Global risk bounds and adaptation in univariate convex regression
Global risk bounds and adaptation in univariate convex regression
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DOI：
10.1007/s00440-014-0595-3
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发表时间：
2015-10-01
影响因子：
2
通讯作者：
Sen, Bodhisattva
中科院分区：
文献类型：
--
作者：
Guntuboyina, Adityanand;Sen, Bodhisattva
本文考虑凸回归函数的非参数估计问题。研究了自然平方误差损失下最小二乘估计的风险问题。我们表明，风险总是有界的模对数因子，而小得多时，是很好的近似分段仿射凸函数，没有太多的仿射片（在这种情况下，风险是最多的对数因子）。另一方面，当有曲率，我们表明，没有估计可以有风险小于一个常数的倍数在一个非常强的意义上证明了一个“本地”极大极小下界。我们还研究了模型误设定的情况下，我们表明，LSE表现出相同的全球行为提供的损失是从最接近的凸投影的真实回归函数。在推导风险界的过程中，我们证明了一元凸函数空间局部邻域的度量熵的新结果。这些结果，这可能是独立的利益，展示了一元凸函数的空间的非均匀性，形成鲜明对比，经典的函数空间的光滑性约束的基础上。
We consider the problem of nonparametric estimation of a convex regression function . We study the risk of the least squares estimator (LSE) under the natural squared error loss. We show that the risk is always bounded from above by modulo logarithmic factors while being much smaller when is well-approximable by a piecewise affine convex function with not too many affine pieces (in which case, the risk is at most up to logarithmic factors). On the other hand, when has curvature, we show that no estimator can have risk smaller than a constant multiple of in a very strong sense by proving a "local" minimax lower bound. We also study the case of model misspecification where we show that the LSE exhibits the same global behavior provided the loss is measured from the closest convex projection of the true regression function. In the process of deriving our risk bounds, we prove new results for the metric entropy of local neighborhoods of the space of univariate convex functions. These results, which may be of independent interest, demonstrate the non-uniform nature of the space of univariate convex functions in sharp contrast to classical function spaces based on smoothness constraints.