We study the problem of solving matrix games of the form $\max_{\mathbf{w}\in\mathcal{W}}\min_{\mathbf{p}\inΔ}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is some 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 [Nemirovski and Yudin, 1983]) is not well-understood. In this work, we first identify different oracle models which are implicitly used by prior algorithms, amounting 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. Specifically, we start by showing that algorithms for linear separability based on one-sided multiplications must require $Ω(γ_A^{-2})$ iterations, where $γ_A$ is the margin, as matched by the Perceptron algorithm. We then prove that accelerated algorithms for this task, which utilize multiplications from both sides, must require $\tildeΩ(γ_{A}^{-2/3})$ iterations, establishing the first oracle complexity barrier for such algorithms. Finally, by adapting our lower bound to $\ell_1$ geometry, we prove that computing an $ε$-approximate Nash equilibrium requires $\tildeΩ(ε^{-2/3})$ iterations, which is an exponential improvement over the previously best-known lower bound due to Hadiji et al. [2024].

翻译：我们研究求解形式为$\max_{\mathbf{w}\in\mathcal{W}}\min_{\mathbf{p}\inΔ}\mathbf{p}^{\top}A\mathbf{w}$的矩阵博弈问题，其中$A$为某矩阵，$Δ$为概率单纯形。该问题囊括了寻找线性分类器及计算零和博弈中纳什均衡等典型任务。然而可能令人惊讶的是，其内在复杂度（以[Nemirovski and Yudin, 1983]提出的标准预言复杂度框架形式化）尚未被充分理解。本工作中，我们首先识别出先前算法隐式采用的不同预言模型，这些模型相当于从单侧或双侧将矩阵$A$与向量相乘。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。具体而言，我们首先证明基于单侧乘法运算的线性可分性算法必须需要$Ω(γ_A^{-2})$次迭代（其中$γ_A$为间隔），这与感知机算法的迭代次数匹配。接着我们证明，对此任务采用双侧乘法运算的加速算法必须需要$\tildeΩ(γ_{A}^{-2/3})$次迭代，这为此类算法建立了首个预言复杂度障碍。最后，通过将下界适配至$\ell_1$几何空间，我们证明计算$ε$近似纳什均衡需要$\tildeΩ(ε^{-2/3})$次迭代，这相对于Hadiji等人[2024]提出的先前最佳下界实现了指数级改进。