Ergodic convergence of a stochastic proximal point algorithm

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates (xn) given by xn+1 = (I+λn A(xin+1,,.,))-1(xn) where (A(s,,.,):s∈ E) is a collection of maximal monotone operators on a separable Hilbert space, (xin) is an independent identically distributed sequence of random variables on E and (λn) is a positive sequence in ℓ2backslash ℓ1. The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation mathbb E(A(xi1,,.,)) under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form min mathbb E(f(xi1,x)) w.r.t. x∈ bigcapi=1m Xi where f is a normal convex integrand and (Xi) is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas.