Deterministic computation of quantiles in a Lipschitz framework

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 is an element of N*, and f : [0, 1](d)-> 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 is an element of (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 (rho (N)) with alpha is an element of (0,1)) and a polynomial deterministic convergence rate for d >1(O(N- 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.