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August 2019 Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions
Pierre Alquier, Vincent Cottet, Guillaume Lecué
Ann. Statist. 47(4): 2117-2144 (August 2019). DOI: 10.1214/18-AOS1742

Abstract

We obtain estimation error rates and sharp oracle inequalities for regularization procedures of the form \begin{equation*}\hat{f}\in\mathop{\operatorname{argmin}}_{f\in F}\Bigg(\frac{1}{N}\sum_{i=1}^{N}\ell_{f}(X_{i},Y_{i})+\lambda \Vert f\Vert \Bigg)\end{equation*} when $\Vert \cdot \Vert $ is any norm, $F$ is a convex class of functions and $\ell$ is a Lipschitz loss function satisfying a Bernstein condition over $F$. We explore both the bounded and sub-Gaussian stochastic frameworks for the distribution of the $f(X_{i})$’s, with no assumption on the distribution of the $Y_{i}$’s. The general results rely on two main objects: a complexity function and a sparsity equation, that depend on the specific setting in hand (loss $\ell$ and norm $\Vert \cdot \Vert $).

As a proof of concept, we obtain minimax rates of convergence in the following problems: (1) matrix completion with any Lipschitz loss function, including the hinge and logistic loss for the so-called 1-bit matrix completion instance of the problem, and quantile losses for the general case, which enables to estimate any quantile on the entries of the matrix; (2) logistic LASSO and variants such as the logistic SLOPE, and also shape constrained logistic regression; (3) kernel methods, where the loss is the hinge loss, and the regularization function is the RKHS norm.

Citation

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Pierre Alquier. Vincent Cottet. Guillaume Lecué. "Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions." Ann. Statist. 47 (4) 2117 - 2144, August 2019. https://doi.org/10.1214/18-AOS1742

Information

Received: 1 January 2018; Revised: 1 June 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082281
MathSciNet: MR3953446
Digital Object Identifier: 10.1214/18-AOS1742

Subjects:
Primary: 60K35, 62G08
Secondary: 62C20, 62G05, 62G20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • August 2019
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