Optimal estimators of cross-partial derivatives and surrogates of functions

TL;DR

This paper develops optimal estimators for cross-partial derivatives of functions using randomized sampling, achieving fast convergence rates and avoiding the curse of dimensionality, with applications in sensitivity analysis and surrogate modeling.

Contribution

It introduces a novel set of estimators for all cross-partial derivatives that are optimal in convergence and bias properties, applicable to high-dimensional functions.

Findings

Estimators reach the optimal convergence rate of O(N^{-1})

Biases do not suffer from the curse of dimensionality for a broad class of functions

Simulations confirm the accuracy and stability of the proposed estimators and surrogates.

Abstract

Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at $N$ randomized points and using a set of $L$ constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on $NL$ model runs, reach the optimal rates of convergence (i.e., $\mathcal{O}(N^{-1})$), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for i) computing the main and upper-bounds of sensitivity indices, and ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.