We develop an implementable stochastic proximal point (SPP) method for a
class of weakly convex, composite optimization problems. The proposed
stochastic proximal point algorithm incorporates a variance reduction mechanism
and the resulting SPP updates are solved using an inexact semismooth Newton
framework. We establish detailed convergence results that take the inexactness
of the SPP steps into account and that are in accordance with existing
convergence guarantees of (proximal) stochastic variance-reduced gradient
methods. Numerical experiments show that the proposed algorithm competes
favorably with other state-of-the-art methods and achieves higher robustness
with respect to the step size selection