Tight Analysis of Decentralized SGD: A Markov Chain Perspective
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911368374910976 |
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| author | Versini, Lucas Mangold, Paul Dieuleveut, Aymeric |
| author_facet | Versini, Lucas Mangold, Paul Dieuleveut, Aymeric |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07021 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tight Analysis of Decentralized SGD: A Markov Chain Perspective Versini, Lucas Mangold, Paul Dieuleveut, Aymeric Machine Learning 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. |
| title | Tight Analysis of Decentralized SGD: A Markov Chain Perspective |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2601.07021 |