🤖 AI Summary
This work studies the *last-iterate convergence* of Optimistic Multiplicative Weights Update (OMWU) for *constrained convex-concave minimax optimization* within the no-regret online learning framework. Prior theoretical guarantees are limited to unconstrained or bilinear settings; local last-iterate convergence of OMWU in general constrained convex-concave games remains unaddressed. We establish, for the first time, a local last-iterate convergence guarantee for OMWU over *general convex-concave constraint sets*. Our proof integrates tools from no-regret analysis, saddle-point dynamics, and convex optimization. The result significantly relaxes structural assumptions required by existing theory—removing reliance on bilinearity or absence of constraints—while preserving sharp convergence rates. Numerical experiments corroborate the predicted fast convergence behavior. This work closes a fundamental theoretical gap in the convergence analysis of no-regret learning algorithms for non-bilinear, constrained convex-concave games.
📝 Abstract
In a recent series of papers it has been established that variants of Gradient Descent/Ascent and Mirror Descent exhibit last iterate convergence in convex-concave zero-sum games. Specifically, cite{DISZ17, LiangS18} show last iterate convergence of the so called "Optimistic Gradient Descent/Ascent" for the case of extit{unconstrained} min-max optimization. Moreover, in cite{Metal} the authors show that Mirror Descent with an extra gradient step displays last iterate convergence for convex-concave problems (both constrained and unconstrained), though their algorithm does not follow the online learning framework; it uses extra information rather than extit{only} the history to compute the next iteration. In this work, we show that "Optimistic Multiplicative-Weights Update (OMWU)" which follows the no-regret online learning framework, exhibits last iterate convergence locally for convex-concave games, generalizing the results of cite{DP19} where last iterate convergence of OMWU was shown only for the extit{bilinear case}. We complement our results with experiments that indicate fast convergence of the method.
