Heavy-Ball Method under Randomized Schedules

Abstract: We study how predefined randomized parameter schedules accelerate the heavy-ball method on general smooth convex objectives. Our analysis distinguishes two levels of randomization: sampling gradient-evaluation times within intervals whose boundaries are deterministic, and additionally randomizing the time boundaries themselves. With deterministic time boundaries, we construct fixed-time and anytime schedules that achieve an expected last-iterate function value gap of order $\mathcal{O}(1/K<sup>{4/3})$; the anytime schedule also satisfies the same rate almost surely. We then use randomized time boundaries and obtain the improved last-iterate rate $\mathcal{O}(1/K<sup>{3/2})$, both in expectation and almost surely. To the best of our knowledge, this is the first global nonasymptotic convergence guarantee for the heavy-ball method on general smooth convex functions that improves polynomially over the classical $\mathcal{O}(1/K)$ rate. Our result shows that the heavy-ball method achieves a strictly better convergence rate than the best known result $\mathcal{O}(1/K<sup>{\log_2(1+\sqrt{2})})$ attainable by plain gradient descent with silver stepsize schedules.