Linear regression through PAC-Bayesian truncation

Jean-Yves Audibert 1, 2 Olivier Catoni 3, 4
1 IMAGINE [Marne-la-Vallée]
LIGM - Laboratoire d'Informatique Gaspard-Monge, CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
4 CLASSIC - Computational Learning, Aggregation, Supervised Statistical, Inference, and Classification
DMA - Département de Mathématiques et Applications - ENS Paris, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt
Abstract : 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.
Type de document :
Pré-publication, Document de travail
2011
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https://hal.archives-ouvertes.fr/hal-00522536
Contributeur : Jean-Yves Audibert <>
Soumis le : dimanche 11 septembre 2011 - 17:26:49
Dernière modification le : lundi 28 janvier 2019 - 09:04:30
Document(s) archivé(s) le : lundi 12 décembre 2011 - 02:21:46

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  • HAL Id : hal-00522536, version 2
  • ARXIV : 1010.0072

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Jean-Yves Audibert, Olivier Catoni. Linear regression through PAC-Bayesian truncation. 2011. 〈hal-00522536v2〉

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