Accelerated Decentralized Optimization with Local Updates for Smooth and Strongly Convex Objectives
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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_ + Δ_)√(κ/γ)(ϵ^-1)) to reach a precision ϵ where γ is the spectral gap of the graph, τ_ the maximum communication delay and Δ_ the maximum computation time. Therefore, it matches the rate of SSDA, which is optimal when τ_ = Ω(Δ_). Applying ESDACD to quadratic local functions leads to an accelerated randomized gossip algorithm of rate O( √(θ_ gossip/n)) where θ_ 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.
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