The Oracle Complexity of Simplex-based Matrix Games

TL;DR

This paper investigates the inherent complexity of solving matrix games using simplex-based algorithms, establishing lower bounds on the number of iterations needed and clarifying the difficulty of canonical tasks like Nash equilibrium computation.

Contribution

It introduces new oracle models for matrix game algorithms, proves lower bounds on iteration complexity, and resolves the oracle complexities for key canonical tasks.

Findings

Two-sided matrix-vector multiplication algorithms require at least rac{}{} ext{ iterations for psilon solutions.

Established lower bounds match recent algorithms for Nash equilibrium and linear separator tasks.

Separated different oracle models, clarifying their impact on complexity.

Abstract

We study the problem of solving matrix games of the form $\min_{\mathbf{p}\in\Delta}\max_{\mathbf{w}\in\mathcal{W}}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is a matrix and $\Delta$ 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 $\tilde{\Omega}(\epsilon^{-2/3})$ iterations to return an $\epsilon$-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.