The Oracle Complexity of Simplex-based Matrix Games: Linear Separability and Nash Equilibria
Guy Kornowski, Ohad Shamir
TL;DR
This work studies the oracle complexity of solving matrix games of the form $\max_{\mathbf{w}\in\mathcal{W}}\min_{\mathbf{p}\in\Delta}\mathbf{p}^\top A\mathbf{w}$, connecting canonical tasks like linear separability and zero-sum Nash equilibria to the Nemirovski–Yudin framework. It formalizes two oracle models—one-sided access with $\mathbf{w}\mapsto A\mathbf{w}$ and row queries, and two-sided access with $\mathbf{p}^\top A$ and $A\mathbf{w}$—and proves distinct lower bounds: $\Omega(\gamma_A^{-2})$ under one-sided access and $\tilde{\Omega}(\gamma_A^{-2/3})$ under two-sided access for linear separability. The results extend to $\ell_1$ geometry to show $\tilde{\Omega}(\epsilon^{-2/3})$ iterations for achieving an $\epsilon$-suboptimal Nash equilibrium, representing a substantial improvement over prior bounds and establishing a separation between oracle models. These findings illuminate intrinsic limits of accelerated methods in size-free regimes and motivate future work on randomized algorithms and tighter upper-lower bound gaps in high-dimensional matrix games.
Abstract
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].