Nonasymptotic oracle inequalities and alternating direction method of multipliers algorithm for adaptive lasso penalized multiplicative regression

In this paper, we mainly consider the regularized estimation problem of parameters in the multiplicative regression model, where the response variable is always positive. Hao, Lin, and Zhao, Comput. Stat. Data An., 103 (2016) investigated an adaptive variable selection method via the least product relative error (LPRE) criterion and lasso-type penalty with fixed or diverging number of covariates and showed the resultant estimator achieves the oracle property. However, the alternating direction method of multipliers (ADMM) algorithm proposed by the authors is based on the least square approximation of the LPRE loss function, where a well-behaved initial estimator must be determined in advance, and the convergence is not validated. Through careful introduction of auxiliary variables and a three-block reformulation, our ADMM algorithm eliminates sensitivity to initial values while ensuring convergence. In addition, by virtue of the symmetric Bregman (SB) divergence and natural extensions of compatibility and weak cone invertibility factors, we establish nonasymptotic oracle inequalities for the ${\ell }_1$ estimation error and prediction error measured by the SB divergence of the lasso penalized LPRE estimator. The proposed method is shown to be very efficient owing to the fact that almost each derived subproblem has a closed-form solution. Extensive simulation studies are conducted to evaluate the finite-sample performance of the proposal. Finally, a real data set is analyzed to illustrate the practical utility of our proposed method.