@inproceedings{809762db77fd4f608e956cdeae382488,
title = "Bounds on the Prediction Error of Penalized Least Squares Estimators with Convex Penalty",
abstract = "This paper considers the penalized least squares estimator with arbitrary convex penalty. When the observation noise is Gaussian, we show that the prediction error is a subgaussian random variable concentrated around its median. We apply this concentration property to derive sharp oracle inequalities for the prediction error of the LASSO, the group LASSO, and the SLOPE estimators, both in probability and in expectation. In contrast to the previous work on the LASSO-type methods, our oracle inequalities in probability are obtained at any confidence level for estimators with tuning parameters that do not depend on the confidence level. This is also the reason why we are able to establish sparsity oracle bounds in expectation for the LASSO-type estimators, while the previously known techniques did not allow for the control of the expected risk. In addition, we show that the concentration rate in the oracle inequalities is better than it was commonly understood before.",
keywords = "Group LASSO, LASSO estimator, Oracle inequality, Penalized least squares, SLOPE estimator",
author = "Pierre Bellec and Alexandre Tsybakov",
note = "Publisher Copyright: {\textcopyright} 2017, Springer International Publishing AG.; International Conference on Modern problems of stochastic analysis and statistics, in honor On the occasion of Valentin Konakov{\textquoteright}s 70th birthday, 2016 ; Conference date: 29-05-2016 Through 02-06-2016",
year = "2017",
month = jan,
day = "1",
doi = "10.1007/978-3-319-65313-6\_13",
language = "English",
isbn = "9783319653129",
series = "Springer Proceedings in Mathematics and Statistics",
publisher = "Springer New York LLC",
pages = "315--333",
editor = "Vladimir Panov",
booktitle = "Modern Problems of Stochastic Analysis and Statistics - Selected Contributions in Honor of Valentin Konakov",
}