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I-LAMM for sparse learning: simultaneous control of algorithmic complexity and statistical error. (English) Zbl 1392.62215

This paper proposes a general computational framework for solving nonconvex optimisation problems such as the penalized M-estimator \(\mathrm{argmin}_{\beta\in{\mathbb R}^d}\{ {\mathcal L}(\beta) + {\mathcal R}_{\lambda}(\beta)\}\), where \({\mathcal L}(\beta)\) is a smooth loss function, \({\mathcal R}_{\lambda}(\beta)\) is a sparsity-inducing penalty with a regularization parameter \(\lambda\). The proposed strategy enables the simultaneous control of the algorithmic complexity and the statistical error when fitting high-dimensional models appearing in various problems including low rank matrix completion problems, high-dimensional graphical models and quantile regression.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62C20 Minimax procedures in statistical decision theory
62G08 Nonparametric regression and quantile regression
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