Optimal minimization of an unknown function in a nonparametric multivariate regression model thanks to a dimension reduction approach

Abstract

In this paper, we propose a novel approach for estimating the minimum of a smooth function and its location from observations corresponding to a multivariate regression function depending a priori on d variables but actually only on r < d active variables and corrupted by some additional noise. Our method consists of two steps: The rst one is a variable selection approach which is used for identifying the r active variables on which f depends and the second one consists in estimating the minimum of the function and its location. The estimation of the minimizers is obtained by using a projected gradient descent where the gradient is estimated using a local polynomial approximation of the regression function limited to its active variables obtained in the rst step. The estimation of the minimum is obtained by evaluating the estimator of the regression function using a local polynomial approach at the estimator of one of the minimizers previously obtained. We establish non asymptotic upper bounds for the quadratic risk of the estimators of the minimizers and of the minimum and prove that they reach the optimal rate that could be expected as if the active variables were known beforehand up to a factor smaller than a power of a logarithmic term.