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December 1999 Adaptive model selection using empirical complexities
Gábor Lugosi, Andrew B. Nobel
Ann. Statist. 27(6): 1830-1864 (December 1999). DOI: 10.1214/aos/1017939241


Given $n$ independent replicates of a jointly distributed pair $(X, Y) \in \mathscr{R}^d \times \mathscr{R}$, we wish to select from a fixed sequence of model classes $\mathscr{F}_1, \mathscr{F}_2,\dots$ a deterministic prediction rule $f: \mathscr{R}^d \to \mathscr{R}$ whose risk is small.We investigate the possibility of empirically assessing the complexity of each model class, that is, the actual difficulty of the estimation problem within each class. The estimated complexities are in turn used to define an adaptive model selection procedure, which is based on complexity penalized empirical risk.

The available data are divided into two parts. The first is used to form an empirical cover of each model class, and the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error. An estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk. A distinguishing feature of the approach is that the complexity of each model class is assessed empirically, based on the size of its empirical cover.

Finite sample performance bounds are established for the estimates, and these bounds are applied to several nonparametric estimation problems. The estimates are shown to achieve a favorable trade-off between approximation and estimation error and to perform as well as if the distribution-dependent complexities of the model classes were known beforehand. In addition, it is shown that the estimate can be consistent, and even possess near optimal rates of convergence, when each model class has an infinite VC or pseudo dimension.

For regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence.


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Gábor Lugosi. Andrew B. Nobel. "Adaptive model selection using empirical complexities." Ann. Statist. 27 (6) 1830 - 1864, December 1999.


Published: December 1999
First available in Project Euclid: 4 April 2002

zbMATH: 0962.62034
MathSciNet: MR1765619
Digital Object Identifier: 10.1214/aos/1017939241

Primary: 62G07, 62G20
Secondary: 62H30

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 6 • December 1999
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