Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bayesian sampling using interacting particles

Shi Chen, Zhiyan Ding, Qin Li

TL;DR

The paper studies Bayesian sampling with interacting particle ensembles, linking PDE descriptions of target measures to finite-particle dynamics via mean-field theory. It develops and analyzes two complementary frameworks: a coupling-based mean-field analysis for ensemble Kalman methods (EKI, EKS) and a martingale/compactness-based mean-field treatment for a Boltzmann-type simulator with Nanbu and Bird collision schemes. The results provide convergence rates and conditions under which the particle ensembles approximate the target distribution, including rate distinctions in high dimensions and gradient-free operational variants. Numerical experiments across Gaussian and non-convex targets illustrate practical convergence and stability of the proposed interacting-particle samplers.

Abstract

Bayesian sampling is an important task in statistics and machine learning. Over the past decade, many ensemble-type sampling methods have been proposed. In contrast to the classical Markov chain Monte Carlo methods, these new methods deploy a large number of interactive samples, and the communication between these samples is crucial in speeding up the convergence. To justify the validity of these sampling strategies, the concept of interacting particles naturally calls for the mean-field theory. The theory establishes a correspondence between particle interactions encoded in a set of coupled ODEs/SDEs and a PDE that characterizes the evolution of the underlying distribution. This bridges numerical algorithms with the PDE theory used to show convergence in time. Many mathematical machineries are developed to provide the mean-field analysis, and we showcase two such examples: The coupling method and the compactness argument built upon the martingale strategy. The former has been deployed to show the convergence of ensemble Kalman sampler and ensemble Kalman inversion, and the latter will be shown to be immensely powerful in proving the validity of the Vlasov-Boltzmann simulator.

Bayesian sampling using interacting particles

TL;DR

The paper studies Bayesian sampling with interacting particle ensembles, linking PDE descriptions of target measures to finite-particle dynamics via mean-field theory. It develops and analyzes two complementary frameworks: a coupling-based mean-field analysis for ensemble Kalman methods (EKI, EKS) and a martingale/compactness-based mean-field treatment for a Boltzmann-type simulator with Nanbu and Bird collision schemes. The results provide convergence rates and conditions under which the particle ensembles approximate the target distribution, including rate distinctions in high dimensions and gradient-free operational variants. Numerical experiments across Gaussian and non-convex targets illustrate practical convergence and stability of the proposed interacting-particle samplers.

Abstract

Bayesian sampling is an important task in statistics and machine learning. Over the past decade, many ensemble-type sampling methods have been proposed. In contrast to the classical Markov chain Monte Carlo methods, these new methods deploy a large number of interactive samples, and the communication between these samples is crucial in speeding up the convergence. To justify the validity of these sampling strategies, the concept of interacting particles naturally calls for the mean-field theory. The theory establishes a correspondence between particle interactions encoded in a set of coupled ODEs/SDEs and a PDE that characterizes the evolution of the underlying distribution. This bridges numerical algorithms with the PDE theory used to show convergence in time. Many mathematical machineries are developed to provide the mean-field analysis, and we showcase two such examples: The coupling method and the compactness argument built upon the martingale strategy. The former has been deployed to show the convergence of ensemble Kalman sampler and ensemble Kalman inversion, and the latter will be shown to be immensely powerful in proving the validity of the Vlasov-Boltzmann simulator.
Paper Structure (16 sections, 8 theorems, 87 equations, 6 figures, 4 algorithms)

This paper contains 16 sections, 8 theorems, 87 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Mean-field limit for ensemble based sampling methods
  3. Unified coupling method framework
  4. Ensemble Kalman inversion
  5. Algorithm
  6. Mean-field analysis for EKI
  7. Ensemble Kalman sampler
  8. Algorithm
  9. Boltzmann simulator
  10. Vlasov-Boltzmann and its properties
  11. Nanbu and Bird algorithm
  12. Mean-field analysis for Boltzmann simulator
  13. Weak formulation and the Martingale problem
  14. Weak formulation and the Martingale problems for the $N$-particle systems
  15. The mean-field analysis
  16. ...and 1 more sections

Key Result

Theorem 2.2

Suppose that $\mathcal{G}$ satisfies the weak nonlinear assumptionFor technical reasons, we can only prove the mean-field under the weak nonlinearity assumption. In particular, we assume that $\mathcal{G}$ is weak nonlinear, meaning that there is a matrix $A\in\mathbb{R}^{d\times d}$ such that $\mat

Figures (6)

  • Figure 1: The process of algorithm design (left to right) typically involves three steps: designing a PDE, designing its particle method, discretization. The analysis procedure (right to left) is consistent with the design process in reverse. This paper only focuses on the second step of algorithm analysis, which involves applying mean-field analysis to validate the mean-field limit of multiple-particle algorithms.
  • Figure 2: The general routine of the mean-field analysis from coupling perspective. The auxiliary particles $\widetilde{x}_i(t)$ evolve according to a coupled ODE/SDE with $\rho(t)$ generated from the mean-field limiting PDE. The coupling method shows that $x_i$ is close to $\widetilde{x}_i$, which implies that $\widetilde{\rho}_N$ is close to $\rho_N$. This, combined with the fact that $\widetilde{\rho}_N\approx \rho$ from Fournier2015, ultimately justifies $\rho_N\approx\rho$.
  • Figure 3: Left Two Plots: The target distribution $\rho^\ast \propto \exp(-\frac{x^2}{2})$ and the histogram of samples obtained from the Nanbu sampler (Leftmost) and the Bird sampler (Second-Left). Right Two Plots: The regularized relative entropy $KL_X^\delta$ (Second-Right) and $KL^\delta$ (Rightmost) of the Bird sampler and the Nanbu sampler. The baseline value is computed by using $N=1000$ samples obtained by the Inverse Transform method.
  • Figure 4: Left Two Plots: The target distribution $\rho^\ast \propto \exp(-(x-1)^2(x+1)^2)$ and the histogram of samples obtained from the Nanbu sampler (Leftmost) and the Bird sampler (Second-Left). Right Two Plots: The regularized relative entropy $KL_X^\delta$ (Second-Right) and $KL^\delta$ (Rightmost) of the Bird sampler and the Nanbu sampler. The baseline value is computed by using $N=1000$ samples obtained by the Inverse Transform method.
  • Figure 5: Left Two Plots: The contour of the target distribution $\rho^\ast \propto \exp(-0.5(x^2+y^2))$ and the samples obtained from the Nanbu sampler (Leftmost) and the Bird sampler (Second-Left). Right Two Plots: The regularized relative entropy $KL_X^\delta$ (Second-Right) and $KL^\delta$ (Rightmost) of the Bird sampler and the Nanbu sampler. The baseline value is computed by using $N=1000$ samples obtained by the Inverse Transform method.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 2.2: Mean-field limit for EKI
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: Mean-field limit of EKS
  • Theorem 3.2
  • proof
  • Theorem 3.3: Theorem 1 in Ki:2014boltzmann
  • Remark 3.4
  • Theorem 3.7: Theorem 4.5 in GrMe:1997stochastic
  • ...and 5 more