Deterministic computation of quantiles in a Lipschitz framework

In this article, we focus on computing the quantiles of a random variable $f\left(X\right)$, where $X$ is a ${[0,1]}^d$-valued random variable, $d\in N^{\ast }$, and $f:{[0,1]}^d\to R$ 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 $f\left(X\right)$ 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 $d=1$ ($O\left({\rho }^N\right)$ with $\rho \in (0,1)$) and a polynomial deterministic convergence rate for $d>1$ ($O\left(N^{-\frac{1}{d-1}}\right)$) and show the optimality of those rates. Furthermore, we design two algorithms, depending on whether the Lipschitz constant of $f$ is known or unknown.