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Convex estimators such as the Lasso, the matrix Lasso and the group Lassohave been studied extensively in the last two decades, demonstrating greatsuccess in both theory and practice. Two quantities are introduced, the noisebarrier and the large scale bias, that provides insights on the performance ofthese convex regularized estimators. It is now well understood that the Lassoachieves fast prediction rates, provided that the correlations of the designsatisfy some Restricted Eigenvalue or Compatibility condition, and providedthat the tuning parameter is large enough. Using the two quantities introducedin the paper, we show that the compatibility condition on the design matrix isactually unavoidable to achieve fast prediction rates with the Lasso. The Lassomust incur a loss due to the correlations of the design matrix, measured interms of the compatibility constant. This results holds for any design matrix,any active subset of covariates, and any tuning parameter. It is now well knownthat the Lasso enjoys a dimension reduction property: the prediction error isof order $\lambda\sqrt k$ where $k$ is the sparsity; even if the ambientdimension $p$ is much larger than $k$. Such results require that the tuningparameters is greater than some universal threshold. We characterize sharpphase transitions for the tuning parameter of the Lasso around a criticalthreshold dependent on $k$. If $\lambda$ is equal or larger than this criticalthreshold, the Lasso is minimax over $k$-sparse target vectors. If $\lambda$ isequal or smaller than critical threshold, the Lasso incurs a loss of order$\sigma\sqrt k$ --which corresponds to a model of size $k$-- even if the targetvector has fewer than $k$ nonzero coefficients. Remarkably, the lower boundsobtained in the paper also apply to random, data-driven tuning parameters. Theresults extend to convex penalties beyond the Lasso.

The noise barrier and the large signal bias of the Lasso and other convex estimators