  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.
