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The lasso and related sparsity inducing algorithms have been the target ofsubstantial theoretical and applied research. Correspondingly, many results areknown about their behavior for a fixed or optimally chosen tuning parameterspecified up to unknown constants. In practice, however, this oracle tuningparameter is inaccessible so one must use the data to select one. Commonstatistical practice is to use a variant of cross-validation for this task.However, little is known about the theoretical properties of the resultingpredictions with such data-dependent methods. We consider the high-dimensionalsetting with random design wherein the number of predictors $p$ grows with thenumber of observations $n$. Under typical assumptions on the data generatingprocess, similar to those in the literature, we recover oracle rates up to alog factor when choosing the tuning parameter with cross-validation. Underweaker conditions, when the true model is not necessarily linear, we show thatthe lasso remains risk consistent relative to its linear oracle. We alsogeneralize these results to the group lasso and square-root lasso andinvestigate the predictive and model selection performance of cross-validationvia simulation.

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