Convergence analysis of the stochastic reflected forward–backward splitting algorithm

Abstract

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate $${\mathcal {O}}(log(n)/n)$$ in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex–concave saddle point problems, we derive the rate of the primal–dual gap. In particular, we also obtain $${\mathcal {O}}(1/n)$$ rate convergence of the primal–dual gap in the deterministic setting.