重球法在弱条件下实现局部加速收敛，突破传统认知。
Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality
- 基于微分几何视角重审Polyak-Lojasiewicz不等式，揭示收敛机制。
- 在离散时间下，小邻域内对广泛超参数实现局部收敛，含激进动量与步长。
- 首次证明非凸四次可微函数上重球法具有渐近局部加速性，适合优化研究者。
本文分析了Polyak重球法在连续与离散时间下对满足Polyak-Lojasiewicz不等式的非凸C⁴目标函数的收敛性。在这一弱假设下，我们恢复了Polyak在1964年针对强凸情形得出的渐近收敛速率。结果表明，该方法在此类函数上表现出渐近局部加速特性。特别地，在离散时间设置中，只要迭代进入极小值集的足够小邻域，即可对广泛超参数（包括导致全局收敛失败的激进动量与步长）实现局部收敛。我们的方法摒弃传统Lyapunov论证，转而采用Rebjock & Boumal（2025）提出的新型微分几何视角来解析PL不等式。
原文摘要 · Abstract (English)
In this work, we analyze the convergence of Polyak's heavy ball method in both continuous and discrete time for non-convex $C^4$-objective functions satisfying the Polyak-Lojasiewicz inequality. Under this weak assumption, we recover the asymptotic convergence rates originally derived by Polyak in [Polyak, U.S.S.R. Comput. Math. and Math. Phys., 1964] for strongly convex objectives. Our results demonstrate that the heavy ball method exhibits asymptotic local acceleration on this class of functions. In particular, in the discrete time setting, we prove local convergence of the iterates to a minimum once the method enters a sufficiently small neighborhood of the set of minima, for a broad range of hyperparameters, including aggressive choices for the momentum parameter and the step-size for which global convergence is known to fail. Instead of the usually employed Lyapunov-type arguments, our approach leverages a new differential geometric perspective of the Polyak-Lojasiewicz inequality proposed in [Rebjock and Boumal, Math. Program., 2025].
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