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Stochastic gradient methods with momentum are widely used in applications andat the core of optimization subroutines in many popular machine learninglibraries. However, their sample complexities have not been obtained forproblems beyond those that are convex or smooth. This paper establishes theconvergence rate of a stochastic subgradient method with a momentum term ofPolyak type for a broad class of non-smooth, non-convex, and constrainedoptimization problems. Our key innovation is the construction of a specialLyapunov function for which the proven complexity can be achieved without anytuning of the momentum parameter. For smooth problems, we extend the knowncomplexity bound to the constrained case and demonstrate how the unconstrainedcase can be analyzed under weaker assumptions than the state-of-the-art.Numerical results confirm our theoretical developments.

Convergence of a Stochastic Gradient Method with Momentum for Non-Smooth Non-Convex Optimization