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Stochastic Alternating Direction Method of Multipliers

Hua Ouyang, Niao He, Long Tran, Alexander Gray
JMLR W&CP 28 (1) : 80–88, 2013


The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from large-scale and data-distributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with non-smooth composite objective functions. To solve this problem, we propose a stochastic ADMM algorithm. Our algorithm applies to a more general class of convex and nonsmooth objective functions, beyond the smooth and separable least squares loss used in lasso. We also demonstrate the rates of convergence for our algorithm under various structural assumptions of the stochastic function: \(O(1/\sqrt{t})\) for convex functions and \(O(\log t/t)\) for strongly convex functions. Compared to previous literature, we establish the convergence rate of ADMM for convex problems in terms of both the objective value and the feasibility violation. A novel application named Graph-Guided SVM is proposed to demonstrate the usefulness of our algorithm.

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