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 $κ$ 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 $O((τ_{\max} + Δ_{\max})\sqrt{κ/γ}\ln(ε^{-1}))$ to reach a precision $ε$ where $γ$ is the spectral gap of the graph, $τ_{\max}$ the maximum communication delay and $Δ_{\max}$ the maximum computation time. Therefore, it matches the rate of $SSDA$, which is optimal when $τ_{\max} = Ω\left(Δ_{\max}\right)$. Applying $ESDACD$ to quadratic local functions leads to an accelerated randomized gossip algorithm of rate $O( \sqrt{θ_{\rm gossip}/n})$ where $θ_{\rm gossip}$ 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.