Near-Optimal Decentralized Stochastic Convex Optimization over Networks

Abstract

We study decentralized stochastic smooth convex optimization, where $M$ workers minimize an average objective using local stochastic gradients and neighbor-only communication over a fixed gossip network. A central question in this setting is to determine the largest number of workers that can be used under a total budget of $N$ gradient samples while still preserving the centralized $O(1/\sqrt N)$ statistical rate. We introduce an accelerated decentralized method that preserves this rate for up to $\smash{M\lesssim \sqrt{\rho}\,N^{3/4}}$ workers, where $\rho$ is the spectral gap of the gossip network, improving the best prior maximal scaling of $\smash{M\lesssim \rho\sqrt N}$. The method is based on a one-step-delayed stochastic acceleration scheme that enables workers to interleave minibatching with accelerated gossip while controlling residual disagreement, and its guarantee depends only logarithmically on the optimum-local heterogeneity. We also establish a matching lower bound for linear-span decentralized first-order methods, showing that the method is optimal up to logarithmic factors.