  The problem of posterior inference is central to Bayesian statistics and a
wealth of Markov Chain Monte Carlo (MCMC) methods have been proposed to obtain
asymptotically correct samples from the posterior. As datasets in applications
grow larger and larger, scalability has emerged as a central problem for MCMC
methods. Stochastic Gradient Langevin Dynamics (SGLD) and related stochastic
gradient Markov Chain Monte Carlo methods offer scalability by using stochastic
gradients in each step of the simulated dynamics. While these methods are
asymptotically unbiased if the stepsizes are reduced in an appropriate fashion,
in practice constant stepsizes are used. This introduces a bias that is often
ignored. In this paper we study the mean squared error of Lipschitz functionals
in strongly log- concave models with i.i.d. data of growing data set size and
show that, given a batchsize, to control the bias of SGLD the stepsize has to
be chosen so small that the computational cost of reaching a target accuracy is
roughly the same for all batchsizes. Using a control variate approach, the cost
can be reduced dramatically. The analysis is performed by considering the
algorithms as noisy discretisations of the Langevin SDE which correspond to the
Euler method if the full data set is used. An important observation is that the
1scale of the step size is determined by the stability criterion if the
accuracy is required for consistent credible intervals. Experimental results
confirm our theoretical findings.
