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제목 [학술세미나] [학과세미나] 5월 23일(수) 17시 특별세미나 안내
작성일 2018-05-21 10:31:07
내용 [학과세미나] 5월 23일(수) 17시 특별세미나 안내


▪ 제목 : Isotonic regression in general dimensions
▪ 연사 : Richard Samworth (Cambridge university)
▪ 일시 : 2018년 5월 23일(수) PM 17:00 – 18:00
▪ 장소 : 25동 405호

초 록

We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_2$ loss, up to poly-logarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{min(1,2/d)}$, again up to poly-logarithmic factors. Previous results are confined to the case $d \leq 2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to poly-logarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.
 
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