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 (x_n) given by x_n+1= (I + λ_nA(ξ_n+1; : ))-1(xn); where (A(s; : ) : s ∈ E) is a collection of maximal monotone operators on a separable Hilbert space, (ξn) is an independent identically distributed sequence of random variables on E; and (λ_n) is a positive sequence in ℓ²\ℓ¹. The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation E (A(ξ₁, . )) under the assumption that the latter is maximal. TWe consider applications to stochastic optimization problems of the form min E(f(ξ1; x)) w.r.t. x ∈∩^m _i=1 X_i, where f is a normal convex integrand and (X_i) 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 [Incremental Constraint Projection-Proximal Methods for Nonsmooth Convex Optimization, Tech. report, MIT, Cambridge, MA, 2013].