Risk-consistency of cross-validation with lasso-type procedures

The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified up to unknown constants. In practice, however, this oracle tuning parameter is inaccessible, so one must instead use the data to choose a tuning parameter. Common statistical practice is to use one of a few variants of cross-validation for this task. However, very little is known about the theoretical properties of the resulting predictions using data-dependent methods. We consider the high-dimensional setting with random design wherein the number of predictors $p$ grows with the number of observations $n$. We show that the lasso remains risk consistent relative to its linear oracle even when the tuning parameter is chosen via cross-validation and the true model is not necessarily linear. We generalize these results to the group lasso and $\sqrt{\mbox{lasso}}$ and compare the performance of cross-validation to other tuning parameter selection methods via simulations.