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SGD with momentum (SGDM) has been widely applied in many machine learningtasks, and it is often applied with dynamic stepsizes and momentum weightstuned in a stagewise manner. Despite of its empirical advantage over SGD, therole of momentum is still unclear in general since previous analyses on SGDMeither provide worse convergence bounds than those of SGD, or assume Lipschitzor quadratic objectives, which fail to hold in practice. Furthermore, the roleof dynamic parameters has not been addressed. In this work, we show that SGDMconverges as fast as SGD for smooth objectives under both strongly convex andnonconvex settings. We also establish \textit{the first} convergence guaranteefor the multistage setting, and show that the multistage strategy is beneficialfor SGDM compared to using fixed parameters. Finally, we verify thesetheoretical claims by numerical experiments.

An Improved Analysis of Stochastic Gradient Descent with Momentum