[Paper Review] Stratification and Optimal Resampling for Sequential Monte Carlo

Yichao Li, Wenshuo Wang+2 more|arXiv (Cornell University)|Apr 4, 2020

Markov Chains and Monte Carlo Methods29 references4 citations

TL;DR

This paper proposes Hilbert curve-based stratified resampling for sequential Monte Carlo (SMC) methods, demonstrating that it reduces resampling variance and improves convergence rates in multidimensional settings. It proves that for d > 1, the mean square error of sequential quasi-Monte Carlo with n particles can achieve O(n^{-1-4/[d(d+4)]}), the first known rate below o(n^{-1}).

ABSTRACT

Sequential Monte Carlo (SMC), also known as particle filters, has been widely accepted as a powerful computational tool for making inference with dynamical systems. A key step in SMC is resampling, which plays the role of steering the algorithm towards the future dynamics. Several strategies have been proposed and used in practice, including multinomial resampling, residual resampling (Liu and Chen 1998), optimal resampling (Fearnhead and Clifford 2003), stratified resampling (Kitagawa 1996), and optimal transport resampling (Reich 2013). We show that, in the one dimensional case, optimal transport resampling is equivalent to stratified resampling on the sorted particles, and they both minimize the resampling variance as well as the expected squared energy distance between the original and resampled empirical distributions; in the multidimensional case, the variance of stratified resampling after sorting particles using Hilbert curve (Gerber et al. 2019) in $\mathbb{R}^d$ is $O(m^{-(1+2/d)})$, an improved rate compared to the original $O(m^{-(1+1/d)})$, where $m$ is the number of resampled particles. This improved rate is the lowest for ordered stratified resampling schemes, as conjectured in Gerber et al. (2019). We also present an almost sure bound on the Wasserstein distance between the original and Hilbert-curve-resampled empirical distributions. In light of these theoretical results, we propose the stratified multiple-descendant growth (SMG) algorithm, which allows us to explore the sample space more efficiently compared to the standard i.i.d. multiple-descendant sampling-resampling approach as measured by the Wasserstein metric. Numerical evidence is provided to demonstrate the effectiveness of our proposed method.

Motivation & Objective

• To improve resampling efficiency in sequential Monte Carlo (SMC) methods by reducing variance and enhancing convergence rates in high-dimensional state spaces.
• To investigate the theoretical properties of stratified resampling when particles are ordered via Hilbert space-filling curves in R^d.
• To establish a theoretical foundation for the improved convergence rate of sequential quasi-Monte Carlo (SQMC) when combined with Hilbert-curve-based resampling.
• To demonstrate that optimal transport resampling is equivalent to stratified resampling on sorted particles in one dimension, minimizing resampling variance and energy distance.
• To provide almost sure bounds on the Wasserstein distance between original and resampled empirical distributions using Hilbert curve ordering.

Proposed method

• Applies stratified resampling after sorting particles using a Hilbert curve in R^d to preserve spatial structure and improve sampling efficiency.
• Proves that the variance of Hilbert-curve-resampled particles decays as O(m^{-(1+2/d)}), improving upon the standard O(m^{-(1+1/d)}) rate for unordered stratified resampling.
• Establishes equivalence between optimal transport resampling and stratified resampling on sorted particles in one dimension, both minimizing resampling variance and expected squared energy distance.
• Derives an almost sure bound on the Wasserstein distance between the original and Hilbert-curve-resampled empirical distributions.
• Combines Hilbert-curve resampling with low-discrepancy sequences to achieve a mean square error of O(n^{-1-4/[d(d+4)]}) in sequential quasi-Monte Carlo.
• Uses a matrix resampling framework to generalize resampling to unequal weights and analyzes variance under Lipschitz and boundedness conditions on the target and proposal densities.

Experimental results

Research questions

• RQ1 Can Hilbert curve-based resampling reduce resampling variance in multidimensional SMC, and if so, by how much?
• RQ2 What is the theoretical convergence rate of sequential quasi-Monte Carlo when using Hilbert-curve-resampled particles with low-discrepancy sequences?
• RQ3 Is optimal transport resampling equivalent to stratified resampling on sorted particles in one dimension, and do both minimize resampling variance and energy distance?
• RQ4 How does the spatial ordering of particles via Hilbert curves affect the variance of resampling compared to standard stratified resampling?
• RQ5 Can the proposed resampling scheme achieve a mean square error rate below o(n^{-1}) in high-dimensional SMC settings?

Key findings

• In one-dimensional cases, optimal transport resampling is equivalent to stratified resampling on sorted particles, and both minimize resampling variance and expected squared energy distance.
• For d > 1, the variance of Hilbert-curve-resampled particles decays as O(m^{-(1+2/d)}), which is an improvement over the standard O(m^{-(1+1/d)}) rate of unordered stratified resampling.
• The improved variance rate O(m^{-(1+2/d)}) is the lowest possible for ordered stratified resampling schemes, confirming a conjecture by Gerber et al. (2019).
• An almost sure bound on the Wasserstein distance between the original and Hilbert-curve-resampled empirical distributions is established.
• When combined with a specific low-discrepancy set, sequential quasi-Monte Carlo with Hilbert-curve resampling achieves a mean square error of O(n^{-1-4/[d(d+4)]}), the first known rate below o(n^{-1}).
• In a stochastic volatility model with d=2, simulations show the empirical MSE slope approaches -4/3, matching the theoretical rate O(n^{-1-4/[2(2+4)]}) = O(n^{-1-1/3}) = O(n^{-4/3}).