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AUDIOVISUAL MATHEMATICS LIBRARY

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The price of competition: effect size heterogeneity matters in high dimensions!

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Virtualconference
Auteurs : Wang, Hua (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : In high-dimensional regression, the number of explanatory variables with nonzero effects - often referred to as sparsity - is an important measure of the difficulty of the variable selection problem. As a complement to sparsity, this paper introduces a new measure termed effect size heterogeneity for a finer-grained understanding of the trade-off between type I and type II errorsor, equivalently, false and true positive rates using the Lasso. Roughly speaking, a regression coefficient vector has higher effect size heterogeneity than another vector (of the same sparsity) if the nonzero entries of the former are more heterogeneous than those of the latter in terms of magnitudes. From the perspective of this new measure, we prove that in a regime of linear sparsity, false and true positive rates achieve the optimal trade-off uniformly along the Lasso path when this measure is maximum in the sense that all nonzero effect sizes have very differentmagnitudes, and the worst-case trade-off is achieved when it is minimum in the sense that allnonzero effect sizes are about equal. Moreover, we demonstrate that the Lasso path produces anoptimal ranking of explanatory variables in terms of the rank of the first false variable when the effect size heterogeneity is maximum, and vice versa. Metaphorically, these two findings suggest that variables with comparable effect sizes—no matter how large they are—would compete with each other along the Lasso path, leading to an increased hardness of the variable selection problem. Our proofs use techniques from approximate message passing theory as well as a novel argument for estimating the rank of the first false variable.

Keywords : Lasso; Lasso path; false variable; false discovery rate; true positive rate; approximate message passing

Codes MSC :
62F03 - Hypothesis testing
 62J05 - Linear regression
 62J07 - "Ridge regression; James-Stein estimators"
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2146/Slides/Wang.pdf
    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 15/06/2020
    Date de captation : 02/06/2020
    Sous collection : Research talks
    arXiv category : Statistics Theory ; Machine Learning
    Domaine : Probability & Statistics
    Format : MP4 (.mp4) - HD
    Durée : 00:34:07
    Audience : Researchers
    Download : https://videos.cirm-math.fr/2020-06-02_Wang.mp4
Informations sur la Rencontre

Nom de la rencontre : Mathematical Methods of Modern Statistics 2 / Méthodes mathématiques en statistiques modernes 2
Organisateurs de la rencontre : Bogdan, Malgorzata ; Graczyk, Piotr ; Panloup, Fabien ; Proïa, Frédéric ; Roquain, Etienne
Dates : 15/06/2020 - 19/06/2020
Année de la rencontre : 2020
URL Congrès : https://www.cirm-math.com/cirm-virtual-e...

Données de citation

DOI : 10.24350/CIRM.V.19644303
Citer cette vidéo: Wang, Hua (2020). The price of competition: effect size heterogeneity matters in high dimensions!. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19644303
URI : http://dx.doi.org/10.24350/CIRM.V.19644303

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