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SUMMARY:Isotonic regression in general dimensions - Richard Samworth (Univ
 ersity of Cambridge)
DTSTART:20180319T143000Z
DTEND:20180319T153000Z
UID:TALK102580@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Co-authors: Qiyang Han		(University of Washington)\, Tengyao W
 ang		(University of Cambridge)\, Sabyasachi Chatterjee		(University of Ill
 inois)<br><br>We study the least squares regression function estimator ove
 r 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 la
 ttice design\, we establish that the estimator achieves the minimax rate o
 f order $n^{&minus\;min\\{2/(d+2)\,1/d\\}}$ in the empirical $L_2$ loss\, 
 up to poly-logarithmic factors. Further\, we prove a sharp oracle inequali
 ty\, which reveals in particular that when the true regression function is
  piecewise constant on $k$ hyperrectangles\, the least squares estimator e
 njoys a faster\, adaptive rate of convergence of $(k/n)^{min(1\,2/d)}$\, a
 gain up to poly-logarithmic factors. Previous results are confined to the 
 case $d\\leq 2$. Finally\, we establish corresponding bounds (which are ne
 w even in the case $d=2$) in the more challenging random&nbsp\;design sett
 ing. There are two surprising features of these results: first\, they demo
 nstrate that it is possible for a global empirical risk minimisation proce
 dure to be rate optimal up to poly-logarithmic factors even when the corre
 sponding 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.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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