## Consistency and Fluctuations For Stochastic Gradient Langevin Dynamics

*Yee Whye Teh, Alexandre H. Thiery, Sebastian J. Vollmer*; 17(7):1−33, 2016.

### 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 $(\delta_m)_{m \geq 0}$. 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 $(\delta_m)_{m \geq 0}$. 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 $\mathcal{O}(m^{-1/3})$.

[abs][pdf][bib]