ProxSTORM -- A Stochastic Trust-Region Algorithm for Nonsmooth Optimization

Abstract: We develop a stochastic trust-region algorithm for minimizing the sum of a possibly nonconvex Lipschitz-smooth function that can only be evaluated stochastically and a nonsmooth, deterministic, convex function. This algorithm, which we call ProxSTORM, generalizes STORM [1, 2] -- a stochastic trust-region algorithm for the unconstrained optimization of smooth functions -- and the inexact deterministic proximal trust-region algorithm in [3]. We generalize and, in some cases, simplify problem assumptions so that they reduce to more succinct version of assumptions on STORM when the convex term is zero. Our analysis follows the STORM framework by employing martingales, but again simplifies certain steps and proving global convergence and an expected complexity bound in the more general setting of a possibly nonsmooth term. To demonstrate that the method is numerically viable, we apply the algorithm to $\ell^1$-regularized neural network training and also to topology optimization.