SI-ADMM: A stochastic inexact admm framework for resolving structured stochastic convex programs

Abstract

We consider the resolution of the structured stochastic convex program: min E[ f (x;x )]+E[ g(y;x )] such that Ax+By =b. To exploit problem structure and allow for developing distributed schemes, we propose an inexact stochastic generalization in which the subproblems are solved inexactly via stochastic approximation schemes. Based on this framework, we prove the following: (i) when the inexactness sequence satisfies suitable summability properties, the proposed stochastic inexact ADMM (SI-ADMM) scheme produces a sequence that converges to the unique solution almost surely; (ii) if the inexactness is driven to zero at a polynomial (geometric) rate, the sequence converges to the unique solution in a mean-squared sense at a prescribed polynomial (geometric) rate.