[论文解读] Tight Analysis of Decentralized SGD: A Markov Chain Perspective

Lucas Versini, Mangold, Paul+1 more|arXiv (Cornell University)|Jan 11, 2026

Stochastic Gradient Optimization Techniques被引用 0

一句话总结

The paper analyzes Decentralized SGD (DSGD) with constant step size by viewing iterates as a Markov chain, deriving first-order bias/variance expansions, showing linear speed-up in the number of clients, and providing non-asymptotic convergence bounds.

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.

研究动机与目标

• Motivate a precise, first-principles analysis of DSGD under stochastic noise.
• Develop a Markov chain framework to study DSGD bias and variance at stationarity.
• Characterize how decentralization, heterogeneity, and topology impact DSGD.
• Provide non-asymptotic convergence bounds and insights into speed-up and sample complexity.

提出的方法

• Interpret DSGD iterates as a Markov chain and prove geometric ergodicity to a stationary distribution.
• Derive first-order expansions of the bias and variance at stationarity separating decentralization/heterogeneity from stochasticity.
• Obtain non-asymptotic convergence bounds for local iterates showing linear speed-up in the number of clients.
• Analyze deterministic DGD to obtain explicit first-order bias expansions with respect to the step size.
• Extend analyses to quadratic and general smooth strongly convex objectives using matrix decompositions (e.g., consensus/disagreement projections, Gramians like G, H, B).
• Introduce Richardson-Romberg extrapolation for decentralized learning to cancel first-order bias.

实验结果

研究问题

• RQ1 What is the stationary behavior (bias and variance) of DSGD with constant step size when viewed as a Markov chain?
• RQ2 How do decentralization, heterogeneity, and network topology contribute to DSGD's bias and variance at stationarity?
• RQ3 Can DSGD achieve linear speed-up in the number of clients without averaging, and how do stochastic gradients affect this?
• RQ4 What non-asymptotic convergence guarantees can be established for DSGD’s local iterates?
• RQ5 How can Richardson-Romberg extrapolation be leveraged to reduce first-order bias in decentralized settings?

主要发现

• DSGD iterates converge to a stationary distribution in Wasserstein distance under constant step size.
• The first-order bias decomposes into a decentralization/heterogeneity component and a stochasticity component.
• DSGD variance at stationarity decreases with the number of clients, yielding a linear speed-up independent of topology at first order.
• Non-asymptotic bounds show DSGD with linear speed-up for local iterates, with topology affecting higher-order terms.
• For quadratic objectives, stochasticity does not add bias; for general smooth strongly convex objectives, stochasticity introduces an additional first-order bias.
• The network topology influences the stationary mean and higher-order bias/variance, but the leading variance term is topology-agnostic at first order.