Convex regression in multidimensions: Suboptimality of least squares estimators
在高斯误差的非参数回归模型下，当维度d≥5时，最小二乘估计在平方误差损失下估计d维凸函数是次优的，其风险阶数为n^{-2/d}，而极小化风险为n^{-4/(d+4)}。
Under the usual nonparametric regression model with Gaussian errors, Least Squares Estimators (LSEs) over natural subclasses of convex functions are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random design) and (iii) convex functions supported on a polytope (in fixed design). For each of these classes, the risk of the LSE is proved to be of the order n−2/d (up to logarithmic factors) while the minimax risk is n−4/(d+4), when d≥5. In addition, the first rate of convergence results (worst case and adaptive) for the unrestricted convex LSE are established in fixed design for polytopal domains for all d≥1. Some new metric entropy results for convex functions are also proved, which are of independent interest.