Convergence of the Iterates of the Stochastic Proximal Gradient Method

We propose a novel study of the stochastic proximal gradient method for minimizing the sum of two convex functions, one of which is smooth. Under suitable assumptions and without requiring any boundedness or control of the variance of the random variables, we derive the almost sure convergence and the convergence in the mean of the iterates to a solution of the minimization problem. The results are applied to classification and convex feasibility problems.