  In this work, we present a globalized stochastic semismooth Newton method for
solving stochastic optimization problems involving smooth nonconvex and
nonsmooth convex terms in the objective function. We assume that only noisy
gradient and Hessian information of the smooth part of the objective function
is available via calling stochastic first and second order oracles. The
proposed method can be seen as a hybrid approach combining stochastic
semismooth Newton steps and stochastic proximal gradient steps. Two inexact
growth conditions are incorporated to monitor the convergence and the
acceptance of the semismooth Newton steps and it is shown that the algorithm
converges globally to stationary points in expectation. Moreover, under
standard assumptions and utilizing random matrix concentration inequalities, we
prove that the proposed approach locally turns into a pure stochastic
semismooth Newton method and converges r-superlinearly with high probability.
We present numerical results and comparisons on $\ell_1$-regularized logistic
regression and nonconvex binary classification that demonstrate the efficiency
of our algorithm.
