The Annals of Statistics

Adaptive model selection using empirical complexities

Gábor Lugosi and Andrew B. Nobel

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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.

Article information

Ann. Statist., Volume 27, Number 6 (1999), 1830-1864.

First available in Project Euclid: 4 April 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Complexity regularization classification pattern recognition regression estimation curve fitting minimum description length


Lugosi, Gábor; Nobel, Andrew B. Adaptive model selection using empirical complexities. Ann. Statist. 27 (1999), no. 6, 1830--1864. doi:10.1214/aos/1017939241.

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