Recursive Estimation of a Failure Probability for a Lipschitz Function

TL;DR

This paper introduces a recursive, optimal algorithm for efficiently estimating very low failure probabilities of Lipschitz functions using minimal evaluations, leveraging adaptive sampling and MCMC techniques.

Contribution

It presents a novel recursive algorithm that adaptively selects regions of interest to accurately estimate low failure probabilities with fewer function evaluations.

Findings

Algorithm effectively reduces the number of g evaluations needed.

Accurate estimation of very low failure probabilities demonstrated.

Method adapts to the function's Lipschitz properties for efficiency.

Abstract

Let g : $\Omega$ = [0, 1] d $\rightarrow$ R denote a Lipschitz function that can be evaluated at each point, but at the price of a heavy computational time. Let X stand for a random variable with values in $\Omega$ such that one is able to simulate, at least approximately, according to the restriction of the law of X to any subset of $\Omega$. For example, thanks to Markov chain Monte Carlo techniques, this is always possible when X admits a density that is known up to a normalizing constant. In this context, given a deterministic threshold T such that the failure probability p := P(g(X) > T) may be very low, our goal is to estimate the latter with a minimal number of calls to g. In this aim, building on Cohen et al. [9], we propose a recursive and optimal algorithm that selects on the fly areas of interest and estimate their respective probabilities.