An LP-Based Approach for Bilinear Saddle Point Problem with Instance-dependent Guarantee and Noisy Feedback

TL;DR

This paper introduces a novel LP-based algorithm for estimating Nash equilibria in two-player zero-sum matrix games with noisy feedback, providing the first instance-dependent sample complexity bounds for general matrices.

Contribution

The paper proposes a new LP-based method that achieves instance-dependent sample complexity bounds for Nash equilibrium estimation in noisy, general-dimension matrix games.

Findings

First instance-dependent sample complexity bound for noisy matrix games

Algorithm identifies NE support and computes NE with controlled bias

Sample complexity depends on problem-specific constants

Abstract

In this work, we study the sample complexity of obtaining a Nash equilibrium (NE) estimate in two-player zero-sum matrix games with noisy feedback. Specifically, we propose a novel algorithm that repeatedly solves linear programs (LPs) to obtain an NE estimate with bias at most $\varepsilon$ with a sample complexity of $O\left(\frac{m_1 m_2}{\varepsilon\min\{\delta^2,\sigma_0^2,\sigma^3\}} \log\frac{m_1 m_2}{\varepsilon}\right)$ for general $m_1 \times m_2$ game matrices, where $\sigma$, $\sigma_0$, $\delta$ are some problem-dependent constants. To our knowledge, this is the first instance-dependent sample complexity bound for finding an NE estimate with $\varepsilon$ bias in general-dimension matrix games with noisy feedback and potentially non-unique equilibria. Our algorithm builds on recent advances in online resource allocation and operates in two stages: (1) identifying the support set of an NE, and (2) computing the unique NE restricted to this support. Both stages rely on a careful analysis of LP solutions derived from noisy samples.