Langevin Markov Chain Monte Carlo with stochastic gradients

Abstract: Monte Carlo sampling techniques have broad applications in machine learning, Bayesian posterior inference, and parameter estimation. Often the target distribution takes the form of a product distribution over a dataset with a large number of entries. For sampling schemes utilizing gradient information it is cheaper for the derivative to be approximated using a random small subset of the data, introducing extra noise into the system. We present a new discretization scheme for underdamped Langevin dynamics when utilizing a stochastic (noisy) gradient. This scheme is shown to bias computed averages to second order in the stepsize while giving exact results in the special case of sampling a Gaussian distribution with a normally distributed stochastic gradient.