Estimating the Minimizer and the Minimum Value of a Regression Function under Passive Design

Arya Akhavan; Davit Gogolashvili; Alexandre B. Tsybakov

We propose a new method for estimating the minimizer

$\boldsymbol{x}^*$ and the minimum value

$f^*$ of a smooth and strongly convex regression function

$f$ from the observations contaminated by random noise. Our estimator

$\boldsymbol{z}_n$ of the minimizer

$\boldsymbol{x}^*$ is based on a version of the projected gradient descent with the gradient estimated by a regularized local polynomial algorithm. Next, we propose a two-stage procedure for estimation of the minimum value

$f^*$ of regression function

$f$ . At the first stage, we construct an accurate enough estimator of

$\boldsymbol{x}^*$ , which can be, for example,

$\boldsymbol{z}_n$ . At the second stage, we estimate the function value at the point obtained in the first stage using a rate optimal nonparametric procedure. We derive non-asymptotic upper bounds for the quadratic risk and optimization risk of

$\boldsymbol{z}_n$ , and for the risk of estimating

$f^*$ . We establish minimax lower bounds showing that, under certain choice of parameters, the proposed algorithms achieve the minimax optimal rates of convergence on the class of smooth and strongly convex functions.

Estimating the Minimizer and the Minimum Value of a Regression Function under Passive Design

Abstract