Support recovery and sup-norm convergence rates for sparse pivotal estimation

Mathurin Massias (1) , Quentin Bertrand (1) , Alexandre Gramfort (1) , Joseph Salmon (2)

1 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
2 IMAG - Institut Montpelliérain Alexander Grothendieck

In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal es-timator 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 estima-tors. 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.

Domaines: Machine Learning [stat.ML]; Théorie [stat.TH]

Mathurin Massias, Quentin Bertrand, Alexandre Gramfort, Joseph Salmon. Support recovery and sup-norm convergence rates for sparse pivotal estimation. AISTATS2020, Aug 2020, Virtual, Italy. ⟨hal-02444978⟩