HAL : hal-00522536, version 2

arXiv : 1010.0072

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v1 (01-10-2010)

v2 (12-09-2011)

Linear regression through PAC-Bayesian truncation

Jean-Yves Audibert 1, 2, Olivier Catoni 3, 4

(30/09/2010)

We consider the problem of predicting as well as the best linear combination of d given functions in least squares regression under L^\infty constraints on the linear combination. When the input distribution is known, there already exists an algorithm having an expected excess risk of order d/n, where n is the size of the training data. Without this strong assumption, standard results often contain a multiplicative log(n) factor, complex constants involving the conditioning of the Gram matrix of the covariates, kurtosis coefficients or some geometric quantity characterizing the relation between L^2 and L^\infty-balls and require some additional assumptions like exponential moments of the output. This work provides a PAC-Bayesian shrinkage procedure with a simple excess risk bound of order d/n holding in expectation and in deviations, under various assumptions. The common surprising factor of these results is their simplicity and the absence of exponential moment condition on the output distribution while achieving exponential deviations. The risk bounds are obtained through a PAC-Bayesian analysis on truncated differences of losses. We also show that these results can be generalized to other strongly convex loss functions.

1 :	IMAGINE
	CSTB – Ecole des Ponts ParisTech – Université Paris-Est Créteil Val-de-Marne (UPEC)
2 :	SIERRA (INRIA Paris - Rocquencourt)
	INRIA : PARIS - ROCQUENCOURT – Ecole normale supérieure de Paris - ENS Paris – CNRS : UMR8548
3 :	Département de Mathématiques et Applications (DMA)
	CNRS : UMR8553 – Ecole normale supérieure de Paris - ENS Paris
4 :	CLASSIC (INRIA Paris - Rocquencourt)
	Ecole normale supérieure de Paris - ENS Paris – INRIA

Domaine

:

Mathématiques/Statistiques Statistiques/Théorie

Mots Clés : Linear regression – Generalization error – Shrinkage – PAC-Bayesian theorems – Risk bounds – Robust statistics – Resistant estimators – Gibbs posterior distributions – Randomized estimators – Statistical learning theory

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Contributeur : Jean-Yves Audibert <>

Soumis le : Dimanche 11 Septembre 2011, 17:26:49

Dernière modification le : Lundi 12 Septembre 2011, 10:41:41