Journal Article

ROBUST LINEAR LEAST SQUARES REGRESSION

Audibert, Jean-Yves, Catoni, Olivier

2011

Overview

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 the 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.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 62J05 / 62J07 / Branch & bound algorithms / Covariance matrices / Eigenvalues / Estimators / Exact sciences and technology / General topics / generalization error / Gibbs posterior distributions / Input output / Least squares / Linear inference, regression / Linear regression / Mathematical functions / Mathematical moments / Mathematics / PAC-Bayesian theorems / Parameter estimation / Probability and statistics / Random variables / randomized estimators / Regression analysis / resistant estimators / Risk / risk bounds / robust statistics / Sciences and techniques of general use / shrinkage / Signal noise / statistical learning theory / Statistics / Statistics Theory / Studies