**Keywords:**distributed optimization, federated learning, communication compression, acceleration/momentum

**TL;DR:**We provide the first gradient-type method provably combining the benefits of both compression (compressed communication in each round) and acceleration (much fewer communication rounds) for distributed convex optimization.

**Abstract:**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 [31, 32] and Adam [14]. In order to combine the benefits of communication compression and convergence acceleration, we propose a \emph{compressed and accelerated} gradient method based on ANITA [20] 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 [12] 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).

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