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Constrained Ensemble Langevin Monte Carlo

Zhiyan Ding, Qin Li

TL;DR

Constrained Ensemble Langevin Monte Carlo (CEnLMC) addresses the gradient-cost bottleneck in Langevin Monte Carlo by leveraging ensemble neighbor information to approximate gradients. Directly replacing gradients with ensemble surrogates (EnLMC) is shown to be unstable due to variance blow-up, whereas the constrained version (CEnLMC) applies ensemble updates only in stable regions, preserving exponential convergence to the target distribution $p(x) \propto e^{-f(x)}$ up to an ensemble error. Thework provides non-asymptotic error bounds, practical parameter-tuning guidance for stability and gradient-saving, and numerical experiments that demonstrate substantial gradient reduction while maintaining sampling accuracy. Overall, CEnLMC offers a principled path to gradient-free, fast-converging sampling in high-dimensional Bayesian problems by fusing ensemble information with Langevin dynamics.

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.

Constrained Ensemble Langevin Monte Carlo

TL;DR

Constrained Ensemble Langevin Monte Carlo (CEnLMC) addresses the gradient-cost bottleneck in Langevin Monte Carlo by leveraging ensemble neighbor information to approximate gradients. Directly replacing gradients with ensemble surrogates (EnLMC) is shown to be unstable due to variance blow-up, whereas the constrained version (CEnLMC) applies ensemble updates only in stable regions, preserving exponential convergence to the target distribution up to an ensemble error. Thework provides non-asymptotic error bounds, practical parameter-tuning guidance for stability and gradient-saving, and numerical experiments that demonstrate substantial gradient reduction while maintaining sampling accuracy. Overall, CEnLMC offers a principled path to gradient-free, fast-converging sampling in high-dimensional Bayesian problems by fusing ensemble information with Langevin dynamics.

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.

Paper Structure

This paper contains 19 sections, 11 theorems, 143 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Two main ingredients
  3. Langevin Monte Carlo (LMC)
  4. Ensemble mean gradient approximation
  5. Algorithms and properties
  6. Ensemble LMC, a direct combination
  7. Constrained Ensemble LMC, a modification
  8. Properties of CEnLMC
  9. Convergence of CEnLMC
  10. Numerical saving of CEnLMC
  11. Numerical experiment
  12. Proof of theoretical results
  13. Proof of Theorem \ref{['thm:blowupvariance']}
  14. Analysis of CEnLMC
  15. Proof of Theorem \ref{['thm:convergenceofCEnLMC_z']}
  16. ...and 4 more sections

Key Result

Theorem 3.1

Assume $\left\{x^m_i\right\}^N_{i=1}$ are generated from Algorithm alg:EnOLMC, then for $d=1$ and $f(x) = x^2/2$, we have: for any $m>0$, $1\leq i\leq N$

Figures (6)

  • Figure 1: Example $1$: Evolution of samples using CEnLMC. $N=10^4$.
  • Figure 2: Example $1$: Evolution of samples using LMC and MALA. $N=10^4$.
  • Figure 3: Example $1$: Evolution of $\mathcal{R}_m$ when $N=2\times10^3, 6\times10^3$ or $10^4$.
  • Figure 4: Example $2$: Evolution of samples using CEnLMC when $N=10^4$
  • Figure 5: Example $2$: Evolution of samples using LMC and MALA when $N=10^4$
  • ...and 1 more figures

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 1
  • Theorem 3.3
  • Remark 3
  • Theorem 3.4
  • Corollary 2
  • Remark 4
  • ...and 15 more