On Dimension-Free Stochastic Surrogates and Estimators of Cross-Partial Derivatives and the Hessian Matrix

This study introduces stochastic surrogates of all the cross-partial derivatives of functions using L evaluations of functions at randomized points. Such randomized points are constructed using the class of ${\mathcal{l}}_p$-spherical distributions or equivalent distributions. For the cross-partial derivatives of a given order $\left|u\right|\in \{2,\dots ,d\}$, the proposed surrogates and the corresponding estimators of cross-partial derivatives enjoy the parametric rate of convergence and dimension-free mean squared errors when $d\ll p$, leading to breaking down the curse of dimensionality. Imposing $p\ll d$ allows to break down the curse of dimensionality for only the cross-partial derivatives of orders given by $\left|u\right|\ll 1+{\displaystyle \frac{d}{2log\left(d\right)}}$. Also, the L-point-based Hessian surrogate and estimator are proposed, including the convergence analysis. A particular choice of p allows to achieve the dimension-free mean squared errors. Analytical examples and simulations have been provided to show the efficiency of such surrogates and estimators.