---
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 high-dimensional statistical audio analysis and a sparse Bayesian logistic regression with random effects problem.'
author:
- 'Valentin De Bortoli [^1]'
- 'Alain Durmus [^2]'
- 'Marcelo Pereyra [^3]'
- 'Ana Fernandez Vidal [^4]'
bibliography:
- 'main.bib'
title: 'Efficient stochastic optimisation by unadjusted Langevin Monte Carlo. Application to maximum marginal likelihood and empirical Bayesian estimation.'
---

Theoretical convergence analysis for SOUL, and generalisation to other inexact MCMC kernels (SOUK) {#sec:convergence-results}
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In this section we state our main theoretical results for SOUL. For completeness, we first present the results in a general stochastic optimisation setting and by considering a generic inexact MCMC sampler, and then show that our results apply to the specific MLE optimisation problem , and to the specific Langevin algorithm used in SOUL.

[^1]: Email: valentin.debortoli@cmla.ens-cachan.fr

[^2]: Email: alain.durmus@cmla.ens-cachan.fr

[^3]: Email: m.pereyra@hw.ac.uk

[^4]: Email: af69@hw.ac.uk
