Efficient stochastic optimisation by unadjusted Langevin Monte Carlo: Application to maximum marginal likelihood and empirical Bayesian estimation

Valentin De Bortoli, Alain Durmus, Marcelo Pereyra, Ana F. Vidal

Research output: Contribution to journalArticlepeer-review

Abstract

Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain Monte Carlo algorithms within a stochastic approximation scheme. This paper proposes to address these difficulties by using unadjusted Langevin algorithms to construct the stochastic approximation. This leads to a highly efficient stochastic optimisation methodology with favourable convergence properties that can be quantified explicitly and easily checked. The proposed methodology is demonstrated with three experiments, including a challenging application to statistical audio analysis and a sparse Bayesian logistic regression with random effects problem.

Original languageEnglish
Article number29
JournalStatistics and Computing
Volume31
Issue number3
Early online date19 Mar 2021
DOIs
Publication statusPublished - May 2021

Keywords

  • Empirical Bayesian inference
  • Markov chain Monte Carlo methods
  • Maximum marginal likelihood estimation
  • Recursive estimation
  • Stochastic approximation
  • Unadjusted Langevin Algorithm

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics

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