揭示了动量在极小极大博弈中与最小化截然不同的行为机制。
Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games
- 通过连续时间分析，比较了同步与交替更新下的动量效果。
- 较小动量更稳定，使算法在更广步长范围内收敛，交替更新更快。
- 动量引导轨迹走向低梯度区域，适用于博弈类优化场景。
自Polyak开创性工作以来，动量（HB）在最小化问题中被广泛研究，但在极小极大博弈中的作用仍不明确。作为实际极小极大算法（如Adam）的关键组件，这一空白限制了其性能。本文对极小极大博弈中同时与交替更新的动量进行了连续时间分析。局部上，证明较小动量增强算法稳定性，扩大收敛步长范围，交替更新通常收敛更快。全局上，研究了动量的隐式正则化效应，发现较小动量引导算法轨迹趋向损失函数曲面的浅梯度区域，且交替更新会放大此效应。令人惊讶的是，这些现象与最小化问题中较大动量产生类似效果的情况完全不同。结果揭示了极小极大博弈中动量与最小化之间的根本差异，数值实验进一步验证了理论发现。
原文摘要 · Abstract (English)
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.
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