Deterministic computation of quantiles in a Lipschitz framework

TL;DR

This paper introduces a deterministic algorithm for computing bounds on quantiles of Lipschitz functions of high-dimensional random variables, achieving exponential or polynomial convergence depending on the dimension, with proven optimality.

Contribution

The paper presents a novel deterministic method for quantile computation with proven convergence rates and optimality, applicable when function evaluation costs are high and the distribution is known.

Findings

Achieves exponential convergence rate in one dimension.

Achieves polynomial convergence rate in higher dimensions.

Demonstrates the optimality of the convergence rates.

Abstract

In this article, we focus 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 evaluation is high, while the law of $X$ is assumed to be known. In this context, we propose a deterministic algorithm to compute deterministic lower and upper bounds for the quantile of $f(X)$ at a given level $\alpha \in (0,1)$. With a fixed budget of $N$ function calls, we demonstrate that our algorithm achieves an exponential deterministic convergence rate for $d=1$ ($\mathcal{O}( \rho^N)$ with $\rho \in (0,1)$) and a polynomial deterministic convergence rate for $d>1$ ($\mathcal{O}(N^{-\frac{1}{d-1}})$) and show the optimality of those rates. Furthermore, we design two algorithms, depending on whether the Lipschitz constant of $f$ is known or unknown.