On the local convergence of a stochastic semismooth Newton method for nonsmooth nonconvex optimization

In this work, we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function. We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic first- and second-order oracles. The approach combines stochastic semismooth Newton steps, stochastic proximal gradient steps and a globalization strategy based on growth conditions. We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates. Under standard local assumptions, we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.