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De-biasing arbitrary convex regularizers and asymptotic normality

By Pierre C. Bellec

Appears in collection : Mathematical Methods of Modern Statistics 2 / Méthodes mathématiques en statistiques modernes 2

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.

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Citation data

  • DOI 10.24350/CIRM.V.19640503
  • Cite this video Bellec, Pierre C. (05/06/2020). De-biasing arbitrary convex regularizers and asymptotic normality. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19640503
  • URL https://dx.doi.org/10.24350/CIRM.V.19640503

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

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