Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases

Navarro, Fabien; Saumard, Adrien

doi:10.1051/ps/2017005

Navarro, Fabien ¹ ; Saumard, Adrien ¹

¹ CREST, ENSAI, Campus de Ker-Lann, rue Blaise Pascal, BP 37203, 35172 Bruz Cedex, France.

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 412-451

Résumé

We investigate the optimality for model selection of the so-called slope heuristics, $V$ -fold cross-validation and $V$ -fold penalization in a heteroscedatic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics – when the optimal penalty shape is known – and $V$ -fold penalization. Furthermore, $V$ -fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1 - V^{- 1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$ -fold cross-validation and $V$ -fold penalization have comparable efficiency.

Reçu le : 2016-08-25
Accepté le : 2017-03-08

MR Zbl

DOI : 10.1051/ps/2017005

Classification : 62G08, 62G09
Keywords: Nonparametric regression, heteroscedastic noise, random design, model selection, cross-validation, wavelets

Affiliations des auteurs :

Navarro, Fabien ¹ ; Saumard, Adrien ¹

¹ CREST, ENSAI, Campus de Ker-Lann, rue Blaise Pascal, BP 37203, 35172 Bruz Cedex, France.

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     author = {Navarro, Fabien and Saumard, Adrien},
     title = {Slope heuristics and {V-Fold} model selection in heteroscedastic regression using strongly localized bases},
     journal = {ESAIM: Probability and Statistics},
     pages = {412--451},
     year = {2017},
     publisher = {EDP Sciences},
     volume = {21},
     doi = {10.1051/ps/2017005},
     mrnumber = {3743921},
     zbl = {1395.62093},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2017005/}
}

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Navarro, Fabien; Saumard, Adrien. Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 412-451. doi: 10.1051/ps/2017005

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