  Motivated by a wide variety of applications, ranging from stochastic
optimization to dimension reduction through variable selection, the problem of
estimating gradients accurately is of crucial importance in statistics and
learning theory. We consider here the classic regression setup, where a real
valued square integrable r.v. $Y$ is to be predicted upon observing a (possibly
high dimensional) random vector $X$ by means of a predictive function $f(X)$ as
accurately as possible in the mean-squared sense and study a
nearest-neighbour-based pointwise estimate of the gradient of the optimal
predictive function, the regression function $m(x)=\mathbb{E}[Y\mid X=x]$.
Under classic smoothness conditions combined with the assumption that the tails
of $Y-m(X)$ are sub-Gaussian, we prove nonasymptotic bounds improving upon
those obtained for alternative estimation methods. Beyond the novel theoretical
results established, several illustrative numerical experiments have been
carried out. The latter provide strong empirical evidence that the estimation
method proposed works very well for various statistical problems involving
gradient estimation, namely dimensionality reduction, stochastic gradient
descent optimization and quantifying disentanglement.
