Solving Matrix Games with Near-Optimal Matvec Complexity
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arXiv
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| Formato: | Preprint |
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2026
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| author | Karmarkar, Ishani O'Carroll, Liam Sidford, Aaron |
| author_facet | Karmarkar, Ishani O'Carroll, Liam Sidford, Aaron |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_02347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving Matrix Games with Near-Optimal Matvec Complexity Karmarkar, Ishani O'Carroll, Liam Sidford, Aaron Optimization and Control Data Structures and Algorithms Computer Science and Game Theory 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. |
| title | Solving Matrix Games with Near-Optimal Matvec Complexity |
| topic | Optimization and Control Data Structures and Algorithms Computer Science and Game Theory |
| url | https://arxiv.org/abs/2601.02347 |