SI-ADMM: A Stochastic Inexact ADMM Framework for Stochastic Convex Programs

We consider the structured stochastic convex program requiring the minimization of \mathbb {E}_\xi [\tilde{f}(x,\xi)]+\mathbb {E}_\xi [\tilde{g}(y,\xi)] subject to the constraint Ax + By = b. Motivated by the need for decentralized schemes, we propose a stochastic inexact alternating direction method of multiplier (SI-ADMM) framework where subproblems are solved inexactly via stochastic approximation schemes. we propose a stochastic inexact alternating direction method of multiplier (SI-ADMM) framework where subproblems are solved inexactly via stochastic approximation schemes. Based on this framework, we prove the following: 1) under suitable assumptions on the associated batch-size of samples utilized at each iteration, the SI-ADMM scheme produces a sequence that converges to the unique solution almost surely; 2) if the number of gradient steps (or equivalently, the number of sampled gradients) utilized for solving the subproblems in each iteration increases at a geometric rate, the mean-squared error diminishes to zero at a prescribed geometric rate; and 3) the overall iteration complexity in terms of gradient steps (or equivalently samples) is found to be consistent with the canonical level of \mathcal {O}(1/\epsilon). Preliminary applications on LASSO and distributed regression suggest that the scheme performs well compared to its competitors.