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Applying standard Markov chain Monte Carlo (MCMC) algorithms to large datasets is computationally infeasible. The recently proposed stochastic gradientLangevin dynamics (SGLD) method circumvents this problem in three ways: itgenerates proposed moves using only a subset of the data, it skips theMetropolis-Hastings accept-reject step, and it uses sequences of decreasingstep sizes. In \cite{TehThierryVollmerSGLD2014}, we provided the mathematicalfoundations for the decreasing step size SGLD, including consistency and acentral limit theorem. However, in practice the SGLD is run for a relativelysmall number of iterations, and its step size is not decreased to zero. Thepresent article investigates the behaviour of the SGLD with fixed step size. Inparticular we characterise the asymptotic bias explicitly, along with itsdependence on the step size and the variance of the stochastic gradient. Onthat basis a modified SGLD which removes the asymptotic bias due to thevariance of the stochastic gradients up to first order in the step size isderived. Moreover, we are able to obtain bounds on the finite-time bias,variance and mean squared error (MSE). The theory is illustrated with aGaussian toy model for which the bias and the MSE for the estimation of momentscan be obtained explicitly. For this toy model we study the gain of the SGLDover the standard Euler method in the limit of large data sets.

(Non-) asymptotic properties of Stochastic Gradient Langevin Dynamics