Optimal last-iterate convergence in matrix games with bandit feedback using the log-barrier

2026-04-16

AI Summary

This work addresses the challenge of achieving optimal last-iterate convergence in zero-sum matrix games with bandit feedback, where existing uncoupled algorithms fall short. The authors propose an online mirror descent method based on log-barrier regularization, analyzed through a dual-space framework, which for the first time attains an exploitability gap convergence rate of Õ(t⁻¹/⁴) with high probability in the uncoupled setting—matching the known theoretical lower bound Ω(t⁻¹/⁴). The approach is further extended to extensive-form games while preserving this optimal convergence rate, thereby significantly advancing the theoretical foundations of uncoupled multi-agent learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of learning minimax policies in zero-sum matrix games. Fiegel et al. (2025) recently showed that achieving last-iterate convergence in this setting is harder when the players are uncoupled, by proving a lower bound on the exploitability gap of Omega(t^{-1/4}). Some online mirror descent algorithms were proposed in the literature for this problem, but none have truly attained this rate yet. We show that the use of a log-barrier regularization, along with a dual-focused analysis, allows this O-tilde(t^{-1/4}) convergence with high-probability. We additionally extend our idea to the setting of extensive-form games, proving a bound with the same rate.

Problem

• last-iterate convergence
• zero-sum matrix games
• bandit feedback
• uncoupled learning
• exploitability gap

Innovation

• log-barrier regularization
• last-iterate convergence
• bandit feedback
• zero-sum matrix games
• online mirror descent