[YPNG] YPNG 29 Jan 2016
sekhar
sekhar.tatikonda at yale.edu
Mon Jan 25 08:54:45 EST 2016
Hi Everyone,
Our first YPNG of the Spring 2016 semester will be held this coming Friday.
Pierre Bellec will talk about:
Title: Almost parametric rates and adaptive confidence sets in shape
constrained problems
Abstract: We study an estimation problem over a known convex polyhedron.
This polyhedron
represents the shape constraint, and the goal is to recover an unknown
parameter from this
polyhedron which is obscured by Gaussian noise. The talk focuses on the
particular situation
where the unknown parameter belongs to a low-dimensional face of the
polyhedron. Is it
easier to estimate the true parameter in this case? How does the face
dimension impact the
rate of estimation? Is it possible to construct confidence sets that
adapt to the face dimension?
We will first answer these questions for the polyhedral cone of
nondecreasing sequences, which
corresponds to univariate isotonic regression. If the true parameter
belongs to a k-dimensional
face, the Least-Squares estimator converges with an almost parametric
rate of order k/n -- and
this rate is optimal in a minimax sense. This stil holds if the true
parameter is close to a
k-dimensional face, in this case the result takes the form of oracle
inequalities or regret bounds.
Thus the Least-Squares estimator automatically adapts to the unknown
dimension k. Then, a
natural problem is to estimate k, or construct data-driven confidence
sets that contain the true
parameter with high probability. Ideally, the expected diameter of this
confidence set should be
of the same order as the optimal rate. We will see that the construction
of such confidence sets
is possible adaptively, without the knowledge of the unknown dimension k.
Finally, we will present the probabilistic arguments used to derive
these results and another
polyhedron for which a similar phenomenon appears.
See you Friday in the Stat's classroom at 11am.
Regards,
sekhar
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