Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games
This work addresses a gap in understanding momentum for min-max optimization, which is incremental but important for improving algorithms like Adam in adversarial training.
The paper tackles the role of heavy ball momentum in min-max games, showing that smaller momentum enhances stability and convergence locally, and guides trajectories to shallower loss regions globally, with alternating updates performing better, differing from minimization where larger momentum is beneficial.
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.