Constrained Ensemble Langevin Monte Carlo
This work provides a more computationally efficient variant of Langevin Monte Carlo, which is beneficial for researchers and practitioners in fields requiring sampling from complex distributions, by reducing the need for explicit gradient calculations.
This paper addresses the high computational cost of gradient computation in classical Langevin Monte Carlo (LMC) by proposing an ensemble approach. They found that a direct ensemble approximation of gradients leads to instability due to high variance, but a constrained ensemble approximation, termed Constrained Ensemble Langevin Monte Carlo (CELMC), successfully removes most gradient computations while resembling classical LMC.
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.