  A fundamental problem in the high-dimensional regression is to understand the
tradeoff between type I and type II errors or, equivalently, false discovery
rate (FDR) and power in variable selection. To address this important problem,
we offer the first complete tradeoff diagram that distinguishes all pairs of
FDR and power that can be asymptotically realized by the Lasso with some choice
of its penalty parameter from the remaining pairs, in a regime of linear
sparsity under random designs. The tradeoff between the FDR and power
characterized by our diagram holds no matter how strong the signals are. In
particular, our results improve on the earlier Lasso tradeoff diagram of
arXiv:1511.01957 by recognizing two simple but fundamental constraints on the
pairs of FDR and power. The improvement is more substantial when the regression
problem is above the Donoho--Tanner phase transition. Finally, we present
extensive simulation studies to confirm the sharpness of the complete Lasso
tradeoff diagram.
