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arXiv · 2601.07021

Tight Analysis of Decentralized SGD: A Markov Chain Perspective

Lucas Versini·Paul Mangold·Aymeric Dieuleveut
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Abstract

We propose a novel analysis of the Decentralized Stochastic Gradient Descent (DSGD) algorithm with constant step size, interpreting the iterates of the algorithm as a Markov chain. We show that DSGD converges to a stationary distribution, with its bias, to first order, decomposable into two components: one due to decentralization (growing with the graph's spectral gap and clients' heterogeneity) and one due to stochasticity. Remarkably, the variance of local parameters is, at the first-order, inversely proportional to the number of clients, regardless of the network topology and even when clients' iterates are not averaged at the end. As a consequence of our analysis, we obtain non-asymptotic convergence bounds for clients' local iterates, confirming that DSGD has linear speed-up in the number of clients, and that the network topology only impacts higher-order terms.

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Lucas Versini, Paul Mangold, Aymeric Dieuleveut. 2026-01-11. Tight Analysis of Decentralized SGD: A Markov Chain Perspective. https://arxiv.org/abs/2601.07021

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