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

• URL: http://arxiv.org/abs/2001.05401v3
• Date: Thu, 3 Sep 2020 16:58:48 GMT
• Title: Support recovery and sup-norm convergence rates for sparse pivotal estimation
• Authors: Mathurin Massias and Quentin Bertrand and Alexandre Gramfort and Joseph Salmon
• Abstract summary: In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. We show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators.
• Score: 79.13844065776928
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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