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# Adaptive model selection using empirical complexities

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## Abstract

Given $n$ independent replicates of a jointly distributed pair $(X,Y) \in {\cal R}^d \times {\cal R}$, we wish to select from a fixed sequence of model classes ${\cal F}_1, {\cal F}_2, \ldots$ a deterministic prediction rule $f: {\cal R}^d \to {\cal R}$ whose risk is small. We investigate the possibility of empirically assessing the {\em 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 non-parametric estimation problems. The estimates are shown to achieve a favorable tradeoff 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.

## Suggested Citation

• Gábor Lugosi & Andrew B. Nobel, 1998. "Adaptive model selection using empirical complexities," Economics Working Papers 323, Department of Economics and Business, Universitat Pompeu Fabra.
• Handle: RePEc:upf:upfgen:323
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File URL: https://econ-papers.upf.edu/papers/323.pdf
File Function: Whole Paper

## Citations

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Cited by:

1. Peter Bickel & Bo Li & Alexandre Tsybakov & Sara Geer & Bin Yu & Teófilo Valdés & Carlos Rivero & Jianqing Fan & Aad Vaart, 2006. "Regularization in statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 271-344, September.
2. Stéphane Boucheron & Gábor Lugosi & Pascal Massart, 1999. "A sharp concentration inequality with applications," Economics Working Papers 376, Department of Economics and Business, Universitat Pompeu Fabra.
3. Yang, Yuhong, 2000. "Combining Different Procedures for Adaptive Regression," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 135-161, July.
4. Marta Horvath & Gábor Lugosi, 1996. "A data-dependent skeleton estimate and a scale-sensitive dimension for classification," Economics Working Papers 199, Department of Economics and Business, Universitat Pompeu Fabra.

### Keywords

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

### JEL classification:

• G20 - Financial Economics - - Financial Institutions and Services - - - General
• H30 - Public Economics - - Fiscal Policies and Behavior of Economic Agents - - - General

### NEP fields

This paper has been announced in the following NEP Reports:

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