First order expansion of convex regularized estimators
This provides theoretical tools for analyzing high-dimensional regularized estimators, which is incremental for statisticians and machine learning researchers working on inference in sparse models.
The paper tackles the problem of deriving first-order expansions for convex penalized estimators in high-dimensional regression, showing that the expansion approximates the estimator with error smaller than the estimation error, leading to precise MSE characterizations and inference results for methods like Lasso and Group-Lasso.
We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function $h$ and the corresponding penalized estimator $\hatβ$, we construct a quantity $η$, the first order expansion of $\hatβ$, such that the distance between $\hatβ$ and $η$ is an order of magnitude smaller than the estimation error $\|\hatβ - β^*\|$. In this sense, the first order expansion $η$ can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of $\hatβ$ is asymptotically the same as the risk of $η$ which leads to a precise characterization of the MSE of $\hatβ$; this characterization takes a particularly simple form for isotropic design. Such first order expansion also leads to inference results based on $\hatβ$. We provide sufficient conditions for the existence of such first order expansion for three regularizers: the Lasso in its constrained form, the lasso in its penalized form, and the Group-Lasso. The results apply to general loss functions under some conditions and those conditions are satisfied for the squared loss in linear regression and for the logistic loss in the logistic model.