Stochastic Forward-Backward Splitting for Monotone Inclusions

We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward-backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as L. Rosasco DIBRIS, Università di Genova Genova, Italy [email protected] L. Rosasco, S. Villa (corresponding author) and B. C. Vũ Laboratory for Computational and Statistical Learning Istituto Italiano di Tecnologia and Massachusetts Institute of Technology, Cambridge, USA [email protected], [email protected] 2 Lorenzo Rosasco et al. well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of so called accelerated methods. Stochastic quasi Fejér’s sequences are a key technical tool to prove almost sure convergence.