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On the Prediction Performance of the Lasso

Author

Listed:
  • Arnak S. Dalalyan

    (CREST-ENSAE)

  • Mohamed Hebiri

    (Université Paris Est)

  • Johannes Lederer

    (Cornell University)

Abstract

Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter leads to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocre irrespective of the choice of the tuning parameter. For the illustration of our approach with an important application, we deduce nearly optimal rates for the least-squares estimator with total variation penalty

Suggested Citation

  • Arnak S. Dalalyan & Mohamed Hebiri & Johannes Lederer, 2014. "On the Prediction Performance of the Lasso," Working Papers 2014-05, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2014-05
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    Cited by:

    1. Pawan Gupta & Marianna Pensky, 2018. "Solution of Linear Ill-Posed Problems Using Random Dictionaries," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 178-193, May.
    2. Sheng Xu & Zhou Fan, 2021. "Iterative Alpha Expansion for estimating gradient‐sparse signals from linear measurements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 271-292, April.
    3. Alexandre Belloni & Mingli Chen & Oscar Hernan Madrid Padilla & Zixuan & Wang, 2019. "High Dimensional Latent Panel Quantile Regression with an Application to Asset Pricing," Papers 1912.02151, arXiv.org, revised Aug 2022.
    4. Pierre Bellec & Alexandre Tsybakov, 2015. "Sharp oracle bounds for monotone and convex regression through aggregation," Working Papers 2015-04, Center for Research in Economics and Statistics.
    5. Wanling Xie & Hu Yang, 2023. "Group sparse recovery via group square-root elastic net and the iterative multivariate thresholding-based algorithm," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(3), pages 469-507, September.
    6. Tanin Sirimongkolkasem & Reza Drikvandi, 2019. "On Regularisation Methods for Analysis of High Dimensional Data," Annals of Data Science, Springer, vol. 6(4), pages 737-763, December.
    7. Jacob Bien & Irina Gaynanova & Johannes Lederer & Christian L. Müller, 2019. "Prediction error bounds for linear regression with the TREX," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 451-474, June.
    8. Gold, David & Lederer, Johannes & Tao, Jing, 2020. "Inference for high-dimensional instrumental variables regression," Journal of Econometrics, Elsevier, vol. 217(1), pages 79-111.
    9. Laura Freijeiro‐González & Manuel Febrero‐Bande & Wenceslao González‐Manteiga, 2022. "A Critical Review of LASSO and Its Derivatives for Variable Selection Under Dependence Among Covariates," International Statistical Review, International Statistical Institute, vol. 90(1), pages 118-145, April.
    10. Tung Duy Luu & Jalal Fadili & Christophe Chesneau, 2020. "Sharp oracle inequalities for low-complexity priors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 353-397, April.

    More about this item

    Keywords

    multiple linear regression; sparse recovery; total variation penalty; oracle inequalities;
    All these keywords.

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