  The classical Langevin Monte Carlo method looks for samples from a target
distribution by descending the samples along the gradient of the target
distribution. The method enjoys a fast convergence rate. However, the numerical
cost is sometimes high because each iteration requires the computation of a
gradient. One approach to eliminate the gradient computation is to employ the
concept of ``ensemble." A large number of particles are evolved together so the
neighboring particles provide gradient information to each other. In this
article, we discuss two algorithms that integrate the ensemble feature into LMC
and the associated properties.
  In particular, we find that if one directly surrogates the gradient using the
ensemble approximation, the algorithm, termed Ensemble Langevin Monte Carlo, is
unstable due to a high variance term. If the gradients are replaced by the
ensemble approximations only in a constrained manner, to protect from the
unstable points, the algorithm, termed Constrained Ensemble Langevin Monte
Carlo, resembles the classical LMC up to an ensemble error but removes most of
the gradient computation.
