A globally convergent stochastic semismooth Newton method for strongly convex problems in Hilbert spaces

We propose and analyze a globalized stochastic semismooth Newton method for stochastic optimization problems in Hilbert spaces where the objective is strongly convex with Lipschitz continuous, Newton differentiable gradient. Gradient and generalized Hessian information are only available through mini-batching. In each iteration, a full semismooth Newton trial step is accepted or rejected based on a stochastic residual test and rejected steps are replaced by damped steps in the same Newton direction. Throughout, we admit gradient estimators with affinely growing variance, as arises in optimization problems with random partial differential equations as constraints. We prove non-asymptotic global R-linear convergence in expectation as well as almost sure convergence. For sufficiently fast-growing batch sizes, the method transitions almost surely into a regime of pure Newton steps and converges R-superlinearly almost surely. The method is applied to the optimal control of a random elliptic equation featuring a Huber tracking term as well as a random parabolic equation with a Moreau--Yosida penalization of an almost sure state constraint.