# Optimal estimators of cross-partial derivatives and surrogates of functions

arXiv: https://arxiv.org/abs/2407.11035 · HTML: https://mlsanity.com/paper/2407.11035/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 56 references that are in the map.
- First posted on arXiv 5 Jul 2024.

## Builds on (most cited first, 6 of 6)

- [Highly-Smooth Zero-th Order Online Optimization Vianney Perchet](https://mlsanity.com/paper/1605.08165/) (2016)
- [Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits](https://mlsanity.com/paper/2006.07862/) (2020)
- [Hessian Estimation via Stein's Identity in Black-Box Problems](https://mlsanity.com/paper/2104.01317/) (2021)
- [Efficient Implementation of Second-Order Stochastic Approximation Algorithms in High-Dimensional Problems](https://mlsanity.com/paper/1906.09533/) (2019)
- [Kernel-based measures of association between inputs and outputs based on ANOVA](https://mlsanity.com/paper/2311.14894/) (2023)
- [Poincar'e inequalities on intervals -- application to sensitivity analysis](https://mlsanity.com/paper/1612.03689/) (2016)

## Abstract (arXiv)

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., 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.