Second order Poincaré inequalities and de-biasing arbitrary convex regularizers when p/n → γ
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A new Central Limit Theorem (CLT) is developed for random variables of the form ξ=z^ f(z) - div f(z) where z∼ N(0,I_n). 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_0,β〉 of the unknown coefficient vector β, this analysis leads to asymptotic normality of the de-biased estimate for most normalized directions a_0, 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. For the group Lasso, a simple condition is given that grants asymptotic normality for a fixed direction a_0. For the lasso, this condition reduces to λ^2Σ^-1a_0_1^2/R̅→0 where R̅ is the noiseless prediction risk.
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