FALSE DISCOVERIES OCCUR EARLY ON THE LASSO PATH By

In regression settings where explanatory variables have very low correlations and where thereare relatively few effects each of large magnitude, it is commonly believed that the Lasso shall beable to find the important variables with few errors—if any. In contrast, this paper shows thatthis is not the case even when the design variables are stochastically independent. In a regimeof linear sparsity, we demonstrate that true features and null features are always interspersedon the Lasso path, and that this phenomenon occurs no matter how strong the effect sizes are.We derive a sharp asymptotic trade-off between false and true positive rates or, equivalently,between measures of type I and type II errors along the Lasso path. This trade-off states thatif we ever want to achieve a type II error (false negative rate) under a given threshold, thenanywhere on the Lasso path the type I error (false positive rate) will need to exceed a giventhreshold so that we can never have both errors at a low level at the same time. Our analysisuses tools from approximate message passing (AMP) theory as well as novel elements to dealwith a possibly adaptive selection of the Lasso regularizing parameter.