Compressed Gradient Tracking Methods for Decentralized Optimization with Linear Convergence

TL;DR

This paper introduces novel compressed gradient tracking algorithms for decentralized optimization, achieving linear convergence with both unbiased and biased compression, and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Contribution

The paper proposes a new compressed gradient tracking algorithm compatible with various compression operators, including biased ones, and establishes its linear convergence for strongly convex functions.

Findings

C-GT achieves linear convergence with general compression operators.

EF-C-GT improves efficiency for biased compression.

Numerical results validate theoretical claims and demonstrate effectiveness.

Abstract

Communication compression techniques are of growing interests for solving the decentralized optimization problem under limited communication, where the global objective is to minimize the average of local cost functions over a multi-agent network using only local computation and peer-to-peer communication. In this paper, we first propose a novel compressed gradient tracking algorithm (C-GT) that combines gradient tracking technique with communication compression. In particular, C-GT is compatible with a general class of compression operators that unifies both unbiased and biased compressors. We show that C-GT inherits the advantages of gradient tracking-based algorithms and achieves linear convergence rate for strongly convex and smooth objective functions. In the second part of this paper, we propose an error feedback based compressed gradient tracking algorithm (EF-C-GT) to further improve the algorithm efficiency for biased compression operators. Numerical examples complement the theoretical findings and demonstrate the efficiency and flexibility of the proposed algorithms.