Particle Metropolis-adjusted Langevin algorithms

TL;DR

This paper introduces a Langevin-based sampling algorithm for particle MCMC methods, analyzing its theoretical properties and providing guidelines for tuning based on the accuracy of gradient estimates.

Contribution

It develops a new particle Langevin algorithm, analyzes its asymptotic behavior, and offers practical tuning guidelines for high-dimensional problems.

Findings

Algorithm performance depends on gradient estimate accuracy.

Optimal scaling is $O(n^{-1/6})$ when gradient errors are controlled.

Provides guidelines for tuning Monte Carlo samples and step-size.

Abstract

This paper proposes a new sampling scheme based on Langevin dynamics that is applicable within pseudo-marginal and particle Markov chain Monte Carlo algorithms. We investigate this algorithm's theoretical properties under standard asymptotics, which correspond to an increasing dimension of the parameters, $n$. Our results show that the behaviour of the algorithm depends crucially on how accurately one can estimate the gradient of the log target density. If the error in the estimate of the gradient is not sufficiently controlled as dimension increases, then asymptotically there will be no advantage over the simpler random-walk algorithm. However, if the error is sufficiently well-behaved, then the optimal scaling of this algorithm will be $O(n^{-1/6})$ compared to $O(n^{-1/2})$ for the random walk. Our theory also gives guidelines on how to tune the number of Monte Carlo samples in the likelihood estimate and the proposal step-size.