On the Use of Minimum Penalties in Statistical Learning

TL;DR

This paper introduces the MinPEN framework for joint estimation of regression coefficients and relationships between outcome variables in multivariate models, using a novel minimum function penalty, with theoretical guarantees and practical applications.

Contribution

The paper proposes the MinPEN framework that simultaneously estimates relationships and regression coefficients using a new penalty, extending to various exponential family models with theoretical and empirical validation.

Findings

High-dimensional convergence rates established

Model selection consistency demonstrated

Effective in binomial response models

Abstract

Modern multivariate machine learning and statistical methodologies estimate parameters of interest while leveraging prior knowledge of the association between outcome variables. The methods that do allow for estimation of relationships do so typically through an error covariance matrix in multivariate regression which does not scale to other types of models. In this article we proposed the MinPEN framework to simultaneously estimate regression coefficients associated with the multivariate regression model and the relationships between outcome variables using mild assumptions. The MinPen framework utilizes a novel penalty based on the minimum function to exploit detected relationships between responses. An iterative algorithm that generalizes current state of the art methods is proposed as a solution to the non-convex optimization that is required to obtain estimates. Theoretical results such as high dimensional convergence rates, model selection consistency, and a framework for post selection inference are provided. We extend the proposed MinPen framework to other exponential family loss functions, with a specific focus on multiple binomial responses. Tuning parameter selection is also addressed. Finally, simulations and two data examples are presented to show the finite sample properties of this framework.