Accelerated Decentralized Optimization with Local Updates for Smooth and Strongly Convex Objectives

In this paper, we study the problem of minimizing a sum of smooth and strongly convex functions split over the nodes of a network in a decentralized fashion. We propose the algorithm ESDACD, a decentralized accelerated algorithm that only requires local synchrony. Its rate depends on the condition number kappa of the local functions as well as the network topology and delays. Under mild assumptions on the topology of the graph, ESDACD takes a time O((τmax + Deltamax)√kappa/γln(ε-1)) to reach a precision ε where γ is the spectral gap of the graph, τmax the maximum communication delay and Deltamax the maximum computation time. Therefore, it matches the rate of SSDA, which is optimal when τmax = Omegaleft(Deltamaxright). Applying ESDACD to quadratic local functions leads to an accelerated randomized gossip algorithm of rate O( √θrm gossip/n) where θrm gossip is the rate of the standard randomized gossip. To the best of our knowledge, it is the first asynchronous gossip algorithm with a provably improved rate of convergence of the second moment of the error. We illustrate these results with experiments in idealized settings.