English
De-biasing arbitrary convex regularizers and asymptotic normality
Virtualconference
Auteurs : Bellec, Pierre C. (Auteur de la conférence)
CIRM (Editeur )
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2146/Slides/Bellec_slides.pdf
CIRM (Editeur )
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Résumé :
A new Central Limit Theorem (CLT) is developed for random variables of the form ξ=z⊤f(z)−divf(z) where z∼N(0,In).
The normal approximation is proved to hold when the squared norm of f(z) dominates the squared Frobenius norm of ∇f(z) in expectation.
Applications of this CLT are given for the asymptotic normality of de-biased estimators in linear regression with correlated design and convex penalty in the regime p/n→γ∈(0,∞). For the estimation of linear functions ⟨a,β⟩ of the unknown coefficient vector β, this analysis leads to asymptotic normality of the de-biased estimate for most normalized directions a0, where "most" is quantified in a precise sense. This asymptotic normality holds for any coercive convex penalty if γ<1 and for any strongly convex penalty if γ≥1. In particular the penalty needs not be separable or permutation invariant.
Mots-Clés : M-estimators; convex regularization; asymptotic normality; confidence intervals; de-biasing
Codes MSC : The normal approximation is proved to hold when the squared norm of f(z) dominates the squared Frobenius norm of ∇f(z) in expectation.
Applications of this CLT are given for the asymptotic normality of de-biased estimators in linear regression with correlated design and convex penalty in the regime p/n→γ∈(0,∞). For the estimation of linear functions ⟨a,β⟩ of the unknown coefficient vector β, this analysis leads to asymptotic normality of the de-biased estimate for most normalized directions a0, where "most" is quantified in a precise sense. This asymptotic normality holds for any coercive convex penalty if γ<1 and for any strongly convex penalty if γ≥1. In particular the penalty needs not be separable or permutation invariant.
Mots-Clés : M-estimators; convex regularization; asymptotic normality; confidence intervals; de-biasing
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2146/Slides/Bellec_slides.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de Publication : 15/06/2020 Date de Captation : 05/06/2020 Sous Collection : Research talks Catégorie arXiv : Statistics Theory ; Computational Geometry Domaine(s) : Probabilités & Statistiques Format : MP4 (.mp4) - HD Durée : 00:46:32 Audience : Chercheurs Download : https://videos.cirm-math.fr/ 2020-06-05_Bellec.mp4 
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Informations sur la Rencontre
Nom de la Rencontre : Mathematical Methods of Modern Statistics 2 / Méthodes mathématiques en statistiques modernes 2Organisateurs 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 de la Rencontre : https://www.cirm-math.com/cirm-virtual-...
Données de citation
DOI : 10.24350/CIRM.V.19640503Citer cette vidéo: Bellec, Pierre C. (2020). De-biasing arbitrary convex regularizers and asymptotic normality. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19640503 URI : http://dx.doi.org/10.24350/CIRM.V.19640503 |
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Bibliographie
- BELLEC, Pierre C. et ZHANG, Cun-Hui. Second order Poincar\'e inequalities and de-biasing arbitrary convex regularizers when $ p/n\to\gamma$. arXiv preprint arXiv:1912.11943, 2019. - https://arxiv.org/abs/1912.11943
- BELLEC, Pierre C. et ZHANG, Cun-Hui. De-biasing the lasso with degrees-of-freedom adjustment. arXiv preprint arXiv:1902.08885, 2019. - https://arxiv.org/abs/1902.08885
