Last-Iterate Convergence with Full and Noisy Feedback in Two-Player Zero-Sum Games

TL;DR

This paper introduces M2WU, a new algorithm for two-player zero-sum games that guarantees last-iterate convergence to Nash equilibria even with noisy feedback, outperforming existing methods.

Contribution

The paper proposes M2WU, a mutation-driven multiplicative weights update algorithm that ensures last-iterate convergence in both full and noisy feedback scenarios, with theoretical and empirical validation.

Findings

M2WU converges to a stationary point near Nash equilibrium under noisy feedback.

M2WU achieves exact Nash equilibrium through iterative mutation adjustments.

Empirical results show M2WU outperforms MWU and OMWU in exploitability and convergence speed.

Abstract

This paper proposes Mutation-Driven Multiplicative Weights Update (M2WU) for learning an equilibrium in two-player zero-sum normal-form games and proves that it exhibits the last-iterate convergence property in both full and noisy feedback settings. In the former, players observe their exact gradient vectors of the utility functions. In the latter, they only observe the noisy gradient vectors. Even the celebrated Multiplicative Weights Update (MWU) and Optimistic MWU (OMWU) algorithms may not converge to a Nash equilibrium with noisy feedback. On the contrary, M2WU exhibits the last-iterate convergence to a stationary point near a Nash equilibrium in both feedback settings. We then prove that it converges to an exact Nash equilibrium by iteratively adapting the mutation term. We empirically confirm that M2WU outperforms MWU and OMWU in exploitability and convergence rates.