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.

翻译：我们研究在玩家策略被约束在简单集合内时，计算具有有界支付矩阵 $A \in \mathbb{R}^{m \times n}$ 的双人双线性博弈的 $ε$-近似纳什均衡问题。我们在两种经过充分研究的情形下，提供了在 $\tilde{O}(ε^{-2/3})$ 次矩阵向量乘法内解决此问题的算法：$\ell_1$-$\ell_1$（或零和）博弈，其中双方玩家的策略均位于概率单纯形内；以及 $\ell_2$-$\ell_1$ 博弈（涵盖硬间隔支持向量机），其中玩家的策略分别位于单位欧几里得球和概率单纯形内。这些结果改进了先前由 [KOS '25] 实现的 $\ell_1$-$\ell_1$ 博弈 $\tilde{O}(ε^{-8/9})$ 和 $\ell_2$-$\ell_1$ 博弈 $\tilde{O}(ε^{-7/9})$ 的最优复杂度。在这两种设定下，我们的结果几乎是理论最优的，因为它们与 [KS '25] 给出的下界在多项式对数因子内相匹配。