Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 2198-2216.doi: 10.1007/s10473-021-0624-0

• Articles • Previous Articles    

PENALIZED LEAST SQUARE IN SPARSE SETTING WITH CONVEX PENALTY AND NON GAUSSIAN ERRORS

Doualeh ABDILLAHI-ALI, Nourddine AZZAOUI, Arnaud GUILLIN, Guillaume LE MAILLOUX, Tomoko MATSUI   

  1. 1. Laboratoire de Mathématiques Blaise Pascal, CNRS-UMR 6620, Université Clermont-Auvergne(UCA), 63000 Clermont-Ferrand, France;
    2. epartment of Statistical Modeling, Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa-shi, Tokyo 1908562, Japan
  • Received:2021-05-26 Revised:2021-10-13 Online:2021-12-25 Published:2021-12-27
  • Supported by:
    This work has been (partially) supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.

Abstract: This paper consider the penalized least squares estimators with convex penalties or regularization norms. We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting. The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap (or Poincaré) inequality instead of Gaussian distributions. We illustrate our results on a heavy tailed example and a sub Gaussian one; we especially give the explicit bounds of the oracle inequalities for these two special examples.

Key words: penalized least squares, Gaussian errors, convex penalty

CLC Number: 

  • 62N01
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