Tight Analysis of Decentralized SGD: a Markov Chain Perspective

Lucas Versini, Paul Mangold, Aymeric Dieuleveut
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3241-3249, 2026.

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 heterogeneity) and one due to stochasticity. Remarkably, the variance of local parameters is, at the first-order, inversely proportional to the number of agents, 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-versini26a, title = { Tight Analysis of Decentralized SGD: a Markov Chain Perspective }, author = {Versini, Lucas and Mangold, Paul and Dieuleveut, Aymeric}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3241--3249}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/versini26a/versini26a.pdf}, url = {https://proceedings.mlr.press/v300/versini26a.html}, 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 heterogeneity) and one due to stochasticity. Remarkably, the variance of local parameters is, at the first-order, inversely proportional to the number of agents, 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. } }
Endnote
%0 Conference Paper %T Tight Analysis of Decentralized SGD: a Markov Chain Perspective %A Lucas Versini %A Paul Mangold %A Aymeric Dieuleveut %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-versini26a %I PMLR %P 3241--3249 %U https://proceedings.mlr.press/v300/versini26a.html %V 300 %X 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 heterogeneity) and one due to stochasticity. Remarkably, the variance of local parameters is, at the first-order, inversely proportional to the number of agents, 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.
APA
Versini, L., Mangold, P. & Dieuleveut, A.. (2026). Tight Analysis of Decentralized SGD: a Markov Chain Perspective . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3241-3249 Available from https://proceedings.mlr.press/v300/versini26a.html.

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