Near-Optimal Decentralized Stochastic Convex Optimization over Networks

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/√ N) statistical rate. We introduce an accelerated decentralized method that preserves this rate for up to smashMlesssim √ρ,N3/4 workers, where ρ is the spectral gap of the gossip network, improving the best prior maximal scaling of smashMlesssim ρ√ 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.