Sharpness of error bounds for cross-validated Lasso

Ascertain whether the established error bounds for the full-sample Lasso estimator with V-fold cross-validation selection of the penalty parameter, namely ||β̂^{FVCV} − β||_2 = O_P(√(s (log p)^2 / n)) and ||β̂^{FVCV} − β||_1 = O_P(√(s^2 (log p)^5 / n)), are sharp.

Background

The paper discusses V-fold cross-validation for selecting the Lasso penalty parameter, noting that while average estimators admit oracle inequalities, analyzing the full-sample estimator is more intricate. Using a relationship between full-sample and subsample solutions plus degrees-of-freedom arguments, current theory yields bounds for the full-sample Lasso that include additional log p factors compared to the standard high-dimensional rates.

The authors highlight that these bounds may be suboptimal relative to bootstrap- or deviation-based tuning rules, and explicitly state that it is uncertain whether the cross-validation-based bounds are tight, leaving open the question of their sharpness.