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

TL;DR

This paper introduces DISA, a novel distributed optimization algorithm that removes the need for prior knowledge of the linear mapping norm, achieves fast convergence, and outperforms existing methods in numerical tests.

Contribution

DISA is the first dual inexact splitting algorithm that eliminates dependence on the linear mapping norm while maintaining simplicity and proving convergence rates.

Findings

DISA achieves sublinear and linear convergence rates.

DISA outperforms existing primal-dual algorithms in experiments.

The variant with approximate proximal mapping also converges globally.

Abstract

In this paper, 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 with a linear mapping. 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 DISA can be executed without prior knowledge of the norm, and tiny step-sizes can be avoided when the norm is large. Additionally, 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 DISA compared to existing PD-PSAs.