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Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

Patrick L. Combettes (plc***at***ljll.math.upmc.fr)
Jean-Christophe Pesquet (jean-christophe.pesquet***at***univ-paris-est.fr)

Abstract: We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems.

Keywords: convex optimization, forward-backward algorithm, monotone operators, primal-dual algorithm, proximal gradient method, stochastic approximation

Category 1: Stochastic Programming

Category 2: Infinite Dimensional Optimization


Download: [Postscript][PDF]

Entry Submitted: 07/24/2015
Entry Accepted: 07/27/2015
Entry Last Modified: 07/27/2015

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