The linearized alternating direction method of multipliers for sparse group LAD model

Abstract

While the least absolute shrinkage and selection operator (LASSO) became a popular method due to its wide applications in high dimensional settings, some generalized LASSO models were developed. The sparse group LASSO is one of the important lasso-type methods, which aims to solve the linear regression problems with grouped covariates and tends to produce a solution with sparse effects both on a group and within group level. At the same time, we know that the least absolute deviation (LAD) is a useful and robust method when the noise distribution may be heavy-tailed or heterogeneous. In this paper, we combine these two classical ideas together to develop sparse group LAD model. We show that the sparse group LAD estimator achieves near oracle performance, i.e., with high probability, the $L_2$ norm of the estimation error is of order $O(\sqrt{k\text{ log }p/n}).$ Moreover, with the help of the linearization technique we propose the linearized alternating direction method of multipliers to solve the sparse group LAD estimator and establish its convergence. Numerical experiments are reported to illustrate the efficiency of the proposed algorithm.