Communication Compression for Distributed Nonconvex Optimization

TL;DR

This paper introduces three distributed primal-dual algorithms with compressed communication for nonconvex optimization, achieving comparable convergence rates to exact communication methods while significantly reducing communication load.

Contribution

The paper proposes novel distributed algorithms that incorporate communication compression with theoretical convergence guarantees for nonconvex optimization.

Findings

Algorithms achieve sublinear convergence to stationary points with compressed communication.

Algorithms attain linear convergence under Polyak–Łojasiewicz condition.

Numerical results confirm the effectiveness of the proposed methods.

Abstract

This paper considers distributed nonconvex optimization with the cost functions being distributed over agents. Noting that information compression is a key tool to reduce the heavy communication load for distributed algorithms as agents iteratively communicate with neighbors, we propose three distributed primal--dual algorithms with compressed communication. The first two algorithms are applicable to a general class of compressors with bounded relative compression error and the third algorithm is suitable for two general classes of compressors with bounded absolute compression error. We show that the proposed distributed algorithms with compressed communication have comparable convergence properties as state-of-the-art algorithms with exact communication. Specifically, we show that they can find first-order stationary points with sublinear convergence rate $\mathcal{O}(1/T)$ when each local cost function is smooth, where $T$ is the total number of iterations, and find global optima with linear convergence rate under an additional condition that the global cost function satisfies the Polyak--{\L}ojasiewicz condition. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.