Convergence occurs when the symmetric part of the Jacobian is nonzero.

Convergence rates depend on the average eigenvalues of the symmetric part.

Over-parameterization with curvature improves convergence speed in mixed Nash equilibrium computations.

Abstract

We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that such dynamics converge locally when $S \succ 0$ and may diverge when $S=0$, where $S\succeq 0$ is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as $S$ is nonzero (partial curvature) and the eigenvectors of the antisymmetric part $A$ are in general position with respect to the kernel of $S$. We then study the convergence rates when $S \ll A$ and prove that they typically depend on the average of the eigenvalues of $S$, instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods -- which optimize over both weights and supports of the mixed strategies -- generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.