Decentralized Stochastic Subgradient-type Methods with Communication Compression for Nonsmooth Nonconvex Optimization

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 $\smash{M\lesssim \sqrtρ\,N^{3/4}}$ workers, where $ρ$ is the spectral gap of the gossip network, improving the best prior maximal scaling of $\smash{M\lesssim ρ\sqrt N}$. 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.

Decentralized stochastic optimization is a fundamental paradigm for large-scale learning over networks, where agents communicate only with their neighbors and no central coordinator is required. For strongly convex problems, communication efficiency is mainly determined by the condition number $κ=L/μ$ and the network spectral gap $1-β$. Although deterministic decentralized methods can simultaneously achieve accelerated $\sqrtκ$ and $1/\sqrt{1-β}$ dependences, no existing stochastic method attains both improvements at once. In this paper, we propose \emph{Multi-Gossip Accelerated DSGD} (MG-ADSGD), a decentralized stochastic algorithm that combines Nesterov-type primal--dual extrapolation with multi-round fast gossip averaging. The key idea is to couple the gossip depth with the mini-batch size so that additional communication rounds simultaneously improve consensus accuracy and reduce gradient variance. We show that MG-ADSGD achieves the communication complexity $$\widetilde{\mathcal O}\!\left( \frac{σ^2}{μnε}\log\frac{1}ε + \sqrt{\fracκ{1-β}}\log\frac{1}ε \right),$$ where $ε$ denotes the target accuracy, $n$ is the number of nodes, and $σ^2$ is the gradient variance. To the best of our knowledge, this bound yields the best currently available communication complexity for decentralized stochastic strongly convex optimization, up to logarithmic factors that are independent of $ε$.

Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of $O(\sqrt{d/K}+\sqrt d (a/(nK))^{1/3})$ for the $\ell_1$-norm and $O(\sqrt{a/K}+\sqrt a/(nK)^{1/3})$ for the $\ell_2$-norm. Here, $K$ is the iteration number, $n$ is the number of workers, $d$ is the dimension, and $a=1+ω$, with $ω$ denoting the compressor's relative variance. For finite-sum problems with $M$ components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are $O(M+d\sqrt{aM}ε^{-2})$ and $O(M+a\sqrt M\ epsilon^{-2})$ for $\ell_1$ and $\ell_2$ gradient norms at most $ε$, matching the corresponding bounds in centralized settings.