Journal of Convex Analysis 26 (2019), No. 2, 397--436
Copyright Heldermann Verlag 2019
A Constant Step Forward-Backward Algorithm Involving Random Maximal Monotone Operators
Pascal Bianchi
LTCI -- Télécom ParisTech, Uni. Paris-Saclay, 46 rue Barrault, 75634 Paris Cd. 13, France
pascal.bianchi@telecom-paristech.fr
Walid Hachem
CNRS / LIGM (UMR 8049), Université Paris-Est, 5 blvd. Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cd. 2, France
walid.hachem@u-pem.fr
Adil Salim
LTCI -- Télécom ParisTech, Uni. Paris-Saclay, 46 rue Barrault, 75634 Paris Cd. 13, France
adil.salim@telecom-paristech.fr
A stochastic Forward-Backward algorithm with a constant step is studied. At each time step, this algorithm involves an independent copy of a couple of random maximal monotone operators. Defining a mean operator as a selection integral, the differential inclusion built from the sum of the two mean operators is considered. As a first result, it is shown that the interpolated process obtained from the iterates converges narrowly in the small step regime to the solution of this differential inclusion. In order to control the long term behavior of the iterates, a stability result is needed in addition. To this end, the sequence of the iterates is seen as a homogeneous Feller Markov chain whose transition kernel is parameterized by the algorithm step size. The cluster points of the Markov chains invariant measures in the small step regime are invariant for the semiflow induced by the differential inclusion. Conclusions regarding the long run behavior of the iterates for small steps are drawn. It is shown that when the sum of the mean operators is demipositive, the probabilities that the iterates are away from the set of zeros of this sum are small in Ces\`aro mean. The ergodic behavior of these iterates is studied as well. Applications of the proposed algorithm are considered. In particular, a detailed analysis of the random proximal gradient algorithm with constant step is performed.
Keywords: Dynamical systems, narrow convergence of stochastic processes, random maximal monotone operators, stochastic approximation with constant step, stochastic forward-backward algorithm, stochastic proximal point algorithm.
MSC: 47H05, 47N10, 62L20, 34A60
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