A Note on Cross-Validation for Lasso Under Measurement Errors

Variants of the Lasso or (Formula presented.) -penalized regression have been proposed to accommodate for presence of measurement errors in the covariates. Theoretical guarantees of these estimates have been established for some oracle values of the regularization parameters which are not known in practice. Data-driven tuning such as cross-validation has not been studied when covariates contain measurement errors. We demonstrate that in the presence of error-in-covariates, even when using a Lasso-variant that adjusts for measurement error, application of naive leave-one-out cross-validation to select the tuning parameter can be problematic. We provide an example where such a practice leads to estimation inconsistency. We also prove that a simple correction to cross-validation procedure restores consistency. We also study the risk consistency of the two cross-validation procedures and offer guideline on the choice of cross-validation based on the measurement error distributions of the training and the prediction data. The theoretical findings are validated using simulated data. Supplementary materials for this article are available online.