On the role of interaction in sequential Monte Carlo algorithms

TL;DR

This paper proposes a generalized framework for sequential Monte Carlo algorithms that adaptively controls particle interaction via a generalized Effective Sample Size, leading to provably stable algorithms and improved parallelization.

Contribution

It introduces a parameterized resampling mechanism and a generalized ESS concept, providing conditions for stable, parallelizable SMC algorithms with proven convergence.

Findings

Generalized ESS naturally arises in convergence analysis.

Adaptive control of particle interaction ensures time-uniform convergence.

Proves convergence of adaptive resampling particle filters.

Abstract

We introduce a general form of sequential Monte Carlo algorithm defined in terms of a parameterized resampling mechanism. We find that a suitably generalized notion of the Effective Sample Size (ESS), widely used to monitor algorithm degeneracy, appears naturally in a study of its convergence properties. We are then able to phrase sufficient conditions for time-uniform convergence in terms of algorithmic control of the ESS, in turn achievable by adaptively modulating the interaction between particles. This leads us to suggest novel algorithms which are, in senses to be made precise, provably stable and yet designed to avoid the degree of interaction which hinders parallelization of standard algorithms. As a byproduct, we prove time-uniform convergence of the popular adaptive resampling particle filter.