A Linearly Convergent Proximal Gradient Algorithm for Decentralized Optimization

TL;DR

This paper introduces a decentralized proximal gradient algorithm that achieves global linear convergence for composite optimization problems with common non-smooth regularization, improving upon existing sublinear methods.

Contribution

The work proposes a new decentralized proximal gradient method with proven linear convergence under common non-smooth regularization, extending analysis to existing algorithms like EXTRA.

Findings

The proposed algorithm converges linearly to the optimal solution.

Analysis provides convergence bounds for the algorithm and the EXTRA method.

Applicable to machine learning problems with shared non-smooth regularization.

Abstract

Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal decentralized methods are known to converge to the optimal solution with sublinear rates, and it remains unclear whether this family of methods can achieve global linear convergence. To tackle this problem, this work assumes the non-smooth regularization term is common across all networked agents, which is the case for many machine learning problems. Under this condition, we design a proximal gradient decentralized algorithm whose fixed point coincides with the desired minimizer. We then provide a concise proof that establishes its linear convergence. In the absence of the non-smooth term, our analysis technique covers the well known EXTRA algorithm and provides useful bounds on the convergence rate and step-size.