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

Convex estimators such as the Lasso, the matrix Lasso and the group Lasso have been studied extensively in the last two decades, demonstrating great success in both theory and practice. Two quantities are introduced, the noise barrier and the large scale bias, that provides insights on the performance of these convex regularized estimators. It is now well understood that the Lasso achieves fast prediction rates, provided that the correlations of the design satisfy some Restricted Eigenvalue or Compatibility condition, and provided that the tuning parameter is large enough. Using the two quantities introduced in the paper, we show that the compatibility condition on the design matrix is actually unavoidable to achieve fast prediction rates with the Lasso. The Lasso must incur a loss due to the correlations of the design matrix, measured in terms of the compatibility constant. This results holds for any design matrix, any active subset of covariates, and any tuning parameter. It is now well known that the Lasso enjoys a dimension reduction property: the prediction error is of order $\lambda\sqrt k$ where $k$ is the sparsity; even if the ambient dimension $p$ is much larger than $k$. Such results require that the tuning parameters is greater than some universal threshold. We characterize sharp phase transitions for the tuning parameter of the Lasso around a critical threshold dependent on $k$. If $\lambda$ is equal or larger than this critical threshold, the Lasso is minimax over $k$-sparse target vectors. If $\lambda$ is equal 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 target vector has fewer than $k$ nonzero coefficients. Remarkably, the lower bounds obtained in the paper also apply to random, data-driven tuning parameters. The results extend to convex penalties beyond the Lasso.