Deterministic computation of quantiles in a Lipschitz framework - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

Deterministic computation of quantiles in a Lipschitz framework

Yurun Gu
  • Fonction : Auteur

Résumé

In this article, our focus lies on computing the quantiles of a random variable $f(X)$, where $X$ is a $[0,1]^d$-valued random variable, $d \in \mathbb{N}^{\ast}$, and $f:[0,1]^d\to \mathbb{R}$ is a deterministic Lipschitz function. We are particularly interested in scenarios where the cost of a single function call is high, while assuming the law of $X$ is known. In this context, we propose a deterministic algorithm to obtain exact deterministic lower and upper bounds for the quantile of $f(X)$ at a given level $α\in (0,1)$. With a fixed budget of $N$ function calls, we demonstrate that our algorithm achieves exponential convergence rate for $d=1$ ($\sim ρ^N$ with $ρ\in (0,1)$) and polynomial convergence rate for $d>1$ ($\sim N^{-\frac{1}{d-1}}$) and show the optimality of those rates within the class of deterministic algorithms. Furthermore, we design two algorithms based on whether the Lipschitz constant of $f$ is known or unknown.
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Dates et versions

hal-04582532 , version 1 (22-05-2024)

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Citer

Yurun Gu, Clément Rey. Deterministic computation of quantiles in a Lipschitz framework. 2024. ⟨hal-04582532⟩
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