BEER: Fast $O(1/T)$ Rate for Decentralized Nonconvex Optimization with Communication Compression

TL;DR

This paper introduces BEER, a communication-compressed decentralized optimization algorithm that achieves an optimal $O(1/T)$ convergence rate, significantly improving over previous methods especially in heterogeneous data settings.

Contribution

BEER is the first algorithm to attain an $O(1/T)$ convergence rate with communication compression in nonconvex decentralized optimization, matching uncompressed rates even with data heterogeneity.

Findings

BEER converges at an $O(1/T)$ rate, faster than previous $O((G/T)^{2/3})$ rates.

Numerical results confirm BEER's superior performance in heterogeneous data scenarios.

BEER effectively combines communication compression with gradient tracking for improved efficiency.

Abstract

Communication efficiency has been widely recognized as the bottleneck for large-scale decentralized machine learning applications in multi-agent or federated environments. To tackle the communication bottleneck, there have been many efforts to design communication-compressed algorithms for decentralized nonconvex optimization, where the clients are only allowed to communicate a small amount of quantized information (aka bits) with their neighbors over a predefined graph topology. Despite significant efforts, the state-of-the-art algorithm in the nonconvex setting still suffers from a slower rate of convergence $O((G/T)^{2/3})$ compared with their uncompressed counterpart, where $G$ measures the data heterogeneity across different clients, and $T$ is the number of communication rounds. This paper proposes BEER, which adopts communication compression with gradient tracking, and shows it converges at a faster rate of $O(1/T)$. This significantly improves over the state-of-the-art rate, by matching the rate without compression even under arbitrary data heterogeneity. Numerical experiments are also provided to corroborate our theory and confirm the practical superiority of BEER in the data heterogeneous regime.