First order expansion of convex regularized estimators
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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 β̂, we construct a quantity η, the first order expansion of β̂, such that the distance between β̂ and η is an order of magnitude smaller than the estimation error β̂ - 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 β̂ is asymptotically the same as the risk of η which leads to a precise characterization of the MSE of β̂; this characterization takes a particularly simple form for isotropic design. Such first order expansion also leads to inference results based on β̂. 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.
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