Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games

Yi Feng, Kaito Fujii, Stratis Skoulakis, Xiao Wang, Volkan Cevher

TL;DR: We study the dynamical behaviors of heavy ball momentum in min-max games.

Abstract: Since Polyak's pioneering work, heavy ball (HB) momentum has been widely studied in minimization. However, its role in min-max games remains largely unexplored. As a key component of practical min-max algorithms like Adam, this gap limits their effectiveness. In this paper, we present a continuous-time analysis for HB with simultaneous and alternating update schemes in min-max games. Locally, we prove *smaller* momentum enhances algorithmic stability by enabling local convergence across a wider range of step sizes, with alternating updates generally converging faster. Globally, we study the implicit regularization of HB, and find *smaller* momentum guides algorithms trajectories towards shallower slope regions of the loss landscapes, with alternating updates amplifying this effect. Surprisingly, all these phenomena differ from those observed in minimization, where *larger* momentum yields similar effects. Our results reveal fundamental differences between HB in min-max games and minimization, and numerical experiments further validate our theoretical results.

Lay Summary: This paper presents a continuous-time analysis of the Heavy Ball momentum method in the context of min-max games, which are optimization problems involving two players with opposing objectives, such as in GANs or adversarial training. The authors aim to bridge the gap in understanding HB momentum, a common component in algorithms like Adam, for min-max games, which has been less explored compared to its application in minimization tasks.

Primary Area: Theory->Game Theory

Keywords: Min-Max Games, Heavy Ball Momentum, Continuous-Time Analysis

Submission Number: 6923