Regularization and the small-ball method I: Sparse recovery

We obtain bounds on estimation error rates for regularization procedures of the form $\begin{equation*}\hat{f}\in\mathop{\operatorname{argmin}}_{f\in F}(\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-f(X_{i}))^{2}+\lambda \Psi(f))\end{equation*}$ when $\Psi$ is a norm and $F$ is convex. Our approach gives a common framework that may be used in the analysis of learning problems and regularization problems alike. In particular, it sheds some light on the role various notions of sparsity have in regularization and on their connection with the size of subdifferentials of $\Psi$ in a neighborhood of the true minimizer. As “proof of concept” we extend the known estimates for the LASSO, SLOPE and trace norm regularization.We obtain bounds on estimation error rates for regularization procedures of the form \begin{equation*}\hat{f}\in\mathop{\operatorname{argmin}}_{f\in F}(\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-f(X_{i}))^{2}+\lambda \Psi(f))\end{equation*} when $\Psi$ is a norm and $F$ is convex. Our approach gives a common framework that may be used in the analysis of learning problems and regularization problems alike. In particular, it sheds some light on the role various notions of sparsity have in regularization and on their connection with the size of subdifferentials of $\Psi$ in a neighborhood of the true minimizer. As “proof of concept” we extend the known estimates for the LASSO, SLOPE and trace norm regularization.

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笔记与分组

分类依据与纠错

Regularization and the small-ball method I: Sparse recovery

代码、数据与补充材料

正则化与小球方法之一：稀疏恢复

原文摘要

同方向的基础文献

同方向的基础文献