We study the problem of solving matrix games of the form $\min_{\mathbf{p}\inΔ}\max_{\mathbf{w}\in\mathcal{W}}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is a matrix and $Δ$ is the probability simplex. This problem encapsulates canonical tasks such as finding a linear separator and computing Nash equilibria in zero-sum games. However, perhaps surprisingly, its inherent complexity (as formalized in the standard framework of oracle complexity) is not well understood. In this work, we first identify different oracle models that are implicitly used by prior algorithms, corresponding to multiplying the matrix $A$ by a vector from either one or both sides. We then prove complexity lower bounds for algorithms under both access models, which in particular imply a separation between them. As our main result, we prove that in the general $\ell_p$/simplex setting where $\mathcal{W}$ is an $\ell_p$ ball for $p\in[1,\infty]$, any algorithm that utilizes two-sided matrix-vector multiplications requires $\tildeΩ(ε^{-2/3})$ iterations to return an $ε$-suboptimal solution. For any $p\in[1,\infty]$, this is either the first lower bound for such problems, or an exponential improvement over the previously best-known results. Moreover, for the canonical tasks of finding a linear separator and computing a Nash equilibrium, our lower bounds match (up to log factors) recent algorithms of Karmarkar, O'Carroll and Sidford (2026), thereby resolving their oracle complexities in a natural setting.

翻译：我们研究求解形式为 $\min_{\mathbf{p}\inΔ}\max_{\mathbf{w}\in\mathcal{W}}\mathbf{p}^{\top}A\mathbf{w}$ 的矩阵博弈问题，其中 $A$ 是一个矩阵，$Δ$ 是概率单纯形。该问题封装了诸如寻找线性分离器以及计算零和博弈中纳什均衡等典型任务。然而，或许令人惊讶的是，其内在复杂度（在预言机复杂度的标准框架下形式化）尚未得到充分理解。在本工作中，我们首先识别了先前算法隐式使用的不同预言机模型，分别对应于将矩阵 $A$ 与来自一侧或两侧的向量相乘。随后，我们证明了两种访问模型下算法的复杂度下界，这尤其揭示了二者之间的分离性。作为主要结果，我们证明在一般 $\ell_p$/单纯形设定下（其中 $\mathcal{W}$ 是 $p\in[1,\infty]$ 时的 $\ell_p$ 球），任何利用双侧矩阵-向量乘法的算法都需要 $\tildeΩ(ε^{-2/3})$ 次迭代才能返回一个 $ε$-次优解。对于任意 $p\in[1,\infty]$，这要么是此类问题的首个下界，要么是对先前已知最佳结果的指数级改进。此外，对于寻找线性分离器和计算纳什均衡这两个典型任务，我们的下界与 Karmarkar、O'Carroll 和 Sidford（2026）近期提出的算法结果（在对数因子内）相匹配，从而在自然设定下解决了它们的预言机复杂度问题。