An LP-Based Approach for Bilinear Saddle Point Problem with Instance-dependent Guarantee and Noisy Feedback
Jiashuo Jiang, Mengxiao Zhang
TL;DR
This work tackles the problem of estimating a Nash equilibrium for two-player zero-sum matrix games with noisy feedback by formulating the equilibrium computation as a pair of primal/dual linear programs and solving them via an LP-resolving, two-stage approach. The first stage identifies the NE support using empirical LPs and samples, then the second stage computes the NE restricted to that support through an adaptive, online-resource-allocation-inspired resolving procedure. The authors establish instance-dependent and independent sample complexity guarantees, parameterized by problem constants $\delta$, $\sigma$, and $\sigma_0$, and develop a doubling trick and estimation procedures to remove the need for prior knowledge of these constants. The approach extends to the dual player and yields a practical, theoretically grounded method for NE estimation under noisy bandit feedback, with applications across dueling bandits, market making, pricing, and blockchain security. Overall, the paper provides a principled LP-based framework that achieves improved, instance-aware sample efficiency for NE estimation in high-dimensional, noisy settings and offers practical estimation strategies for latent problem 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 $O\left(\frac{m_1 m_2}{\varepsilon\min\{δ^2,σ_0^2,σ^3\}} \log\frac{m_1 m_2}{\varepsilon}\right)$ for general $m_1 \times m_2$ game matrices, where $σ$, $σ_0$, $δ$ 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.