Global risk bounds and adaptation in univariate convex regression

• 成果类型: Article
• 署名作者: Guntuboyina, Adityanand; Sen, Bodhisattva
• 署名单位: University of California System; University of California Berkeley; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0595-3
• 发表日期: 2015
• 页码: 379-411
• 关键词: grenander estimator; Consistency; CONVERGENCE; algorithm; entropy

摘要

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.