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arXiv · 2001.05401

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

Mathurin Massias·Quentin Bertrand·Alexandre Gramfort·Joseph Salmon
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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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Mathurin Massias, Quentin Bertrand, Alexandre Gramfort, Joseph Salmon. 2020-01-15. Support recovery and sup-norm convergence rates for sparse pivotal estimation. https://arxiv.org/abs/2001.05401

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