Consistency and fluctuations for stochastic gradient Langevin dynamics

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 $({\delta }_m{)}_{m\ge 0}$. %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 $({\delta }_m{)}_{m\ge 0}$. 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 ${\delta }_m\asymp m^{-1/3}$, leading to an algorithm whose mean squared error (MSE) decreases at rate $\mathcal{O}(m^{-1/3})$

35 pages, 5 figures

References in corpus (4)

• Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget
• Robust and Scalable Bayes via a Median of Subset Posterior Measures
• An adaptive scheme for the approximation of dissipative systems
• Rate optimality of Random walk Metropolis algorithm in high-dimension with heavy-tailed target distribution

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