We consider the stochastic composition optimization problem proposed in wang2017stochastic , which has applications ranging from estimation to statistical and machine learning. We propose the first ADMM-based algorithm named com-SVR-ADMM, and show that com-SVR-ADMM converges linearly for strongly convex and Lipschitz smooth objectives, and has a convergence rate of @math , which improves upon the @math rate in wang2016accelerating when the objective is convex and Lipschitz smooth. Moreover, com-SVR-ADMM possesses a rate of @math when the objective is convex but without Lipschitz smoothness. We also conduct experiments and show that it outperforms existing algorithms.
The ADMM algorithm, on the other hand, was first proposed in @cite_5 @cite_20 and later reviewed in @cite_14 . Since then, several ADMM-based stochastic algorithms have been proposed, e.g., @cite_2 @cite_7 @cite_11 . However, these algorithms all possess sublinear convergence rates. Therefore, several recent works tried to combine the variance reduction scheme and ADMM to accelerate convergence. For instance, SVRG-ADMM was proposed in @cite_3 . It was shown that SVRG-ADMM converges linearly when the objective is strongly convex, and has a sublinear convergence rate in the general convex case. Another recent work @cite_16 further proved that SVRG-ADMM converges to a stationary point with a rate @math when the objective is nonconvex. In @cite_17 , the authors used acceleration technique in @cite_6 @cite_10 to further improve the convergence rate of SVRG-ADMM. However, all aforementioned variance-reduced ADMM algorithms cannot be directly applied to solving the stochastic composition optimization problem.
@cite_14 @cite_7 @cite_17 @cite_3 @cite_6 @cite_2 @cite_5 @cite_16 @cite_10 @cite_20 @cite_11
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Recently, many variance reduced stochastic alternating direction method of multipliers (ADMM) methods (e.g. SAG-ADMM, SDCA-ADMM and SVRG-ADMM) have made exciting progress such as linear convergence rates for strongly convex problems. However, the best known convergence rate for general convex problems is O(1 T ) as opposed to O(1 T 2 ) of accelerated batch algorithms, where T is the number of iterations. Thus, there still remains a gap in convergence rates between existing stochastic ADMM and batch algorithms. To bridge this gap, we introduce the momentum acceleration trick for batch optimization into the stochastic variance reduced gradient based ADMM (SVRG-ADMM), which leads to an accelerated (ASVRG-ADMM) method. Then we design two different momentum term update rules for strongly convex and general convex cases. We prove that ASVRG-ADMM converges linearly for strongly convex problems. Besides having a low-iteration complexity as existing stochastic ADMM methods, ASVRG-ADMM improves the convergence rate on general convex problems from O(1 T ) to O(1 T 2 ). Our experimental results show the effectiveness of ASVRG-ADMM. In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stochastic variance reduced gradient ADMM (SVRG-ADMM), uses a multi-stage scheme to progressively reduce the variance of stochastic gradients. The second is the nonconvex stochastic average gradient ADMM (SAG-ADMM), which additionally uses the old gradients estimated in the previous iteration. The third called SAGA-ADMM is an extension of the SAG-ADMM method. Moreover, under some mild conditions, we establish the iteration complexity bound of @math of the proposed methods to obtain an @math -stationary solution of the nonconvex optimizations. In particular, we provide a general framework to analyze the iteration complexity of these nonconvex stochastic ADMM methods with variance reduction. Finally, some numerical experiments demonstrate the effectiveness of our methods. 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 √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. In this paper, we propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, the proposed algorithm improves the convergence rate on convex problems from @math to @math , where @math is the number of iterations. This matches the convergence rate of the batch ADMM algorithm, but without the need to visit all the samples in each iteration. Experiments on the graph-guided fused lasso demonstrate that the new algorithm is significantly faster than state-of-the-art stochastic and batch ADMM algorithms.