Robust linear least squares regression

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

1 imagine [Marne-la-Vallée]
2 WILLOW - Models of visual object recognition and scene understanding
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 CLASSIC - Computational Learning, Aggregation, Supervised Statistical, Inference, and Classification

We consider the problem of robustly predicting as well as the best linear combination of d given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. For the ridge estimator and the ordinary least squares estimator, and their variants, we provide new risk bounds of order d/n without logarithmic factor unlike some standard results, where n is the size of the training data. We also provide a new estimator with better deviations in presence of heavy-tailed noise. It is based on truncating differences of losses in a min-max framework and satisfies a d/n risk bound both in expectation and in deviations. The key common surprising factor of these results is the absence of exponential moment condition on the output distribution while achieving exponential deviations. All risk bounds are obtained through a PAC-Bayesian analysis on truncated differences of losses. Experimental results strongly back up our truncated min-max estimator.

Mots clés: Linear regression; Least squares regression; Generalization error; Shrinkage; PAC-Bayesian theorems; Risk bounds; Robust statistics; Resistant estimators; Statistical learning theory

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jean-Yves Audibert, Olivier Catoni. Robust linear least squares regression. 2010. ⟨hal-00522534v1⟩