Title: Improved Last-iterate Convergence Properties for the FLBR Dynamics
Primary area: optimization
Keywords: Nash equilibria, Zero-sum games, Duality gap, last-iterate convergence

Abstract: The recent years have seen a surge of interest in algorithms with last-iterate convergence for 2-player games, motivated in part by applications in machine learning. 
Driven by this, we revisit a variant of Multiplicative Weights Update (MWU), defined recently by Fasoulakis et al. (2022), and denoted as Forward Looking Best Response MWU (FLBR-MWU). These dynamics are based on the approach of extra gradient methods, with the tweak of using a different learning rate in the intermediate step. So far, it has been proved that this algorithm attains asymptotic convergence but no explicit rate has been known. We answer the open question from Fasoulakis et al. by establishing a geometric convergence rate for the duality gap. In particular, we first show such a rate, of the form $O(c^t)$, till we reach an approximate Nash equilibrium, where $c<1$ is independent of the game parameters. We then prove that from that point onwards, the duality gap keeps getting decreased with a geometric rate, albeit with a dependence on the maximum eigenvalue of the Jacobian matrix. Finally, we complement our theoretical analysis with an experimental comparison to OGDA, which ranks among the best last-iterate methods for solving 0-sum games. Although in practice it does not generally outperform OGDA, it is often comparable, with a similar average performance.

TL;DR: The main result of the paper is  an improved analysis with a concrete convergence rate for the method FLBR-MWU for 0-sum games