Constrained Ensemble Langevin Monte Carlo

TL;DR

This paper introduces Constrained Ensemble Langevin Monte Carlo, a gradient-free sampling method that leverages particle ensembles to improve efficiency while maintaining stability, offering a practical alternative to classical Langevin Monte Carlo.

Contribution

It proposes a novel constrained ensemble approach that reduces gradient computations and enhances stability compared to direct ensemble gradient surrogates.

Findings

Constrained ensemble method reduces gradient computation.

Direct ensemble gradient surrogates cause instability.

The proposed method maintains accuracy with fewer gradient evaluations.

Abstract

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.