Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ergodicity and error estimate of laws for a random splitting Langevin Monte Carlo

Lei Li, Chen Wang, Mengchao Wang

TL;DR

This work introduces the randomized splitting Langevin Monte Carlo (RSLMC), a simple random-order Lie–Trotter splitting of Langevin dynamics designed to reduce the first-order bias of standard Langevin Monte Carlo. The authors develop a rigorous framework based on relative entropy and semigroup commutators to quantify finite-time discretization errors, and establish a uniform-in-time bound by proving geometric ergodicity through reflection coupling, yielding $W_1$-convergence of order $O(\tau^2)$. They further obtain sharp finite-time relative-entropy bounds of order $O(\tau^4)$ and derive polynomial-in-$|x|$ bounds for the gradient and Hessian of the log-density, ensuring robust density control and applicability to nonconvex settings. Numerical experiments in 1D and 2D validate the theory, demonstrating fourth-order convergence in KL divergence and confirming the method’s practical efficacy with modest extra cost compared to standard LMC.

Abstract

The random splitting Langevin Monte Carlo could mitigate the first order bias in Langevin Monte Carlo with little extra work compared other high order schemes. We develop in this work an analysis framework for the sampling error under Wasserstein distance regarding the random splitting Langevin Monte Carlo. First, the sharp local truncation error is obtained by the relative entropy approach together with the explicit formulas for the commutator of related semi-groups. The necessary pointwise estimates of the gradient and Hessian of the logarithmic density are established by the Bernstein type approach in PDE theory. Second, the geometric ergodicity is established by accommodation of the reflection coupling. Combining the ergodicity with the local error estimate, we establish a uniform-in-time sampling error bound, showing that the invariant measure of the method approximates the true Gibbs distribution with $O(τ^2)$ accuracy where $τ$ is the time step. Lastly, we perform numerical experiments to validate the theoretical results.

Ergodicity and error estimate of laws for a random splitting Langevin Monte Carlo

TL;DR

This work introduces the randomized splitting Langevin Monte Carlo (RSLMC), a simple random-order Lie–Trotter splitting of Langevin dynamics designed to reduce the first-order bias of standard Langevin Monte Carlo. The authors develop a rigorous framework based on relative entropy and semigroup commutators to quantify finite-time discretization errors, and establish a uniform-in-time bound by proving geometric ergodicity through reflection coupling, yielding -convergence of order . They further obtain sharp finite-time relative-entropy bounds of order and derive polynomial-in- bounds for the gradient and Hessian of the log-density, ensuring robust density control and applicability to nonconvex settings. Numerical experiments in 1D and 2D validate the theory, demonstrating fourth-order convergence in KL divergence and confirming the method’s practical efficacy with modest extra cost compared to standard LMC.

Abstract

The random splitting Langevin Monte Carlo could mitigate the first order bias in Langevin Monte Carlo with little extra work compared other high order schemes. We develop in this work an analysis framework for the sampling error under Wasserstein distance regarding the random splitting Langevin Monte Carlo. First, the sharp local truncation error is obtained by the relative entropy approach together with the explicit formulas for the commutator of related semi-groups. The necessary pointwise estimates of the gradient and Hessian of the logarithmic density are established by the Bernstein type approach in PDE theory. Second, the geometric ergodicity is established by accommodation of the reflection coupling. Combining the ergodicity with the local error estimate, we establish a uniform-in-time sampling error bound, showing that the invariant measure of the method approximates the true Gibbs distribution with accuracy where is the time step. Lastly, we perform numerical experiments to validate the theoretical results.

Paper Structure

This paper contains 25 sections, 25 theorems, 340 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and some basic results
  3. The random splitting Langevin Monte Carlo
  4. Some assumptions and basic results
  5. Ergodicity of the random splitting LMC
  6. Finite time error estimate
  7. Finite time error bound in relative entropy
  8. The bounds on the density
  9. Derivatives of the logarithmic density
  10. First order derivative estimates
  11. Second order derivative estimates
  12. The polynomial bounds of the derivatives for the logarithmic density
  13. The sampling error estimates
  14. Numerical Experiments
  15. One-dimensional test cases
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\rho$ denote the probability density function associated with the SDE eq:SDE. With the operators $\mathcal{L}_1$ and $\mathcal{L}_2$ defined in eq:operaL1 and eq:operaL2, respectively, then the semigroups are given as follows: and that

Figures (4)

  • Figure 1: Relative entropy versus step size for the quadratic--logcosh model (log--log scale). The empirical curve (solid line with markers) is compared with reference slopes $\tau^2$ and $\tau^4$ (dashed).
  • Figure 2: Relative entropy versus step size for the 1D double-well potential (log--log scale). The empirical curve (solid line with markers) is compared with reference slopes $\tau^2$ and $\tau^4$ (dashed).
  • Figure 3: Relative entropy versus step size for the 1D logistic distribution (log--log scale). The empirical curve (solid line with markers) is compared with reference slopes $\tau^2$ and $\tau^4$ (dashed).
  • Figure 4: Relative entropy versus step size for the 2D Gaussian mixture (log--log scale). The empirical curve (solid line with markers) is compared with reference slopes $\tau^2$ and $\tau^4$ (dashed).

Theorems & Definitions (42)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lem:moment_hybrid_refined']} (ii)
  • Proposition 2.1
  • Theorem 2.1: Derivative estimates for log-density
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • ...and 32 more