Title: Continuous-time Analysis of Adam in Min-Max Games: Local Convergence and Implicit Gradient Regularization
Primary area: learning theory
Keywords: Continuous-time Analysis, Min-Max Games, Adam

Abstract: Although Adam has been widely used in solving min-max games, its dynamical behavior in this setting has not been studied as extensively as in standard minimization. In min-max games, each player aims to minimize their own loss function, which may lead to the intuitive assumption that Adam’s properties in minimization naturally extend to min-max games with minor adjustments.  In this paper, we demonstrate that this belief is incorrect. Specifically, we investigate the local convergence and implicit gradient regularization aspects of Adam in min-max games, focusing on how its first- and second-order momentum parameters influence the dynamics. We show that, in both aspects, the momentum parameters affect Adam’s behavior in min-max games in precisely the opposite manner to their effect in minimization. Our methodology is based on the continuous-time analysis techniques commonly used for studying learning algorithms. We further provide numerical experiments to support our theoretical findings.

TL;DR: We proposed continuous-time models for Adam in min-max games and studied their dynamical behaviors.