Leave-one-out cross-validation is risk consistent for lasso

TL;DR

This paper proves that the lasso method remains risk consistent when the tuning parameter is selected through leave-one-out cross-validation, under certain design matrix conditions.

Contribution

It provides the first theoretical validation of risk consistency for lasso with cross-validated tuning parameter selection.

Findings

Lasso is risk consistent with cross-validation under specific conditions.

Theoretical justification for data-driven tuning parameter choice.

Addresses practical limitations of oracle tuning in lasso.

Abstract

The lasso procedure is ubiquitous in the statistical and signal processing literature, and as such, is the target of substantial theoretical and applied research. While much of this research focuses on the desirable properties that lasso possesses---predictive risk consistency, sign consistency, correct model selection---all of it has assumes that the tuning parameter is chosen in an oracle fashion. Yet, this is impossible in practice. Instead, data analysts must use the data twice, once to choose the tuning parameter and again to estimate the model. But only heuristics have ever justified such a procedure. To this end, we give the first definitive answer about the risk consistency of lasso when the smoothing parameter is chosen via cross-validation. We show that under some restrictions on the design matrix, the lasso estimator is still risk consistent with an empirically chosen tuning parameter.