Consistency and fluctuations for stochastic gradient Langevin dynamics

2014/09/01 by Yee Whye Teh, Teh, Yee Whye, Alexandre Thiéry +5 · 17 citations
Computer Science · Mathematics · Medicine · #60J22 #65C40 #Advanced Neuroimaging Techniques and Applications #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #msc:60J22 #msc:65C40 #stat.ML

paper · pdf · doi:10.48550/arxiv.1409.0578

35 pages, 5 figures

openalex publication_date 2014/09/01 · arxiv created 2015/06/12 · arxiv updated 2015/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally expensive. Both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem by generating proposals which are only based on a subset of the data, by skipping the accept-reject step and by using decreasing step-sizes sequence (δ_m)_{m ≥ 0}. %Under appropriate Lyapunov conditions, We provide in this article a rigorous mathematical framework for analysing this algorithm. We prove that, under verifiable assumptions, the algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence (δ_m)_{m ≥ 0}. We leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm: it is asymptotically optimal to use a step-size sequence of the type δ_m \asymp m^-1/3, leading to an algorithm whose mean squared error (MSE) decreases at rate O(m^-1/3)

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