  Monte Carlo sampling for Bayesian posterior inference is a common approach
used in machine learning. The Markov Chain Monte Carlo procedures that are used
are often discrete-time analogues of associated stochastic differential
equations (SDEs). These SDEs are guaranteed to leave invariant the required
posterior distribution. An area of current research addresses the computational
benefits of stochastic gradient methods in this setting. Existing techniques
rely on estimating the variance or covariance of the subsampling error, and
typically assume constant variance. In this article, we propose a
covariance-controlled adaptive Langevin thermostat that can effectively
dissipate parameter-dependent noise while maintaining a desired target
distribution. The proposed method achieves a substantial speedup over popular
alternative schemes for large-scale machine learning applications.
