On the Peaking Phenomenon of the Lasso in Model Selection

I briefly report on some unexpected results that I obtained when optimizing the model parameters of the Lasso. In simulations with varying observations-to-variables ratio n=p, I typically observe a strong peak in the test error curve at the transition point n/p = 1. This peaking phenomenon is well-documented in scenarios that involve the inversion of the sample covariance matrix, and as I illustrate in this note, it is also the source of the peak for the Lasso. The key problem is the parametrization of the Lasso penalty (as e.g. in the current R package lars) and I present a solution in terms of a normalized Lasso parameter.