Convergence of a Stochastic Gradient Method with Momentum for Nonsmooth Nonconvex Optimization

Stochastic gradient methods with momentum are widely used in applications and at the core of optimization subroutines in many popular machine learning libraries. However, their sample complexities have never been obtained for problems that are non-convex and non-smooth. This paper establishes the convergence rate of a stochastic subgradient method with a momentum term of Polyak type for a broad class of non-smooth, non-convex, and constrained optimization problems. Our key innovation is the construction of a special Lyapunov function for which the proven complexity can be achieved without any tunning of the momentum parameter. For smooth problems, we extend the known complexity bound to the constrained case and demonstrate how the unconstrained case can be analyzed under weaker assumptions than the state-of-the-art. Numerical results confirm our theoretical developments.