Sparsity considerations for dependent observations

TL;DR

This paper extends the theoretical understanding of L1-penalized estimators, like LASSO, to dependent data scenarios, providing a foundation for their use in non-iid stochastic optimization problems.

Contribution

It introduces a general L1-penalized estimator framework for dependent observations and analyzes its statistical properties beyond the iid assumption.

Findings

Provides theoretical guarantees for LASSO under dependence

Defines a general estimator for stochastic optimization with dependence

Extends LASSO analysis to non-iid data scenarios

Abstract

The aim of this paper is to provide a comprehensive introduction for the study of L1-penalized estimators in the context of dependent observations. We define a general $\ell_{1}$-penalized estimator for solving problems of stochastic optimization. This estimator turns out to be the LASSO in the regression estimation setting. Powerful theoretical guarantees on the statistical performances of the LASSO were provided in recent papers, however, they usually only deal with the iid case. Here, we study our estimator under various dependence assumptions.