Stochastic methods for composite and weakly convex optimization problems

We consider minimization of stochastic functionals that are compositions of a (potentially) nonsmooth convex function h and smooth function c and, more generally, stochastic weakly convex functionals. We develop a family of stochastic methods—including a stochastic prox-linear algorithm and a stochastic (generalized) subgradient procedure—and prove that, under mild technical conditions, each converges to first order stationary points of the stochastic objective. We provide experiments further investigating our methods on nonsmooth phase retrieval problems; the experiments indicate the practical effectiveness of the procedures.