Unbiased Compression Saves Communication in Distributed Optimization: When and How Much?

TL;DR

This paper provides a theoretical analysis of when unbiased communication compression in distributed optimization can reduce total communication costs, highlighting the importance of independence among compressors and quantifying potential savings.

Contribution

It introduces the first theoretical framework for total communication cost with compression, showing conditions under which unbiased compression reduces overall communication.

Findings

Unbiased compression alone does not guarantee communication savings.

Independent compressors can reduce total communication by up to a factor of Θ(√min{n, κ}).

Lower bounds on communication rounds are established and shown to be tight.

Abstract

Communication compression is a common technique in distributed optimization that can alleviate communication overhead by transmitting compressed gradients and model parameters. However, compression can introduce information distortion, which slows down convergence and incurs more communication rounds to achieve desired solutions. Given the trade-off between lower per-round communication costs and additional rounds of communication, it is unclear whether communication compression reduces the total communication cost. This paper explores the conditions under which unbiased compression, a widely used form of compression, can reduce the total communication cost, as well as the extent to which it can do so. To this end, we present the first theoretical formulation for characterizing the total communication cost in distributed optimization with communication compression. We demonstrate that unbiased compression alone does not necessarily save the total communication cost, but this outcome can be achieved if the compressors used by all workers are further assumed independent. We establish lower bounds on the communication rounds required by algorithms using independent unbiased compressors to minimize smooth convex functions and show that these lower bounds are tight by refining the analysis for ADIANA. Our results reveal that using independent unbiased compression can reduce the total communication cost by a factor of up to $\Theta(\sqrt{\min\{n, \kappa\}})$ when all local smoothness constants are constrained by a common upper bound, where $n$ is the number of workers and $\kappa$ is the condition number of the functions being minimized. These theoretical findings are supported by experimental results.