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## Section: New Results

### Accelerated decentralized optimization with local updates for smooth and strongly convex objectives

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 $O\left(\left({\tau }_{max}+{\Delta }_{max}\right)\sqrt{\kappa /\gamma }ln\left({ϵ}^{-1}\right)\right)$ to reach a precision $ϵ$ where $\gamma$ is the spectral gap of the graph, ${\tau }_{max}$ the maximum communication delay and ${\Delta }_{max}$ the maximum computation time. Therefore, it matches the rate of $SSDA$, which is optimal when ${\tau }_{max}=\Omega \left({\Delta }_{max}\right)$. Applying $ESDACD$ to quadratic local functions leads to an accelerated randomized gossip algorithm of rate $O\left(\sqrt{{\theta }_{\mathrm{gossip}}/n}\right)$ where ${\theta }_{\mathrm{gossip}}$ 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.