Error Bounds for Generalized Group Sparsity

TL;DR

This paper introduces a universal theoretical framework for analyzing error bounds in generalized group sparsity regularization, unifying results for various sparsity-inducing methods.

Contribution

It presents a universal theorem that establishes consistency and convergence rates for a generalized sparse-group Lasso, extending existing results to double sparsity regularization.

Findings

Unified convergence rates for multiple sparsity regularizations

Identification of a generalized norm for dual formulation

Extension of results to double regularization cases

Abstract

In high-dimensional statistical inference, sparsity regularizations have shown advantages in consistency and convergence rates for coefficient estimation. We consider a generalized version of Sparse-Group Lasso which captures both element-wise sparsity and group-wise sparsity simultaneously. We state one universal theorem which is proved to obtain results on consistency and convergence rates for different forms of double sparsity regularization. The universality of the results lies in an generalization of various convergence rates for single regularization cases such as LASSO and group LASSO and also double regularization cases such as sparse-group LASSO. Our analysis identifies a generalized norm of $\epsilon$-norm, which provides a dual formulation for our double sparsity regularization.