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Ensemble-Based Annealed Importance Sampling

Haoxuan Chen, Lexing Ying

TL;DR

An ensemble-based version of AIS is proposed by combining it with population-based Monte Carlo methods to improve its efficiency and take advantage of the interaction within the ensemble to encourage the exploration of undiscovered modes.

Abstract

Sampling from a multimodal distribution is a fundamental and challenging problem in computational science and statistics. Among various approaches proposed for this task, one popular method is Annealed Importance Sampling (AIS). In this paper, we propose an ensemble-based version of AIS by combining it with population-based Monte Carlo methods to improve its efficiency. By keeping track of an ensemble instead of a single particle along some continuation path between the starting distribution and the target distribution, we take advantage of the interaction within the ensemble to encourage the exploration of undiscovered modes. Specifically, our main idea is to utilize either the snooker algorithm or the genetic algorithm used in Evolutionary Monte Carlo. We discuss how the proposed algorithm can be implemented and derive a partial differential equation governing the evolution of the ensemble under the continuous time and mean-field limit. We also test the efficiency of the proposed algorithm on various continuous and discrete distributions.

Ensemble-Based Annealed Importance Sampling

TL;DR

An ensemble-based version of AIS is proposed by combining it with population-based Monte Carlo methods to improve its efficiency and take advantage of the interaction within the ensemble to encourage the exploration of undiscovered modes.

Abstract

Sampling from a multimodal distribution is a fundamental and challenging problem in computational science and statistics. Among various approaches proposed for this task, one popular method is Annealed Importance Sampling (AIS). In this paper, we propose an ensemble-based version of AIS by combining it with population-based Monte Carlo methods to improve its efficiency. By keeping track of an ensemble instead of a single particle along some continuation path between the starting distribution and the target distribution, we take advantage of the interaction within the ensemble to encourage the exploration of undiscovered modes. Specifically, our main idea is to utilize either the snooker algorithm or the genetic algorithm used in Evolutionary Monte Carlo. We discuss how the proposed algorithm can be implemented and derive a partial differential equation governing the evolution of the ensemble under the continuous time and mean-field limit. We also test the efficiency of the proposed algorithm on various continuous and discrete distributions.
Paper Structure (22 sections, 1 theorem, 35 equations, 13 figures, 5 algorithms)

This paper contains 22 sections, 1 theorem, 35 equations, 13 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization of the paper
  5. Preliminaries and Notations
  6. Review of Related Sampling Methods
  7. Annealed Importance Sampling (AIS)
  8. Population-Based Monte Carlo Methods
  9. Snooker Algorithm
  10. Genetic Algorithm
  11. Ensemble-Based Annealed Importance Sampling (AIS)
  12. Two Hybrid Algorithms
  13. Mean-Field Limit and Convergence Properties
  14. Numerical Experiments
  15. Continuous Distributions
  16. ...and 7 more sections

Key Result

Lemma 2.3

\newlabellem: snooker alg correctness0 Fix some probability distribution function $\pi:\mathbb{R}^d \rightarrow \mathbb{R}$ and some point $b \in \mathbb{R}^d$. Assume that $a \in \mathbb{R}^d$ is distributed as $\pi(\cdot)$ and let $e_a := a-b$. Furthermore, assume that $r \in \mathbb{R}$ is samp

Figures (13)

  • Figure 1: (a) Contour plot of the target density in \ref{['example: 2D Gaussian Mixture']}; (b) Plots of $\mathcal{L}_{\text{KL}}$ evaluated at evolving weighted samples generated by different testing algorithms in \ref{['example: 2D Gaussian Mixture']}.
  • Figure 2: Evolving Estimations of $\mathbb{E}[f(x,y)]$ evaluated at weighted samples generated by different testing algorithms in \ref{['example: 2D Gaussian Mixture']} for $f_1(x,y) = y$ and $f_2(x,y) = \frac{1}{3}x^2 + \frac{1}{5}y^2$.
  • Figure 3: Scatter plots of the empirical samples returned by different testing algorithms in \ref{['example: 2D Gaussian Mixture']}; (a) MALA + Snooker + BD; (b) MALA + BD; (c) MALA + Reweight; (d) Gaussian MH + Reweight.
  • Figure 4: (a) & (b) Two minimizers $\boldsymbol{x}_{+}^{(1)}$ and $\boldsymbol{x}_{-}^{(1)}$ of the 1D Ginzburg-Landau Energy (\ref{['eqn: continuous energy in GL']}) with $\lambda = 0.05$; (c) Plots of $\mathcal{L}_{\text{KL}}$ evaluated at evolving weighted samples generated by different testing algorithms in the 1D case of \ref{['example: GL distribution']}.
  • Figure 5: Scatter plots of the marginal distribution at $(x_5, x_6)$ returned by different testing algorithms in the 1D case of \ref{['example: GL distribution']}; (a) MALA + Snooker + BD; (b) MALA + BD; (c) MALA + Reweight; (d) Gaussian MH + Reweight.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 3.1
  • Remark 3.2
  • Proof 1