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(root(1+sec(3) /n)L/is an element of+sec(1/is an element of)(1/3)), which improves upon the state-of-the-art non-accelerated rate O((1+sec/n)L/sec+sec(2)+sec+n1sec) of DIANA (Khaled et al., 2020) for distributed general convex problems, where is an element of is the target error, L is the smooth parameter of the objective, n is the number of machines/devices, and ? is the compression parameter (larger ? means more compression can be applied, and no compression implies sec=0). Our results show that as long as the number of devices n is large (often true in distributed/federated learning), or the compression ? is not very high, CANITA achieves the faster convergence rate O(L is an element of--v), i.e., the number of communication rounds is O(L is an element of--v) (vs. O(L is an element of) 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)