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De-biasing arbitrary convex regularizers and asymptotic normality
Virtualconference
Authors : Bellec, Pierre C. (Author of the conference)
CIRM (Publisher )
Additional resources :
https://www.cirm-math.fr/RepOrga/2146/Slides/Bellec_slides.pdf
CIRM (Publisher )
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Abstract :
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.
Keywords : M-estimators; convex regularization; asymptotic normality; confidence intervals; de-biasing
MSC Codes : 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.
Keywords : M-estimators; convex regularization; asymptotic normality; confidence intervals; de-biasing
Additional resources :
https://www.cirm-math.fr/RepOrga/2146/Slides/Bellec_slides.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 15/06/2020 Conference Date : 05/06/2020 Subseries : Research talks arXiv category : Statistics Theory ; Computational Geometry Mathematical Area(s) : Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:46:32 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/ 2020-06-05_Bellec.mp4 
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Information on the Event
Event Title : Mathematical Methods of Modern Statistics 2 / Méthodes mathématiques en statistiques modernes 2Event Organizers : Bogdan, Malgorzata ; Graczyk, Piotr ; Panloup, Fabien ; Proïa, Frédéric ; Roquain, Etienne Dates : 15/06/2020 - 19/06/2020 Event Year : 2020 Event URL : https://www.cirm-math.com/cirm-virtual-...
Citation Data
DOI : 10.24350/CIRM.V.19640503Cite this video as: 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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Bibliography
- 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
