  The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to
approximate Bayesian posterior distributions in statistical learning procedures
with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC)
algorithms, SGLD is not stationary with respect to the posterior distribution;
two sources of error appear: The first error is introduced by an
Euler--Maruyama discretisation of a Langevin diffusion process, the second
error comes from the data subsampling that enables its use in large-scale data
settings. In this work, we consider an idealised version of SGLD to analyse the
method's pure subsampling error that we then see as a best-case error for
diffusion-based subsampling MCMC methods. Indeed, we introduce and study the
Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov
process that follows the Langevin diffusion corresponding to a data subset and
switches this data subset after exponential waiting times. There, we show the
exponential ergodicity of SLGDiff and that the Wasserstein distance between the
posterior and the limiting distribution of SGLDiff is bounded above by a
fractional power of the mean waiting time. We bring our results into context
with other analyses of SGLD.
