Accelerated Variance Reduced Stochastic ADMM

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 per-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.Comment: 16 pages, 5 figures, Appears in Proceedings of the 31th AAAI Conference on Artificial Intelligence (AAAI), San Francisco, California, USA, pp. 2287--2293, 201

Accelerated Method for Stochastic Composition Optimization with Nonsmooth Regularization

Stochastic composition optimization draws much attention recently and has been successful in many emerging applications of machine learning, statistical analysis, and reinforcement learning. In this paper, we focus on the composition problem with nonsmooth regularization penalty. Previous works either have slow convergence rate or do not provide complete convergence analysis for the general problem. In this paper, we tackle these two issues by proposing a new stochastic composition optimization method for composition problem with nonsmooth regularization penalty. In our method, we apply variance reduction technique to accelerate the speed of convergence. To the best of our knowledge, our method admits the fastest convergence rate for stochastic composition optimization: for strongly convex composition problem, our algorithm is proved to admit linear convergence; for general composition problem, our algorithm significantly improves the state-of-the-art convergence rate from $O(T^{-1/2})$ to $O((n_1+n_2)^{{2}/{3}}T^{-1})$. Finally, we apply our proposed algorithm to portfolio management and policy evaluation in reinforcement learning. Experimental results verify our theoretical analysis.Comment: AAAI 201

Scalable Peaceman-Rachford Splitting Method with Proximal Terms

Along with developing of Peaceman-Rachford Splittling Method (PRSM), many batch algorithms based on it have been studied very deeply. But almost no algorithm focused on the performance of stochastic version of PRSM. In this paper, we propose a new stochastic algorithm based on PRSM, prove its convergence rate in ergodic sense, and test its performance on both artificial and real data. We show that our proposed algorithm, Stochastic Scalable PRSM (SS-PRSM), enjoys the $O(1/K)$ convergence rate, which is the same as those newest stochastic algorithms that based on ADMM but faster than general Stochastic ADMM (which is $O(1/\sqrt{K})$). Our algorithm also owns wide flexibility, outperforms many state-of-the-art stochastic algorithms coming from ADMM, and has low memory cost in large-scale splitting optimization problems