DISA: A Dual Inexact Splitting Algorithm for Distributed Convex Composite Optimization

In this article, we propose a novel dual inexact splitting algorithm (DISA) for distributed convex composite optimization problems, where the local loss function consists of a smooth term and a possibly nonsmooth term composed of a linear mapping. The DISA, for the first time, eliminates the dependence of the convergent step-size range on the Euclidean norm of the linear mapping, while inheriting the advantages of the classic primal-dual proximal splitting algorithm (PD-PSA): simple structure and easy implementation. This indicates that the DISA can be executed without prior knowledge of the norm, and tiny step sizes can be avoided when the norm is large. In addition, we prove sublinear and linear convergence rates of DISA under general convexity and metric subregularity, respectively. Moreover, we provide a variant of DISA with approximate proximal mapping and prove its global convergence and sublinear convergence rate. Numerical experiments corroborate our theoretical analyses and demonstrate a significant acceleration of the DISA compared to the existing PD-PSAs