Proceedings of the 2016 Winter Simulation Conference T. M. K. Roeder, P. I. Frazier, R. Szechtman, E. Zhou, T. Huschka, and S. E. Chick, eds. SI-ADMM: A STOCHASTIC INEXACT ADMM FRAMEWORK FOR RESOLVING STRUCTURED STOCHASTIC CONVEX PROGRAMS Yue Xie Uday V. Shanbhag Department of Industrial and Manufacturing Engineering Pennsylvania State University, PA 16803, USA. ABSTRACT We consider the resolution of the structured stochastic convex program: min E[ ˜ f (x, ξ )] + E[ ˜ g(y, ξ )] 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. 1 INTRODUCTION In the context of large datasets, it has become increasingly important to process the data in a parallel and decentralized fashion. Therefore, distributed optimization is often considered an option, and a simple yet powerful algorithm of this kind is the alternating direction method of multipliers (ADMM). ADMM schemes may be traced to mid-70s to the work by Glowinski and Marroco (1975) and subsequently Gabay and Mercier (1976). It has grown immensely in popularity and has been utilized for resolving a host of structured machine learning and image processing problems such as image recovery (Afonso, Bioucas-Dias, and Figueiredo 2010), robust PCA (Lin, Chen, and Ma 2010), low-rank representation (Lin, Liu, and Su 2011); see Boyd, Parikh, Chu, Peleato, and Eckstein (2011) for a comprehensive review. Typically, ADMM is applied towards structured deterministic convex optimization problems of the form: min x,y f (x)+ g(y) subject to Ax + By = b, (1) We consider a stochastic generalization leading to a structured stochastic convex program: min x,y E[ ˜ f (x, ξ )] + E[ ˜ g(y, ξ )] subject to Ax + By = b, (SOpt) where ξ : Ω → R d , ˜ f : R n × R d → R,˜ g : R m × R d → R, A ∈ R p×n , B ∈ R p×m , b ∈ R p , and (Ω, F , P) denotes the probability space. Furthermore, we assume that ˜ f (., ξ ) and ˜ g(., ξ ) are convex in (.) for every ξ ∈ Ξ. A rather popular approach for the solution of (SOpt) is by utilizing Monte-Carlo sampling schemes, such as sample-average approximation (see Shapiro, Dentcheva, and Ruszczy´ nski (2009)) or stochastic approximation schemes (Robbins and Monro 1951). In fact, over the last decade, there has been significant study of stochastic approximation schemes, particularly from the standpoint of the tuning of steplengths (Nemirovski, Juditsky, Lan, and Shapiro (2009); Yousefian, Nedi´ c, and Shanbhag (2012)), the resolution 978-1-5090-4486-3/16/$31.00 ©2016 IEEE 714