  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.
