ROBUST LINEAR LEAST SQUARES REGRESSION

• 成果类型: Article
• 署名作者: Audibert, Jean-Yves; Catoni, Olivier
• 署名单位: Universite Gustave-Eiffel; ESIEE Paris; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS918
• 发表日期: 2011
• 页码: 2766-2794
• 关键词: rates

摘要

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.