  Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful
algorithms for sampling from complex high-dimensional distributions. To sample
from a distribution with density $\pi(\theta)\propto \exp(-U(\theta)) $, LMC
iteratively generates the next sample by taking a step in the gradient
direction $\nabla U$ with added Gaussian perturbations. Expectations w.r.t. the
target distribution $\pi$ are estimated by averaging over LMC samples. In
ordinary Monte Carlo, it is well known that the estimation error can be
substantially reduced by replacing independent random samples by quasi-random
samples like low-discrepancy sequences. In this work, we show that the
estimation error of LMC can also be reduced by using quasi-random samples.
Specifically, we propose to use completely uniformly distributed (CUD)
sequences with certain low-discrepancy property to generate the Gaussian
perturbations. Under smoothness and convexity conditions, we prove that LMC
with a low-discrepancy CUD sequence achieves smaller error than standard LMC.
The theoretical analysis is supported by compelling numerical experiments,
which demonstrate the effectiveness of our approach.
