CANITA: Faster Rates for Distributed Convex Optimization with Communication Compression

TL;DR

CANITA is a novel distributed convex optimization method that combines communication compression with acceleration, achieving faster convergence rates and reducing communication rounds in federated learning scenarios.

Contribution

It introduces CANITA, the first accelerated gradient method with communication compression, improving convergence rates over previous non-accelerated methods like DIANA.

Findings

Achieves the first accelerated rate for compressed distributed optimization.

Outperforms state-of-the-art non-accelerated methods in convergence speed.

Reduces communication rounds significantly in large-scale federated learning.

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 (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).