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Monte Carlo sampling for Bayesian posterior inference is a common approachused in machine learning. The Markov Chain Monte Carlo procedures that are usedare often discrete-time analogues of associated stochastic differentialequations (SDEs). These SDEs are guaranteed to leave invariant the requiredposterior distribution. An area of current research addresses the computationalbenefits of stochastic gradient methods in this setting. Existing techniquesrely on estimating the variance or covariance of the subsampling error, andtypically assume constant variance. In this article, we propose acovariance-controlled adaptive Langevin thermostat that can effectivelydissipate parameter-dependent noise while maintaining a desired targetdistribution. The proposed method achieves a substantial speedup over popularalternative schemes for large-scale machine learning applications.

Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling