论文标题
多元凸回归的有效最小值最佳估计器
Efficient Minimax Optimal Estimators For Multivariate Convex Regression
论文作者
论文摘要
我们研究多元凸回归任务的计算方面$ d \ geq 5 $。我们提供了第一个计算高效的最小值最佳（最大为对数因素）估计器，用于（I）$ L $ -LIPSCHITZ凸回归（II）$γ$结合的凸回归。这些估计量正确性的证明使用了来自不同学科的各种工具，其中包括经验过程理论，随机几何学和潜在理论。这项工作是第一个表明非二声制类别的有效最小估计量的存在，其相应的最小二乘估计量是最小值的亚最佳选择。独立利益的结果。
We study the computational aspects of the task of multivariate convex regression in dimension $d \geq 5$. We present the first computationally efficient minimax optimal (up to logarithmic factors) estimators for the tasks of (i) $L$-Lipschitz convex regression (ii) $Γ$-bounded convex regression under polytopal support. The proof of the correctness of these estimators uses a variety of tools from different disciplines, among them empirical process theory, stochastic geometry, and potential theory. This work is the first to show the existence of efficient minimax optimal estimators for non-Donsker classes that their corresponding Least Squares Estimators are provably minimax sub-optimal; a result of independent interest.
