Résumé
This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using$\ell_p$ -spherical distributions on$\R^d$with$d, p\geq 1$ . The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any$p\geq 1$ . Also, the mean squared errors (MSEs) of the gradient estimators are bounded by$K_0 N^{-1} d$for any$p \in [1, 2]$ , and by$K_1 N^{-1} d^{2/p}$when$2 \leq p \ll d$with$N$the sample size and$K_0, K_1$some constants. Taking$\max\left\{2, \log(d) \right\} < p \ll d$allows for achieving dimension-free upper-bounds of MSEs. In the case where$d\ll p< +\infty$ , the upper-bound$K_2 N^{-1} d^{2-2/p}/ (d+2)^2$is reached with$K_2$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.