  Due to the high communication cost in distributed and federated learning,
methods relying on compressed communication are becoming increasingly popular.
Besides, the best theoretically and practically performing gradient-type
methods invariably rely on some form of acceleration/momentum to reduce the
number of communications (faster convergence), e.g., Nesterov's accelerated
gradient descent (Nesterov, 1983, 2004) and Adam (Kingma and Ba, 2014). In
order to combine the benefits of communication compression and convergence
acceleration, we propose a \emph{compressed and accelerated} gradient method
based on ANITA (Li, 2021) for distributed optimization, which we call CANITA.
Our CANITA achieves the \emph{first accelerated rate}
$O\bigg(\sqrt{\Big(1+\sqrt{\frac{\omega^3}{n}}\Big)\frac{L}{\epsilon}} +
\omega\big(\frac{1}{\epsilon}\big)^{\frac{1}{3}}\bigg)$, which improves upon
the state-of-the-art non-accelerated rate
$O\left((1+\frac{\omega}{n})\frac{L}{\epsilon} +
\frac{\omega^2+\omega}{\omega+n}\frac{1}{\epsilon}\right)$ of DIANA (Khaled et
al., 2020) for distributed general convex problems, where $\epsilon$ is the
target error, $L$ is the smooth parameter of the objective, $n$ is the number
of machines/devices, and $\omega$ is the compression parameter (larger $\omega$
means more compression can be applied, and no compression implies $\omega=0$).
Our results show that as long as the number of devices $n$ is large (often true
in distributed/federated learning), or the compression $\omega$ is not very
high, CANITA achieves the faster convergence rate
$O\Big(\sqrt{\frac{L}{\epsilon}}\Big)$, i.e., the number of communication
rounds is $O\Big(\sqrt{\frac{L}{\epsilon}}\Big)$ (vs.
$O\big(\frac{L}{\epsilon}\big)$ achieved by previous works). As a result,
CANITA enjoys the advantages of both compression (compressed communication in
each round) and acceleration (much fewer communication rounds).
