Towards the interpretation of time-varying regularization parameters in streaming penalized regression models

TL;DR

This paper investigates how various statistical properties of streaming data influence the regularization parameter in l1-penalized models, emphasizing the importance of understanding these factors for better model interpretation and tuning.

Contribution

It provides a comprehensive empirical analysis of factors affecting the regularization parameter in streaming penalized regression, highlighting the complexity beyond sparsity changes.

Findings

Changes in regularization parameter can be driven by sparsity, noise, or model misspecification.

Empirical evidence shows data properties significantly influence parameter tuning.

Applications demonstrate relevance in finance and neuroimaging contexts.

Abstract

High-dimensional, streaming datasets are ubiquitous in modern applications. Examples range from finance and e-commerce to the study of biomedical and neuroimaging data. As a result, many novel algorithms have been proposed to address challenges posed by such datasets. In this work, we focus on the use of $\ell_1$ regularized linear models in the context of (possibly non-stationary) streaming data Recently, it has been noted that the choice of the regularization parameter is fundamental in such models and several methods have been proposed which iteratively tune such a parameter in a~time-varying manner; thereby allowing the underlying sparsity of estimated models to vary. Moreover, in many applications, inference on the regularization parameter may itself be of interest, as such a parameter is related to the underlying \textit{sparsity} of the model. However, in this work, we highlight and provide extensive empirical evidence regarding how various (often unrelated) statistical properties in the data can lead to changes in the regularization parameter. In particular, through various synthetic experiments, we demonstrate that changes in the regularization parameter may be driven by changes in the true underlying sparsity, signal-to-noise ratio or even model misspecification. The purpose of this letter is, therefore, to highlight and catalog various statistical properties which induce changes in the associated regularization parameter. We conclude by presenting two applications: one relating to financial data and another to neuroimaging data, where the aforementioned discussion is relevant.