# Optimal penalty selection for lasso

Are there any analytical results or experimental papers regarding the optimal choice of the coefficient of the $\ell_1$ penalty term. By optimal, I mean a parameter that maximizes the probability of selecting the best model, or that minimizes the expected loss. I am asking because often it is impractical to choose the parameter by cross-validation or bootstrap, either because of a large number of instances of the problem, or because of the size of the problem at hand. The only positive result I am aware of is Candes and Plan, Near-ideal model selection by $\ell_1$ minimization.

Checkout Theorem 5.1 of this Bickel et al.. A statistically optimal choice in terms of the error $\|y-\hat{y}(\lambda)\|_2^2$ is $\lambda = A \sigma_{\text{noise}} \sqrt{\dfrac{\log p}{n}}$ (with high probability), for a constant $A > 2\sqrt{2}$.