High-Dimensional Linear Regression via Implicit Regularization

Abstract: Many statistical estimators for high-dimensional linear regression are M-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that under suitable restricted isometry conditions, overparameterization leads to implicit regularization: if we directly apply gradient descent to the residual sum of squares with sufficiently small initial values, then under some proper early stopping rule, the iterates converge to a nearly sparse rate-optimal solution that improves over explicitly regularized approaches. In particular, the resulting estimator does not suffer from extra bias due to explicit penalties, and can achieve the parametric root-n rate when the signal-to-noise ratio is sufficiently high. We also perform simulations to compare our methods with high dimensional linear regression with explicit regularization. Our results illustrate the advantages of using implicit regularization via gradient descent after overparameterization in sparse vector estimation.