Analysis of adaptive multilevel splitting algorithms in an idealized case

Abstract

Abstract. The Adaptive Multilevel Splitting algorithm [4] 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.

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“…For this adaptive algorithm, a better notion of cost involves the mean number of iterations -which is of order n/k: we prove in [15] that this cost is minimal when k = 1. Therefore our analysis reveals that taking k > 1 in the ideal case produces no improvement compared with k = 1; however it is an important first step for the definition of appropriate algorithms in more complicated situations, for which the unbiasedness is preserved, see Section 3.6.…”

“…The main result from [15] is that the AMS algorithm 1 yields unbiased estimators given by (7) for the probability p, for any values of the parameters n and k: in particular no asymptotic regime is required. Theorem 3.1.…”

“…The fundamental argument in [15] and [16] is the embedding of the estimation problem of p into the problem of estimating the conditional probability p(x) = P(X > a|X > x) for any x ∈ [0, a]; for that purpose, we introduce the estimatorp n,k (x) obtained thanks to Algorithm 1 where in the initialization step the random variables are generated according to L(X|X > x). Then Theorem 3.1 is a consequence of two facts:…”

“…We now review the results from [16] and [15]. The algorithm 1 defined below is well-posed and yields a family of unbiased estimators under the following condition.…”

“…After a brief review of reduction of variance strategies, especially the Multilevel Splitting approach (see [42]), the author introduces different adaptive versions of the latter and states a series of its properties: the goal is to build a theoretically unbiased estimator in an idealized setting and then to explain and to test numerically how the algorithm should be modified to preserve the unbiasedness property in a more general setting. This note is mainly based on the following joint works: [16] and [15].…”