De-biasing convex regularized estimators and interval estimation in linear models

New upper bounds are developed for the $L_2$ distance between $\xi/\text{Var}[\xi]^{1/2}$ and linear and quadratic functions of $z\sim N(0,I_n)$ for random variables of the form $\xi=bz^\top f(z) - \text{div} f(z)$. The linear approximation yields a central limit theorem when the squared norm of $f(z)$ dominates the squared Frobenius norm of $\nabla f(z)$ in expectation. Applications of this normal approximation are given for the asymptotic normality of de-biased estimators in linear regression with correlated design and convex penalty in the regime $p/n \le \gamma$ for constant $\gamma\in(0,{\infty})$. For the estimation of linear functions $\langle a_0,\beta\rangle$ of the unknown coefficient vector $\beta$, 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 convex penalty if $\gamma<1$ and for any strongly convex penalty if $\gamma\ge 1$. In particular the penalty needs not be separable or permutation invariant. By allowing arbitrary regularizers, the results vastly broaden the scope of applicability of de-biasing methodologies to obtain confidence intervals in high-dimensions. In the absence of strong convexity for $p>n$, asymptotic normality of the de-biased estimate is obtained for the Lasso and the group Lasso under additional conditions. For general convex penalties, our analysis also provides prediction and estimation error bounds of independent interest.