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 error 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.

翻译：针对光滑且强凸回归函数$f$的最小点$\boldsymbol{x}^*$与最小值$f^*$的估计问题，本文提出一种基于含噪声观测数据的新方法。所提出的最小点$\boldsymbol{x}^*$的估计量$\boldsymbol{z}_n$采用投影梯度下降法的改进版本，其梯度通过正则化局部多项式算法进行估计。进一步，我们提出两阶段过程来估计回归函数$f$的最小值$f^*$：第一阶段构建$\boldsymbol{x}^*$的足够精确估计量（例如$\boldsymbol{z}_n$），第二阶段采用速率最优的非参数方法估计第一阶段的函数值。我们推导出$\boldsymbol{z}_n$的二次风险与优化误差以及$f^*$估计风险的非渐近上界，并建立极小极大下界，证明在特定参数选择下，所提算法在光滑强凸函数类上达到极小极大最优收敛速率。