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Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo

Lei Li, Jian-Guo Liu, Yuliang Wang

TL;DR

The paper addresses sampling from high-dimensional distributions governed by stochastic dynamics with non-Lipschitz drifts by analyzing implicit Langevin Monte Carlo (iLMC). It develops a second-order relative-entropy error bound through a continuous-time interpolation, introduces a Bernstein-based gradient estimate to control the numerical density, and establishes geometric ergodicity via a reflection-type coupling, followed by a uniform-in-time Wasserstein bound between the discrete and continuous dynamics. The combination of PDE techniques with probabilistic couplings yields a universal framework for analyzing implicit or splitting schemes for SDEs with challenging drifts. These results enhance stability and accuracy guarantees for iLMC, with potential impact on Bayesian inference and diffusion-model sampling in high dimensions.

Abstract

We study the implicit Langevin Monte Carlo (iLMC) method, which simulates the overdamped Langevin equation via an implicit iteration rule. In many applications, iLMC is favored over other explicit schemes such as the (explicit) Langevin Monte Carlo (LMC). LMC may blow up when the drift field $\nabla U$ is not globally Lipschitz, while iLMC has convergence guarantee when the drift is only one-sided Lipschitz. Starting from an adapted continuous-time interpolation, we prove a time-discretization error bound under the relative entropy (or the Kullback-Leibler divergence), where a crucial gradient estimate for the logarithm numerical density is obtained via a sequence of PDE techniques, including Bernstein method. Based on a reflection-type continuous-discrete coupling method, we prove the geometric ergodicity of iLMC under the Wasserstein-1 distance. Moreover, we extend the error bound to a uniform-in-time one by combining the relative entropy error bound and the ergodicity. Our proof technique is universal and can be applied to other implicit or splitting schemes for simulating stochastic differential equations with non-Lipschitz drifts.

Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo

TL;DR

The paper addresses sampling from high-dimensional distributions governed by stochastic dynamics with non-Lipschitz drifts by analyzing implicit Langevin Monte Carlo (iLMC). It develops a second-order relative-entropy error bound through a continuous-time interpolation, introduces a Bernstein-based gradient estimate to control the numerical density, and establishes geometric ergodicity via a reflection-type coupling, followed by a uniform-in-time Wasserstein bound between the discrete and continuous dynamics. The combination of PDE techniques with probabilistic couplings yields a universal framework for analyzing implicit or splitting schemes for SDEs with challenging drifts. These results enhance stability and accuracy guarantees for iLMC, with potential impact on Bayesian inference and diffusion-model sampling in high dimensions.

Abstract

We study the implicit Langevin Monte Carlo (iLMC) method, which simulates the overdamped Langevin equation via an implicit iteration rule. In many applications, iLMC is favored over other explicit schemes such as the (explicit) Langevin Monte Carlo (LMC). LMC may blow up when the drift field is not globally Lipschitz, while iLMC has convergence guarantee when the drift is only one-sided Lipschitz. Starting from an adapted continuous-time interpolation, we prove a time-discretization error bound under the relative entropy (or the Kullback-Leibler divergence), where a crucial gradient estimate for the logarithm numerical density is obtained via a sequence of PDE techniques, including Bernstein method. Based on a reflection-type continuous-discrete coupling method, we prove the geometric ergodicity of iLMC under the Wasserstein-1 distance. Moreover, we extend the error bound to a uniform-in-time one by combining the relative entropy error bound and the ergodicity. Our proof technique is universal and can be applied to other implicit or splitting schemes for simulating stochastic differential equations with non-Lipschitz drifts.

Paper Structure

This paper contains 13 sections, 20 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Setup and a continuous-time interpolation
  3. Error Estimate in Relative Entropy
  4. Gradient estimate
  5. Technique overview
  6. Geometric ergodicity via a reflection-type coupling method
  7. Evolution of the drift step
  8. Evolution of the diffusion step: a reflection-type coupling
  9. Extension to a long-time Wasserstein error bound
  10. Conclusion
  11. Omitted proofs for the relative entropy error estimate
  12. Technical Lemmas used in Section \ref{['sec:ergodicity']}
  13. Some estimates for the Fokker-Planck equation

Key Result

Theorem 1

Fix $T>0$. Denote $\rho^h_s$, $\rho_s$ the laws of $X^h_s$, $X_s$, respectively. Then for small time step $h$ one has Here, $C$ is a positive constant that may depend algebraically on $T$, and $\mathcal{H}$ denotes the relative entropy.

Theorems & Definitions (39)

  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1: Relative entropy error bound of iLMC
  • proof
  • ...and 29 more