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

TL;DR

This paper provides a continuous-time analysis of heavy ball momentum in min-max games, revealing how smaller momentum improves stability and convergence, with distinct behaviors from minimization, supported by theoretical and numerical evidence.

Contribution

It introduces the first continuous-time analysis of heavy ball momentum in min-max games, highlighting how momentum size affects stability, convergence, and implicit regularization in these settings.

Findings

Smaller momentum enhances local stability and convergence.

Alternating updates generally lead to faster convergence.

Smaller momentum guides trajectories towards shallower loss landscape regions.

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.