  Communication is a key bottleneck in distributed training. Recently, an
\emph{error-compensated} compression technology was particularly designed for
the \emph{centralized} learning and receives huge successes, by showing
significant advantages over state-of-the-art compression based methods in
saving the communication cost. Since the \emph{decentralized} training has been
witnessed to be superior to the traditional \emph{centralized} training in the
communication restricted scenario, therefore a natural question to ask is "how
to apply the error-compensated technology to the decentralized learning to
further reduce the communication cost." However, a trivial extension of
compression based centralized training algorithms does not exist for the
decentralized scenario. key difference between centralized and decentralized
training makes this extension extremely non-trivial. In this paper, we propose
an elegant algorithmic design to employ error-compensated stochastic gradient
descent for the decentralized scenario, named $\texttt{DeepSqueeze}$. Both the
theoretical analysis and the empirical study are provided to show the proposed
$\texttt{DeepSqueeze}$ algorithm outperforms the existing compression based
decentralized learning algorithms. To the best of our knowledge, this is the
first time to apply the error-compensated compression to the decentralized
learning.
