Iterated sampling importance resampling with adaptive number of proposals

TL;DR

This work develops an adaptive i-SIR framework that automatically tunes the number of proposals by linking asymptotic efficiency to a continuous-number-of-proposals relaxation. It proves convexity and monotonicity properties of the i-SIR asymptotic variance and offers a practical surrogate via an approximate transition, enabling stochastic-approximation-based adaptation with a strong law of large numbers. Theoretical results are complemented by experiments showing the approximation-based adaptation robustly finds near-optimal efficiency across finite-state and continuous problems, including a Gaussian mixture and Bayesian logistic regression. The approach highlights the potential for parallelizable proposal evaluation to enhance MCMC efficiency while providing a principled method to balance cost and statistical accuracy.

Abstract

Iterated sampling importance resampling (i-SIR) is a Markov chain Monte Carlo (MCMC) algorithm which is based on $N$ independent proposals. As $N$ grows, its samples become nearly independent, but with an increased computational cost. We discuss a method which finds an approximately optimal number of proposals $N$ in terms of the asymptotic efficiency. The optimal $N$ depends on both the mixing properties of the i-SIR chain and the (parallel) computing costs. Our method for finding an appropriate $N$ is based on an approximate asymptotic variance of the i-SIR, which has similar properties as the i-SIR asymptotic variance, and a generalised i-SIR transition having fractional `number of proposals.' These lead to an adaptive i-SIR algorithm, which tunes the number of proposals automatically during sampling. Our experiments demonstrate that our approximate efficiency and the adaptive i-SIR algorithm have promising empirical behaviour. We also present new theoretical results regarding the i-SIR, such as the convexity of asymptotic variance in the number of proposals, which can be of independent interest.

Figures (8)

• Figure 1: Experiment \ref{['ex:1']}: \ref{['figure:example-1-prob-mass-function']} probability mass functions $\pi$ (red) and $q$ (black). \ref{['figure:example-1-asymptotic-variance']} functions $V_{\hat{f}}$ (red), $V_{\hat{g}}$ (orange), $V_{\hat{h}}$ (brown), $V_{\hat{k}}$ (purple), $V_{\hat{l}}$ (pink), $\hat{G}$ (green dashed), $\hat{H}$ (blue dashed) with upper/lower bounds (black dotted). \ref{['figure:example-1-asymptotic-variance-with-c']} loss i.e. the functions in \ref{['figure:example-1-asymptotic-variance']} multiplied by the cost $c(\lambda) = a + \lambda$.
• Figure 2: Experiment \ref{['ex:2']}: \ref{['figure:example-1.2-prob-mass-function']} probability mass functions $\pi$ (red) and $q$ (black). \ref{['figure:example-1.2-asymptotic-variance']} functions $V_{\hat{f}}$ (red), $V_{\hat{g}}$ (orange), $V_{\hat{h}}$ (brown), $V_{\hat{k}}$ (purple), $V_{\hat{l}}$ (pink), $\hat{G}$ (green dashed), $\hat{H}$ (blue dashed) with upper/lower bounds (black dotted). \ref{['figure:example-1.2-asymptotic-variance-with-c']} loss i.e. the functions in \ref{['figure:example-1.2-asymptotic-variance']} multiplied by the cost $c(\lambda) = a + \lambda$.
• Figure 3: Experiment \ref{['ex:3']}: \ref{['figure:example-1.3-prob-mass-function']} probability mass functions $\pi$ (red) and $q$ (black). \ref{['figure:example-1.3-asymptotic-variance']} functions $V_{\hat{f}}$ (red), $V_{\hat{g}}$ (orange), $V_{\hat{h}}$ (brown), $V_{\hat{k}}$ (purple), $V_{\hat{l}}$ (pink), $\hat{G}$ (green dashed), $\hat{H}$ (blue dashed) with upper/lower bounds (black dotted). \ref{['figure:example-1.3-asymptotic-variance-with-c']} loss i.e. the functions in \ref{['figure:example-1.3-asymptotic-variance']} multiplied by the cost $c(\lambda) = a + \lambda$.
• Figure 4: Experiment \ref{['ex:4']}: \ref{['figure:example-1.4-prob-mass-function']} probability mass functions $\pi$ (red) and $q$ (black). \ref{['figure:example-1.4-asymptotic-variance']} functions $V_{\hat{f}}$ (red), $V_{\hat{g}}$ (orange), $V_{\hat{h}}$ (brown), $V_{\hat{k}}$ (purple), $V_{\hat{l}}$ (pink dashed), $\hat{G}$ (green dashed), $\hat{H}$ (blue dashed) with upper/lower bounds (black dotted). \ref{['figure:example-1.4-asymptotic-variance-with-c']} loss i.e. the functions in \ref{['figure:example-1.4-asymptotic-variance']} multiplied by the cost $c(\lambda) = a + \lambda$.
• Figure 5: Experiment 5: \ref{['figure:example-2-prob-mass-function']} probability mass functions $\pi$ (red) and $q$ (black). \ref{['figure:example-2-asymptotic-variance']} functions $V_{\hat{f}}$ (red), $V_{\hat{g}}$ (orange), $V_{\hat{h}}$ (brown dashed), $V_{\hat{k}}$ (purple), $V_{\hat{l}}$ (pink), $\hat{G}$ (green dashed), $\hat{H}$ (blue dashed) with upper/lower bounds (black dotted). \ref{['figure:example-2-asymptotic-variance-with-c']} loss i.e. the functions in \ref{['figure:example-2-asymptotic-variance']} multiplied by the cost $c(\lambda) = a + \lambda$.

Theorems & Definitions (105)

• Proposition 1
• Definition 2
• Theorem 3
• Theorem 4
• Definition 5
• Proposition 6
• Theorem 7
• proof
• Proposition 8
• Remark 9