  The classical Langevin Monte Carlo method looks for i.i.d. samples from a
target distribution by descending along the gradient of the target
distribution. It is popular partially due to its fast convergence rate.
However, the numerical cost is sometimes high because the gradient can be hard
to obtain. One approach to eliminate the gradient computation is to employ the
concept of "ensemble", where a large number of particles are evolved together
so that 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. There are two sides of our discovery:
  1. By directly surrogating the gradient using the ensemble approximation, we
develop Ensemble Langevin Monte Carlo. We show that this method is unstable due
to a potentially small denominator that induces high variance. We provide a
counterexample to explicitly show this instability.
  2. We then change the strategy and enact the ensemble approximation to the
gradient only in a constrained manner, to eliminate the unstable points. The
algorithm is termed Constrained Ensemble Langevin Monte Carlo. We show that,
with a proper tuning, the surrogation takes place often enough to bring the
reasonable numerical saving, while the induced error is still low enough for us
to maintain the fast convergence rate, up to a controllable discretization and
ensemble error.
  Such combination of ensemble method and LMC shed light on inventing
gradient-free algorithms that produce i.i.d. samples almost exponentially fast.
