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

2025-05-26

AI Summary

This work bridges a theoretical gap in understanding Heavy-Ball (HB) momentum in min-max optimization. Despite HB’s widespread use as a core component in modern min-max algorithms (e.g., Adam), its dynamical behavior remains poorly understood. We establish continuous-time models to systematically analyze the stability, convergence, and implicit regularization of HB under both simultaneous and alternating update schemes. Our theoretical analysis reveals a counterintuitive phenomenon: small momentum values—contrary to the conventional “large momentum accelerates convergence” intuition in minimization—broaden the stable step-size region and bias trajectories toward flatter solutions. Alternating updates further amplify this effect and induce preference for regions with shallower gradient slopes. We rigorously substantiate these findings via local stability analysis, derivation of implicit regularization, and comprehensive numerical experiments.

Technology Category

• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems

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.

Problem

• Analyzes heavy ball momentum in min-max games
• Explores local and global effects of momentum size
• Compares min-max game dynamics with minimization scenarios

Innovation

• Continuous-time analysis of HB momentum
• Smaller momentum enhances algorithmic stability
• Alternating updates amplify implicit regularization