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 lp-spherical distributions or equivalent distributions. For the cross-partial derivatives of a given order |u|∈{2,…,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≪p, leading to breaking down the curse of dimensionality. Imposing p≪d allows to break down the curse of dimensionality for only the cross-partial derivatives of orders given by |u|≪1+d2log(d). 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.