Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization

TL;DR

This paper introduces a decentralized algorithm for convex composite optimization that operates with network-independent stepsizes and handles inexact proximal mappings, achieving optimal convergence rates.

Contribution

It presents a novel CTA-based decentralized method with uncoordinated stepsizes and inexact proximal solutions, improving convergence analysis for convex composite problems.

Findings

Achieves an O(1/k) convergence rate for convex problems.

Improves to o(1/k) with exact proximal solutions.

Establishes linear convergence under metric subregularity.

Abstract

This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general convex case, we prove an O(1/k) convergence rate of the proposed algorithm, which can be improved to o(1/k) if the proximal mappings are solved exactly. Furthermore, with metric subregularity, we establish a linear convergence rate for the proposed algorithm. Numerical experiments demonstrate the efficiency of the algorithm.