Asymptotic equivalence of regularization methods in thresholded parameter space

TL;DR

This paper demonstrates that in high-dimensional generalized linear models, various regularization methods become asymptotically equivalent in a thresholded parameter space, with differences in convergence rates depending on the growth rate of dimensionality.

Contribution

It characterizes the asymptotic equivalence of convex and concave regularization methods under general conditions, revealing phase transition phenomena based on dimensionality growth.

Findings

Lasso and concave methods are equivalent for polynomial growth in dimension.

Concave methods outperform Lasso in exponential dimensionality.

Oracle inequalities support the asymptotic equivalence and convergence rate differences.

Abstract

High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates the other, an important question both in theory and to practitioners. In this paper, we characterize the asymptotic equivalence of regularization methods, with general penalty functions, in a thresholded parameter space under the generalized linear model setting, where the dimensionality can grow up to exponentially with the sample size. To assess their performance, we establish the oracle inequalities, as in Bickel, Ritov and Tsybakov (2009), of the global minimizer for these methods under various prediction and variable selection losses. These results reveal an interesting phase transition phenomenon. For polynomially growing dimensionality, the $L_1$-regularization method of Lasso and concave methods are asymptotically equivalent, having the same convergence rates in the oracle inequalities. For exponentially growing dimensionality, concave methods are asymptotically equivalent but have faster convergence rates than the Lasso. We also establish a stronger property of the oracle risk inequalities of the regularization methods, as well as the sampling properties of computable solutions. Our new theoretical results are illustrated and justified by simulation and real data examples.