Dimension-Free Estimators of Gradients of Functions with(out) Non-Independent Variables

This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using ℓp-spherical distributions on Rd with d,p≥1. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any p≥1. Additionally, the mean squared errors (MSEs) of the gradient estimators are bounded by K0N−1d for any p∈[1,2], and by K1N−1d2/p when 2≤p≪d with N the sample size and K0,K1 some constants. Taking max2,log(d)<p≪d allows for achieving dimension-free upper-bounds of MSEs. In the case where d≪p<+∞, the upper-bound K2N−1d2−2/p/(d+2)2 is reached with K2 a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.