Solving Matrix Games with Near-Optimal Matvec Complexity

TL;DR

This paper introduces algorithms for efficiently computing approximate Nash equilibria in certain bilinear games, achieving near-optimal complexity in terms of matrix-vector multiplications, which improves upon previous methods.

Contribution

The paper presents near-optimal algorithms with $ ilde{O}( ext{epsilon}^{-2/3})$ matvec complexity for specific bilinear games, surpassing prior state-of-the-art complexities.

Findings

Achieves $ ilde{O}( ext{epsilon}^{-2/3})$ matvec complexity for $ ext{ell}_1$-$ ext{ell}_1$ games.

Achieves $ ilde{O}( ext{epsilon}^{-2/3})$ matvec complexity for $ ext{ell}_2$-$ ext{ell}_1$ games.

Results are nearly optimal, matching lower bounds up to polylogarithmic factors.

Abstract

We study the problem of computing an $\epsilon$-approximate Nash equilibrium of a two-player, bilinear game with a bounded payoff matrix $A \in \mathbb{R}^{m \times n}$, when the players' strategies are constrained to lie in simple sets. We provide algorithms which solve this problem in $\tilde{O}(\epsilon^{-2/3})$ matrix-vector multiplies (matvecs) in two well-studied cases: $\ell_1$-$\ell_1$ (or zero-sum) games, where the players' strategies are both in the probability simplex, and $\ell_2$-$\ell_1$ games (encompassing hard-margin SVMs), where the players' strategies are in the unit Euclidean ball and probability simplex respectively. These results improve upon the previous state-of-the-art complexities of $\tilde{O}(\epsilon^{-8/9})$ for $\ell_1$-$\ell_1$ and $\tilde{O}(\epsilon^{-7/9})$ for $\ell_2$-$\ell_1$ due to [KOS '25]. In both settings our results are nearly-optimal as they match lower bounds of [KS '25] up to polylogarithmic factors.