[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.


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