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Boosting for high-dimensional linear models. (English) Zbl 1095.62077
Summary: We prove that boosting with the squared error loss, ${L}_{2}$-Boosting, is consistent for very high-dimensional linear models, where the number of predictor variables is allowed to grow essentially as fast as $O$(exp(sample size)), assuming that the true underlying regression function is sparse in terms of the ${\ell }_{1}$-norm of the regression coefficients. In the language of signal processing, this means consistency for de-noising using a strongly overcomplete dictionary if the underlying signal is sparse in terms of the ${\ell }_{1}$-norm. We also propose here an AIC-based method for tuning, namely for choosing the number of boosting iterations. This makes ${L}_{2}$-Boosting computationally attractive since it is not required to run the algorithm multiple times for cross-validation as commonly used so far. We demonstrate ${L}_{2}$-Boosting for simulated data, in particular where the predictor dimension is large in comparison to sample size, and for a difficult tumor-classification problem with gene expression microarray data.
##### MSC:
 62J05 Linear regression 62B10 Statistical information theory 65C60 Computational problems in statistics 49M15 Newton-type methods in calculus of variations 62P10 Applications of statistics to biology and medical sciences 68Q32 Computational learning theory 62J99 Linear statistical inference