A study on tuning parameter selection for the high-dimensional lasso

TL;DR

This paper develops new information criteria for tuning parameter selection in high-dimensional lasso regression, addressing variance estimation challenges and demonstrating superior performance through extensive simulations.

Contribution

It introduces novel risk estimators for tuning parameter selection in high-dimensional lasso, improving upon existing methods by leveraging high-dimensional variance estimation techniques.

Findings

New estimators perform well across various simulation scenarios.

Existing information-theoretic approaches are less effective in high dimensions.

The proposed methods show theoretical justification and practical advantages.

Abstract

High-dimensional predictive models, those with more measurements than observations, require regularization to be well defined, perform well empirically, and possess theoretical guarantees. The amount of regularization, often determined by tuning parameters, is integral to achieving good performance. One can choose the tuning parameter in a variety of ways, such as through resampling methods or generalized information criteria. However, the theory supporting many regularized procedures relies on an estimate for the variance parameter, which is complicated in high dimensions. We develop a suite of information criteria for choosing the tuning parameter in lasso regression by leveraging the literature on high-dimensional variance estimation. We derive intuition showing that existing information-theoretic approaches work poorly in this setting. We compare our risk estimators to existing methods with an extensive simulation and derive some theoretical justification. We find that our new estimators perform well across a wide range of simulation conditions and evaluation criteria.