On Arbitrary Compression for Decentralized Consensus and Stochastic Optimization over Directed Networks

TL;DR

This paper introduces a gradient-based decentralized algorithm that compresses messages arbitrarily to reduce communication costs in directed networks, achieving linear convergence for consensus and explicit rates for stochastic optimization.

Contribution

It presents the first method allowing arbitrary compression ratios in decentralized consensus and stochastic optimization over directed graphs with proven convergence guarantees.

Findings

Linear convergence for consensus problem.

Explicit convergence rates for stochastic optimization.

Effective communication reduction demonstrated in experiments.

Abstract

We study the decentralized consensus and stochastic optimization problems with compressed communications over static directed graphs. We propose an iterative gradient-based algorithm that compresses messages according to a desired compression ratio. The proposed method provably reduces the communication overhead on the network at every communication round. Contrary to existing literature, we allow for arbitrary compression ratios in the communicated messages. We show a linear convergence rate for the proposed method on the consensus problem. Moreover, we provide explicit convergence rates for decentralized stochastic optimization problems on smooth functions that are either (i) strongly convex, (ii) convex, or (iii) non-convex. Finally, we provide numerical experiments to illustrate convergence under arbitrary compression ratios and the communication efficiency of our algorithm.