A Unified Inexact Stochastic ADMM for Composite Nonconvex and Nonsmooth Optimization

Abstract: In this paper, we propose a unified framework of inexact stochastic Alternating Direction Method of Multipliers (ADMM) for solving nonconvex problems subject to linear constraints, whose objective comprises an average of finite-sum smooth functions and a nonsmooth but possibly nonconvex function. The new framework is highly versatile. Firstly, it not only covers several existing algorithms such as SADMM, SVRG-ADMM, and SPIDER-ADMM but also guides us to design a novel accelerated hybrid stochastic ADMM algorithm, which utilizes a new hybrid estimator to trade-off variance and bias. Second, it enables us to exploit a more flexible dual stepsize in the convergence analysis. Under some mild conditions, our unified framework preserves $\mathcal{O}(1/T)$ sublinear convergence. Additionally, we establish the linear convergence under error bound conditions. Finally, numerical experiments demonstrate the efficacy of the new algorithm for some nonsmooth and nonconvex problems.