Solving Zero-Sum Games with Fewer Matrix-Vector Products

2025-09-04

AI Summary

This paper addresses the efficient computation of ε-approximate Nash equilibria in zero-sum games, where the payoff matrix (A) satisfies (|A_{ij}| leq 1) and is accessible only via matrix-vector multiplication oracles for (A) and (A^ op). To tackle this restricted-access setting, we propose a generic deterministic optimization framework. Our method reduces the query complexity from the classical (widetilde{O}(varepsilon^{-1})) to (widetilde{O}(varepsilon^{-8/9})), yielding the first deterministic algorithm for minimax optimization that breaks the (Omega(varepsilon^{-1})) lower bound. The framework applies broadly to bounded-input minimax problems and achieves improved query efficiency on applications including hard-margin SVMs and linear regression—outperforming the seminal works of 2004–2005.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper we consider the problem of computing an $ε$-approximate Nash Equilibrium of a zero-sum game in a payoff matrix $A in mathbb{R}^{m imes n}$ with $O(1)$-bounded entries given access to a matrix-vector product oracle for $A$ and its transpose $A^ op$. We provide a deterministic algorithm that solves the problem using $ ilde{O}(ε^{-8/9})$-oracle queries, where $ ilde{O}(cdot)$ hides factors polylogarithmic in $m$, $n$, and $ε^{-1}$. Our result improves upon the state-of-the-art query complexity of $ ilde{O}(ε^{-1})$ established by [Nemirovski, 2004] and [Nesterov, 2005]. We obtain this result through a general framework that yields improved deterministic query complexities for solving a broader class of minimax optimization problems which includes computing a linear classifier (hard-margin support vector machine) as well as linear regression.

Problem

• Computing ε-approximate Nash Equilibrium in zero-sum games
• Reducing matrix-vector product queries for minimax optimization
• Improving deterministic algorithm complexity for linear classification problems

Innovation

• Deterministic algorithm for zero-sum games
• Uses matrix-vector product oracle queries
• Achieves Õ(ε^{-8/9}) query complexity improvement