  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.
