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Communication efficiency has been widely recognized as the bottleneck forlarge-scale decentralized machine learning applications in multi-agent orfederated environments. To tackle the communication bottleneck, there have beenmany efforts to design communication-compressed algorithms for decentralizednonconvex optimization, where the clients are only allowed to communicate asmall amount of quantized information (aka bits) with their neighbors over apredefined graph topology. Despite significant efforts, the state-of-the-artalgorithm in the nonconvex setting still suffers from a slower rate ofconvergence $O((G/T)^{2/3})$ compared with their uncompressed counterpart,where $G$ measures the data heterogeneity across different clients, and $T$ isthe number of communication rounds. This paper proposes BEER, which adoptscommunication compression with gradient tracking, and shows it converges at afaster rate of $O(1/T)$. This significantly improves over the state-of-the-artrate, by matching the rate without compression even under arbitrary dataheterogeneity. Numerical experiments are also provided to corroborate ourtheory and confirm the practical superiority of BEER in the data heterogeneousregime.

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