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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Isotonic regression in general dimensions - Tengya
o Wang (University of Cambridge)
DTSTART;TZID=Europe/London:20180308T110000
DTEND;TZID=Europe/London:20180308T120000
UID:TALK102205AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/102205
DESCRIPTION:We study the least squares regression function est
imator over the class of real-valued functions on
[0\,1]^d that are increasing in each coordinate.&n
bsp\; For uniformly bounded signals and with a fix
ed\, cubic lattice design\, we establish that the
estimator achieves the minimax rate of order n^{-m
in(2/(d+2)\,1/d)} in the empirical L_2 loss\, up t
o poly-logarithmic factors. \; Further\, we pr
ove a sharp oracle inequality\, which reveals in p
articular that when the true regression function i
s piecewise constant on $k$ hyperrectangles\, the
least squares estimator enjoys a faster\, adaptive
rate of convergence of (k/n)^{min(1\,2/d)}\, agai
n up to poly-logarithmic factors. \; Previous
results are confined to the case d&le\;2. \; F
inally\, we establish corresponding bounds (which
are new even in the case d=2) in the more challeng
ing random design setting. \; There are two su
rprising features of these results: first\, they d
emonstrate that it is possible for a global empiri
cal risk minimisation procedure to be rate optimal
up to poly-logarithmic factors even when the corr
esponding 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. <
br>
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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