A Stochastic Proximal Point Algorithm: Convergence and Application to Convex Optimization

Maximal monotone operators are set-valued mappings which extend (but are not limited to) the notion of subdifferential of a convex function. The proximal point algorithm is a method for finding a zero of a maximal monotone operator. The algorithm consists in fixed point iterations of a mapping called the resolvent which depends on the maximal monotone operator of interest. The paper investigates a stochastic version of the algorithm where the resolvent used at iteration k is associated to one realization of a random maximal monotone operator. We establish the almost sure ergodic convergence of the iterates to a zero of the expectation (in the Aumann sense) of the latter random operator. Application to constrained stochastic optimization is considered.