Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes

Abstract

In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods using variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deterministic or stochastic ADMM. This new stochastic optimization algorithm is shown to converge in expectation with O(1/T ) convergence rate, where T is the number of outer iterations. In addition, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are also discussed. Our preliminary numerical experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications

Convergence results of two-step inertial proximal point algorithm

“…We remark that the idea of Algorithm 6 can be applied to stochastic ADMM or its multi-block extensions in the recent works in [10,11] for solving problems arising in machine learning because the basic model therein is a two-block separable convex optimization problem.…”

“…We observe that since T := I − J A λ is Lipschitz continuous, there is some flexibility in this choice. Therefore, (11) becomes…”