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Stochastic approximation techniques have been used in various contexts indata science. We propose a stochastic version of the forward-backward algorithmfor minimizing the sum of two convex functions, one of which is not necessarilysmooth. Our framework can handle stochastic approximations of the gradient ofthe smooth function and allows for stochastic errors in the evaluation of theproximity operator of the nonsmooth function. The almost sure convergence ofthe iterates generated by the algorithm to a minimizer is established underrelatively mild assumptions. We also propose a stochastic version of a popularprimal-dual proximal splitting algorithm, establish its convergence, and applyit to an online image restoration problem.

Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration