Holistic Generalized Linear Models

TL;DR

The paper addresses constrained model selection for generalized linear models by introducing Holistic Generalized Linear Models (HGLMs) that integrate sparsity and structural constraints via mixed-integer conic optimization. It extends conic optimization approaches to GLMs by reformulating common family-link log-likelihoods with exponential and second-order cones, enabling reliable maximum likelihood estimation under holistic constraints. The authors present the holiglm R package, providing a drop-in glm-like interface that translates constraints into conic problems solved through ROI, and demonstrate applications in fairness-constrained logistic regression and constrained model selection in log-binomial regression. The work offers a practical framework and software for interpretable, constraint-driven GLMs, with potential for broader adoption as conic optimization advances continue to emerge.

Abstract

Holistic linear regression extends the classical best subset selection problem by adding additional constraints designed to improve the model quality. These constraints include sparsity-inducing constraints, sign-coherence constraints and linear constraints. The $\textsf{R}$ package $\texttt{holiglm}$ provides functionality to model and fit holistic generalized linear models. By making use of state-of-the-art conic mixed-integer solvers, the package can reliably solve GLMs for Gaussian, binomial and Poisson responses with a multitude of holistic constraints. The high-level interface simplifies the constraint specification and can be used as a drop-in replacement for the $\texttt{stats::glm()}$ function.