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Decentralized optimization methods with local updates have recently gainedattention for their provable ability to communication acceleration. In thesemethods, nodes perform several iterations of local computations between thecommunication rounds. Nevertheless, this capability is effective only when theloss function is smooth and the network is sufficiently well-connected. In thispaper, we propose a communication-efficient method MG-Skip with probabilisticlocal updates and multi-gossip communications for decentralized composite(smooth + nonsmooth) optimization, whose stepsize is independent of the numberof local updates and the network topology. Without any additional condition fornetwork connectivity, MG-Skip allows for the multi-gossip communications to beskipped in most iterations in the strongly convex setting, while its iterationcomplexity is $\mathcal{O}\left(\kappa \log \frac{1}{\epsilon}\right)$ andcommunication complexity is only$\mathcal{O}\left(\sqrt{\frac{\kappa}{(1-\rho)}} \log\frac{1}{\epsilon}\right)$, where $\kappa$ is the condition number of the lossfunction, $\rho$ reflects the connectivity of the network topology, and$\epsilon$ is the target accuracy. The theoretical results demonstrate thatMG-Skip achieves the optimal communication complexity and confirm the benefitsof local updates in the nonsmooth setup.

A Proximal Gradient Method With Probabilistic Multi-Gossip Communications for Decentralized Composite Optimization