AI 中文总结
针对受噪声污染、仅依赖r个活跃变量的多元回归函数，提出含变量选择与投影梯度下降两步的方法，建立估计量的非渐近上界，达到最优速率。
AI 中文摘要
本文提出一种新方法，用于从对应多元回归函数的观测值中估计光滑函数的最小值及其位置，该回归函数先验依赖于d个变量，但实际仅依赖于r（r<d）个活跃变量，且受附加噪声污染。我们的方法包含两个步骤：第一步是变量选择方法，用于识别函数f所依赖的r个活跃变量；第二步是估计函数的最小值及其位置。极小值点的估计通过投影梯度下降获得，其中梯度使用回归函数的局部多项式近似估计，且该近似仅限定于第一步得到的活跃变量。最小值的估计通过在先前得到的某一极小值点的估计值处，使用局部多项式方法评估回归函数的估计值来实现。我们为极小值点估计量和最小值估计量的二次风险建立了非渐近上界，并证明它们达到了最优速率，该速率与预先已知活跃变量时可预期的最优速率一致，仅相差一个小于对数项幂次的因子。
英文摘要
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.
