Online forward-backward algorithm using random maximal monotone operators

Maximal monotone operators are set-valued mappings which extend (but are not limited to) the notion of subdifferential of a convex function. The forward-backward splitting algorithm is one of the most studied algorithms for finding iteratively a zero of a sum of two maximal monotone operators A and B. The purpose of this talk is to investigate the case where operators A and B are either unobserved or difficult to handle numerically. We propose an online version of the forward-backward algorithm where at each iteration, A and B are replaced with one realization of some random maximal monotone operators whose expectations (in the Aumann sense) coincide with A and B respectively. Under the assumption of a decreasing step size, we prove that the iterates "shadow" the behavior of a continuous-time differential inclusion and converge almost surely to a zero of A+B provided some conditions. Applications to constrained convex optimization are considered