High-dimensional stochastic optimization with the generalized Dantzig estimator

TL;DR

This paper introduces a generalized Dantzig selector for high-dimensional stochastic optimization, demonstrating its theoretical properties and applying it to linear regression with Huber loss, achieving convergence and sign consistency.

Contribution

It extends the Dantzig selector to a generalized form, providing theoretical guarantees and applying it to high-dimensional regression with robust loss functions.

Findings

Sparsity oracle inequalities for the generalized Dantzig selector

Sup-norm convergence rate in high-dimensional regression

Sign concentration property under mutual coherence

Abstract

We propose a generalized version of the Dantzig selector. We show that it satisfies sparsity oracle inequalities in prediction and estimation. We consider then the particular case of high-dimensional linear regression model selection with the Huber loss function. In this case we derive the sup-norm convergence rate and the sign concentration property of the Dantzig estimators under a mutual coherence assumption on the dictionary.