 License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.06201.3
URN: urn:nbn:de:0030-drops-8912
URL: https://drops.dagstuhl.de/opus/volltexte/2007/891/
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### Local Minimax Learning of Approximately Polynomial Functions

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### Abstract

Suppose we have a number of noisy measurements of an unknown real-valued function \$f\$ near
point of interest \$mathbf{x}_0 in mathbb{R}^d\$. Suppose also that nothing can be assumed
about the noise distribution, except for zero mean and bounded covariance matrix. We want
to estimate \$f\$ at \$mathbf{x=x}_0\$ using a general linear parametric family
\$f(mathbf{x};mathbf{a}) = a_0 h_0 (mathbf{x}) ++ a_q h_q (mathbf{x})\$, where
\$mathbf{a} in mathbb{R}^q\$ and \$h_i\$'s are bounded functions on a neighborhood \$B\$ of
\$mathbf{x}_0\$ which contains all points of measurement. Typically, \$B\$ is a Euclidean ball
or cube in \$mathbb{R}^d\$ (more generally, a ball in an \$l_p\$-norm). In the case when the
\$h_i\$'s are polynomial functions in \$x_1,ldots,x_d\$ the model is called
locally-polynomial. In particular, if the \$h_i\$'s form a basis of the linear space of
polynomials of degree at most two, the model is called locally-quadratic (if the degree is
at most three, the model is locally-cubic, etc.). Often, there is information, which is
called context, about the function \$f\$ (restricted to \$B\$ ) available, such as that it
takes values in a known interval, or that it satisfies a Lipschitz condition. The theory of
local minimax estimation with context for locally-polynomial models and approximately
locally polynomial models has been recently initiated by Jones. In the case of local
linearity and a bound on the change of \$f\$ on \$B\$, where \$B\$ is a ball, the solution for
squared error loss is in the form of ridge regression, where the ridge parameter is
identified; hence, minimax justification for ridge regression is given together with
explicit best error bounds. The analysis of polynomial models of degree above 1 leads to
interesting and difficult questions in real algebraic geometry and non-linear optimization.

We show that in the case when \$f\$ is a probability function, the optimal (in the minimax
sense) estimator is effectively computable (with any given precision), thanks to Tarski's
elimination principle.

### BibTeX - Entry

```@InProceedings{jones_et_al:DagSemProc.06201.3,
author =	{Jones, Lee and Rybnikov, Konstantin},
title =	{{Local Minimax Learning of Approximately Polynomial Functions}},
booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
pages =	{1--12},
series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
ISSN =	{1862-4405},
year =	{2007},
volume =	{6201},
editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
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