Accelerated Variance Reduced Stochastic ADMM

TL;DR

This paper introduces an accelerated stochastic ADMM method that incorporates momentum to improve convergence rates, achieving linear convergence for strongly convex problems and an O(1/T^2) rate for general convex problems, narrowing the gap with batch algorithms.

Contribution

The paper proposes ASVRG-ADMM, an accelerated stochastic ADMM algorithm with momentum, providing faster convergence rates for both strongly convex and general convex problems.

Findings

Achieves linear convergence for strongly convex problems.

Improves convergence rate to O(1/T^2) for general convex problems.

Experimental results confirm the effectiveness of the proposed method.

Abstract

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.