Finite Sample Bounds for Sequential Monte Carlo and Adaptive Path Selection Using the $L_2$ Norm

TL;DR

This paper establishes finite sample error bounds for sequential Monte Carlo methods on static spaces using the $L_2$ norm, without requiring importance weight bounds, and demonstrates how adaptive distribution selection improves efficiency.

Contribution

It provides the first finite sample convergence bounds for SMC that do not depend on importance weight bounds and introduces an adaptive distribution selection method with theoretical guarantees.

Findings

Adaptive SMC can be made more efficient with proper distribution selection.

The new bounds justify adaptive SMC approaches like relative effective sample size.

Modified data tempering algorithms satisfy the theoretical guarantees and perform well empirically.

Abstract

We prove a bound on the finite sample error of sequential Monte Carlo (SMC) on static spaces using the $L_2$ distance between interpolating distributions and the mixing times of Markov kernels. This result is unique in that it is the first finite sample convergence result for SMC that does not require an upper bound on the importance weights. Using this bound we show that careful selection of the interpolating distributions can lead to substantial improvements in the computational complexity of the algorithm. This result also justifies the adaptive selection of SMC distributions using the relative effective sample size commonly used in the literature, and we establish conditions guaranteeing the approximation accuracy of the adaptive SMC approach. We show that the commonly used data tempering approach fails to satisfy these conditions, and introduce a modified data tempering algorithm under which our guarantees do hold. We then demonstrate empirically that this procedure provides nearly-optimal sequences of distributions in an automatic fashion for realistic examples.