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Due to the high communication cost in distributed and federated learningproblems, methods relying on compression of communicated messages are becomingincreasingly popular. While in other contexts the best performing gradient-typemethods invariably rely on some form of acceleration/momentum to reduce thenumber of iterations, there are no methods which combine the benefits of bothgradient compression and acceleration. In this paper, we remedy this situationand propose the first accelerated compressed gradient descent (ACGD) methods.In the single machine regime, we prove that ACGD enjoys the rate$O\Big((1+\omega)\sqrt{\frac{L}{\mu}}\log \frac{1}{\epsilon}\Big)$ for$\mu$-strongly convex problems and$O\Big((1+\omega)\sqrt{\frac{L}{\epsilon}}\Big)$ for convex problems,respectively, where $\omega$ is the compression parameter. Our results improveupon the existing non-accelerated rates $O\Big((1+\omega)\frac{L}{\mu}\log\frac{1}{\epsilon}\Big)$ and $O\Big((1+\omega)\frac{L}{\epsilon}\Big)$,respectively, and recover the optimal rates of accelerated gradient descent asa special case when no compression ($\omega=0$) is applied. We further proposea distributed variant of ACGD (called ADIANA) and prove the convergence rate$\widetilde{O}\Big(\omega+\sqrt{\frac{L}{\mu}}+\sqrt{\big(\frac{\omega}{n}+\sqrt{\frac{\omega}{n}}\big)\frac{\omegaL}{\mu}}\Big)$, where $n$ is the number of devices/workers and $\widetilde{O}$hides the logarithmic factor $\log \frac{1}{\epsilon}$. This improves upon theprevious best result $\widetilde{O}\Big(\omega + \frac{L}{\mu}+\frac{\omegaL}{n\mu} \Big)$ achieved by the DIANA method of Mishchenko et al. (2019).Finally, we conduct several experiments on real-world datasets whichcorroborate our theoretical results and confirm the practical superiority ofour accelerated methods.

Acceleration for Compressed Gradient Descent in Distributed and Federated Optimization