Solving Matrix Games with Near-Optimal Matvec Complexity

2026-01-05

AI Summary

This work addresses the computation of ε-approximate Nash equilibria in two-player bilinear zero-sum games with bounded payoff matrices under strategy constraints. Focusing on ℓ₁–ℓ₁ and ℓ₂–ℓ₁ matrix games, the study proposes novel first-order optimization algorithms that exploit the geometric structure of the problems to accelerate convergence. The key contribution lies in significantly reducing the matrix-vector multiplication complexity for ℓ₁–ℓ₁ games from Õ(ε⁻⁸⁄⁹) to Õ(ε⁻²⁄³). Moreover, for ℓ₂–ℓ₁ games—which correspond to hard-margin support vector machines—the algorithm achieves a complexity of Õ(ε⁻²⁄³) (ignoring logarithmic factors), matching the known lower bound for the first time and thereby attaining theoretically optimal efficiency.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications

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.

Problem

• Matrix Games
• Nash Equilibrium
• Zero-sum Games
• Matrix-Vector Products
• Bilinear Optimization

Innovation

• matrix games
• Nash equilibrium
• matrix-vector products
• zero-sum games
• support vector machines