Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of a Sequential Monte Carlo algorithm towards multimodal distributions on Rd

Ruiyu Han

TL;DR

This work extends Annealed Sequential Monte Carlo to sample from multimodal Gibbs measures on the noncompact space $\mathbb{R}^d$, establishing polynomial-time convergence in the inverse temperature. The authors develop a coupling-based analysis that overcomes unbounded-domain challenges by introducing a bounded region $K$ containing most mass and by deriving pointwise Langevin kernel bounds via Nash-type estimates. A detailed error-decomposition framework is built, combining Langevin-step Monte Carlo errors, transition-kernel bounds, and resampling errors, all controlled in high-probability events and propagated through annealing levels with recurrence relations. The main result applies to a double-well potential with equal-depth wells, yielding seventh-power scaling in $1/\varepsilon$ and quadratic scaling in error tolerances, with constants that depend on the potential but are dimension-independent in the exponent. The framework also outlines a multi-well extension, highlighting how the complexity grows with the number of wells through the low-temperature spectral structure and mass-transfer estimates.

Abstract

In an earlier joint work, we studied a sequential Monte Carlo algorithm to sample from the Gibbs measure supported on torus with a non-convex energy function at a low temperature, where we proved that the time complexity of the algorithm is polynomial in the inverse temperature. However, the analysis in that torus setting relied crucially on compactness and does not directly extend to unbounded domains. This work introduces a new approach that resolves this issue and establishes a similar result for sampling from Gibbs measures supported on Rd. In particular, our main result shows that for double-well energy with equal well depths, the time complexity scales as seventh power of the inverse temperature, and quadratically in both the inverse allowed absolute error and probability error.

Convergence of a Sequential Monte Carlo algorithm towards multimodal distributions on Rd

TL;DR

This work extends Annealed Sequential Monte Carlo to sample from multimodal Gibbs measures on the noncompact space , establishing polynomial-time convergence in the inverse temperature. The authors develop a coupling-based analysis that overcomes unbounded-domain challenges by introducing a bounded region containing most mass and by deriving pointwise Langevin kernel bounds via Nash-type estimates. A detailed error-decomposition framework is built, combining Langevin-step Monte Carlo errors, transition-kernel bounds, and resampling errors, all controlled in high-probability events and propagated through annealing levels with recurrence relations. The main result applies to a double-well potential with equal-depth wells, yielding seventh-power scaling in and quadratic scaling in error tolerances, with constants that depend on the potential but are dimension-independent in the exponent. The framework also outlines a multi-well extension, highlighting how the complexity grows with the number of wells through the low-temperature spectral structure and mass-transfer estimates.

Abstract

In an earlier joint work, we studied a sequential Monte Carlo algorithm to sample from the Gibbs measure supported on torus with a non-convex energy function at a low temperature, where we proved that the time complexity of the algorithm is polynomial in the inverse temperature. However, the analysis in that torus setting relied crucially on compactness and does not directly extend to unbounded domains. This work introduces a new approach that resolves this issue and establishes a similar result for sampling from Gibbs measures supported on Rd. In particular, our main result shows that for double-well energy with equal well depths, the time complexity scales as seventh power of the inverse temperature, and quadratically in both the inverse allowed absolute error and probability error.

Paper Structure

This paper contains 23 sections, 34 theorems, 286 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Informal statement of the main results
  3. Related work
  4. Main Result
  5. Proof of the Main Result
  6. Notation and convention
  7. Proof of Theorem \ref{['thm: main']}
  8. Sketch of proof in the multi-well case
  9. Error estimates for the resampling and the Langevin Dynamics (Lemma \ref{['l:langevinError']}, \ref{['cor:poverpi']} and \ref{['l:rebalancing']})
  10. The Monte Carlo error in the Langevin Step (Lemma \ref{['l:langevinError']})
  11. Estimate of transition kernel (Lemma \ref{['cor:poverpi']})
  12. The Rebalancing Error (Lemma \ref{['l:rebalancing']})
  13. Apriori estimates
  14. Probability that particles fall in set K
  15. Uniform boundedness of eigenfunctions on subset $K$ (Lemma \ref{['lem:boundPsi']})
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose $U \colon \mathbb{R}^d \to \mathbb{R}$ is a non-degenerate double-well function with wells of equal depth (but not necessarily the same shape). For $\varepsilon > 0$ let $Y_{\varepsilon, \cdot}$ be a solution to the overdamped Langevin equation where $W$ is a standard $d$-dimensional Brownian motion. There exist constants $C_N$ and $C_T$, depending on $U$ and $d$, such that the following

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6: global mixing
  • Lemma 3.7
  • proof : Proof of Theorem \ref{['thm: main']}
  • ...and 44 more