  In high dimensional sparse regression, pivotal estimators are estimators for
which the optimal regularization parameter is independent of the noise level.
The canonical pivotal estimator is the square-root Lasso, formulated along with
its derivatives as a "non-smooth + non-smooth" optimization problem. Modern
techniques to solve these include smoothing the datafitting term, to benefit
from fast efficient proximal algorithms. In this work we show minimax sup-norm
convergence rates for non smoothed and smoothed, single task and multitask
square-root Lasso-type estimators. Thanks to our theoretical analysis, we
provide some guidelines on how to set the smoothing hyperparameter, and
illustrate on synthetic data the interest of such guidelines.
