Forward Looking Best-Response Multiplicative Weights Update Methods for Bilinear Zero-sum Games

TL;DR

This paper introduces a novel extra gradient learning algorithm for bilinear zero-sum games that guarantees last-iterate convergence to Nash equilibria, outperforming existing methods in convergence speed and practical efficiency.

Contribution

The paper proposes a new variant of optimistic mirror descent with large intermediate steps, ensuring convergence to Nash equilibria in bilinear zero-sum games, with theoretical guarantees and practical improvements.

Findings

Guarantees last-iterate convergence to Nash equilibrium.

Achieves faster convergence compared to existing methods.

Experimental results show significant practical acceleration.

Abstract

Our work focuses on extra gradient learning algorithms for finding Nash equilibria in bilinear zero-sum games. The proposed method, which can be formally considered as a variant of Optimistic Mirror Descent \cite{DBLP:conf/iclr/MertikopoulosLZ19}, uses a large learning rate for the intermediate gradient step which essentially leads to computing (approximate) best response strategies against the profile of the previous iteration. Although counter-intuitive at first sight due to the irrationally large, for an iterative algorithm, intermediate learning step, we prove that the method guarantees last-iterate convergence to an equilibrium. Particularly, we show that the algorithm reaches first an $\eta^{1/\rho}$-approximate Nash equilibrium, with $\rho > 1$, by decreasing the Kullback-Leibler divergence of each iterate by at least $\Omega(\eta^{1+\frac{1}{\rho}})$, for sufficiently small learning rate, $\eta$, until the method becomes a contracting map, and converges to the exact equilibrium. Furthermore, we perform experimental comparisons with the optimistic variant of the multiplicative weights update method, by \cite{Daskalakis2019LastIterateCZ} and show that our algorithm has significant practical potential since it offers substantial gains in terms of accelerated convergence.