  In spite of the wealth of literature on the theoretical properties of the
Lasso, there is very little known when the value of the tuning parameter is
chosen using the data, even though this is what actually happens in practice.
We give a general upper bound on the prediction error of Lasso when the tuning
parameter is chosen using a variant of 2-fold cross-validation. No special
assumption is made about the structure of the design matrix, and the tuning
parameter is allowed to be optimized over an arbitrary data-dependent set of
values. The proof is based on a general principle that may extend to other
kinds of cross-validation as well as to other penalized regression methods.
Based on this result, we propose a new estimate for error variance in high
dimensional regression and prove that it has good properties under minimal
assumptions.
