On Lasso refitting strategies

Abstract : A well-know drawback of l1-penalized estimators is the systematic shrinkage of the large coefficients towards zero. A simple remedy is to treat Lasso as a model-selection procedure and to perform a second refitting step on the selected support. In this work we formalize the notion of refitting and provide oracle bounds for arbitrary refitting procedures of the Lasso solution. One of the most widely used refitting techniques which is based on least-squares may bring a problem of interpretability, since the signs of the refitted estimator might be flipped with respect to the original estimator. This problem arises from the fact that the least-square refitting considers only the support of the Lasso solution, avoiding any information about signs or amplitudes. To this end we define a sign-consistent refitting as an arbitrary refitting procedure, preserving the signs of the first step Lasso solution and provide Oracle inequalities for such estimators. Finally, we consider special refitting strategies: Bregman Lasso and Boosted Lasso. Bregman Lasso has a fruitful property to converge to the sign-consistent least-squares refitting (least-squares with sign constraints), which provides with greater interpretability. We additionally study the Bregman Lasso refitting in the case of orthogonal design, providing with simple intuition behind the proposed method. Boosted Lasso, in contrast, considers information about magnitudes of the first Lasso step and allows to develop better oracle rates for prediction. Finally, we conduct an extensive numerical study to show advantages of one approach over others in different synthetic and semi-real scenarios.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01593888
Contributeur : Evgenii Chzhen <>
Soumis le : mardi 26 septembre 2017 - 19:49:41
Dernière modification le : mardi 10 octobre 2017 - 13:46:17

Identifiants

  • HAL Id : hal-01593888, version 1
  • ARXIV : 1707.05232

Citation

Evgenii Chzhen, Mohamed Hebiri, Joseph Salmon. On Lasso refitting strategies. 2017. 〈hal-01593888〉

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