The Benefit of Group Sparsity in Group Inference with Debiased Scaled Group Lasso

• Activity Number: 87
• Type: Contributed
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Ritwik Mitra* and Cun-Hui Zhang
• Companies: Rutgers University and Rutgers University
• Keywords: Group Inference ; Asymptotic Normality ; Relaxed Projection ; Chi Square Test ; Scaled Group Lasso

Abstract

We study confidence regions and approximate chi-square tests for variable groups in high-dimensional linear regression. When the size of the group is small, low-dimensional projection estimators for individual coefficients can be directly used to construct efficient confidence regions and p-values for the group. However, the existing analyses of low-dimensional projection estimators do not directly carry through for chi- square-based inference of a large group of variables. We develop and analyze a scaled group Lasso and propose to de-bias the scaled group Lasso for efficient chi-square-based statistical inference of variable groups. We prove that the proposed methods capture the benefit of group sparsity under proper conditions, for statistical inference of the noise level and variable groups, large and small. Such benefit is especially strong when the group size is large. Oracle inequalities are provided for the scaled group Lasso in prediction and several estimation losses, and for the group Lasso as well in a weighted mixed loss.