Estimator Augmentation with Applications in High-Dimensional Group Inference

TL;DR

This paper introduces a novel estimator augmentation method for the block Lasso in high-dimensional data, enabling efficient inference on parameter groups through a closed-form joint distribution and Monte Carlo sampling.

Contribution

It develops a new estimator augmentation technique for the block Lasso, providing a closed-form joint distribution and improving inference efficiency in high-dimensional group analysis.

Findings

Importance sampling with estimator augmentation outperforms bootstrap in tail probability estimation.

The method offers new geometric insights into the block Lasso solution space.

Numerical results demonstrate significant efficiency gains in significance testing.

Abstract

To make inference about a group of parameters on high-dimensional data, we develop the method of estimator augmentation for the block Lasso, which is defined via the block norm. By augmenting a block Lasso estimator $\hat{\beta}$ with the subgradient $S$ of the block norm evaluated at $\hat{\beta}$, we derive a closed-form density for the joint distribution of $(\hat{\beta},S)$ under a high-dimensional setting. This allows us to draw from an estimated sampling distribution of $\hat{\beta}$, or more generally any function of $(\hat{\beta},S)$, by Monte Carlo algorithms. We demonstrate the application of estimator augmentation in group inference with the group Lasso and a de-biased group Lasso constructed as a function of $(\hat{\beta},S)$. Our numerical results show that importance sampling via estimator augmentation can be orders of magnitude more efficient than parametric bootstrap in estimating tail probabilities for significance tests. This work also brings new insights into the geometry of the sample space and the solution uniqueness of the block Lasso.