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Solving Matrix Games with Near-Optimal Matvec Complexity

Ishani Karmarkar, Liam O'Carroll, Aaron Sidford

TL;DR

The paper tackles computing $oldsymbol{ extepsilon}$-approximate Nash equilibria in two-player matrix games with bilinear payoffs under simple strategy sets, focusing on matvec-efficient algorithms. It introduces a prox multi-point outer-loop framework with a dyadic, multi-center approach and an amortized analysis that bounds total movement across iterations, enabling a $ ilde{O}(oldsymbol{ extepsilon}^{-2/3})$ matvec complexity for both $ ext{ell}_1 ext{-ell}_1$ and $ ext{ell}_2 ext{-ell}_1$ settings. The approach combines a dynamic bisection procedure for regularization, a smooth-until-proven-guilty inner solver, and a matrix-approximation path to learn $A$ efficiently, achieving near-optimal performance up to polylogarithmic factors. The framework unifies and extends prior methods, providing a scalable path toward solving broad classes of structured convex-concave problems with limited access to the matrix via matvecs, and it includes detailed robustness and stability analyses that support its practical applicability. Overall, the work significantly advances the matvec efficiency frontier for bilinear matrix games and offers techniques likely applicable to broader monotone-operator problems in optimization and game theory.

Abstract

We study the problem of computing an $ε$-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 $\tilde{O}(ε^{-2/3})$ 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 $\tilde{O}(ε^{-8/9})$ for $\ell_1$-$\ell_1$ and $\tilde{O}(ε^{-7/9})$ 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.

Solving Matrix Games with Near-Optimal Matvec Complexity

TL;DR

The paper tackles computing -approximate Nash equilibria in two-player matrix games with bilinear payoffs under simple strategy sets, focusing on matvec-efficient algorithms. It introduces a prox multi-point outer-loop framework with a dyadic, multi-center approach and an amortized analysis that bounds total movement across iterations, enabling a matvec complexity for both and settings. The approach combines a dynamic bisection procedure for regularization, a smooth-until-proven-guilty inner solver, and a matrix-approximation path to learn efficiently, achieving near-optimal performance up to polylogarithmic factors. The framework unifies and extends prior methods, providing a scalable path toward solving broad classes of structured convex-concave problems with limited access to the matrix via matvecs, and it includes detailed robustness and stability analyses that support its practical applicability. Overall, the work significantly advances the matvec efficiency frontier for bilinear matrix games and offers techniques likely applicable to broader monotone-operator problems in optimization and game theory.

Abstract

We study the problem of computing an -approximate Nash equilibrium of a two-player, bilinear game with a bounded payoff matrix , when the players' strategies are constrained to lie in simple sets. We provide algorithms which solve this problem in matrix-vector multiplies (matvecs) in two well-studied cases: - (or zero-sum) games, where the players' strategies are both in the probability simplex, and - 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 for - and for - due to [KOS '25]. In both settings our results are nearly-optimal as they match lower bounds of [KS '25] up to polylogarithmic factors.
Paper Structure (53 sections, 35 theorems, 151 equations, 2 tables)

This paper contains 53 sections, 35 theorems, 151 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Our results.
  3. Techniques.
  4. Paper organization.
  5. Preliminaries
  6. General notation.
  7. Simplices, entropy, and KL divergence.
  8. Problem setups.
  9. Monotone operators and proximal mappings.
  10. Notation for finite multisets $\mathcal{U} \subset \mathcal{Z}$.
  11. Convex-concave functions, gradient mappings, and regret.
  12. Technical overview
  13. Assumptions.
  14. The approach of karmarkar2025solvingzerosumgames for $\ell_2\text{-}\ell_1$ games.
  15. A key challenge faced by the prior work.
  16. ...and 38 more sections

Key Result

Theorem 1.1

There is a deterministic algorithm that computes an $\epsilon$-solution of any $\ell_1$-$\ell_1$ game with $\tilde{O}(\epsilon^{-2/3})$ matvecs to $A$.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: dgf setup
  • Definition 2.2: Product dgf setup and local-norm notation
  • Definition 2.4: Proximal mappings, Definition 2.2 of karmarkar2025solvingzerosumgames, restated
  • Definition 2.5: $\epsilon$-solution and gap function, Definition 1.1 of karmarkar2025solvingzerosumgames, restated
  • Definition 2.6: Regret
  • Lemma 2.7: Lemma 2.3 of karmarkar2025solvingzerosumgames, restated
  • Definition 4.1: $\epsilon'$-$\textsc{DMP}$
  • Lemma 4.2
  • ...and 75 more