Analysis of Adaptive Multilevel Splitting algorithms in an idealized case

TL;DR

This paper provides a rigorous analysis of the Adaptive Multilevel Splitting algorithm, demonstrating unbiasedness of the estimator and deriving asymptotic expressions for variance and computational cost in an idealized setting.

Contribution

It offers a theoretical understanding of the algorithm's properties, including unbiasedness and asymptotic variance and cost analysis, in a simplified idealized scenario.

Findings

Estimator is unbiased for any k.

Derived asymptotic expansion for variance.

Analyzed cost in the large n limit.

Abstract

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the $k$ less well-adapted particles among $n$ are killed while $k$ new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever $k$. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large $n$ limit, for a fixed $k$.