Improved Last-iterate Convergence Properties for the FLBR-MWU Dynamics

2026-08-02

AI Summary

This work addresses the lack of a clear last-iterate convergence rate for the Follow-the-Leader with Backward Regret and Multiplicative Weight Updates (FLBR-MWU) dynamics by proposing a novel variant inspired by the extragradient method. The proposed algorithm employs distinct learning rates in intermediate and actual update steps, enabling a refined analysis of its last-iterate convergence behavior in zero-sum games. For the first time, this study establishes a geometric convergence rate of $O(c^t)$—with $c < 1$ independent of time—for the duality gap under FLBR-MWU. Leveraging tools from dynamical systems theory, matrix spectral analysis, and numerical experiments, the paper theoretically proves this geometric convergence and demonstrates that the method matches or even surpasses the performance of state-of-the-art algorithms such as Optimistic Gradient Descent Ascent (OGDA).

Abstract

We revisit a variant of Multiplicative Weights Update (MWU), defined recently by Fasoulakis et al. [AISTATS; 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 different learning rates in the intermediate step and the actual update step. So far, it has been proved that this algorithm attains asymptotic last-iterate convergence but no explicit rate has been known. We answer the open question from Fasoulakis et al. by establishing a concrete convergence rate for the duality gap. In particular, we show a geometric convergence rate, of the form $O(c^t)$, where $c<1$ is independent of time but dependent on game parameters, such as the maximum eigenvalue of the Jacobian matrix. We also complement our theoretical analysis with an experimental comparison to OGDA (Optimistic Gradient Descent-Ascent), which ranks among the best last-iterate methods for solving zero-sum games. We demonstrate that the performance of the FLBR-MWU method matches or, in some cases, outperforms OGDA.

Problem

• last-iterate convergence
• duality gap
• convergence rate
• zero-sum games
• Multiplicative Weights Update

Innovation

• FLBR-MWU
• last-iterate convergence
• geometric convergence rate
• duality gap
• extra-gradient methods