Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives

TL;DR

This paper introduces a novel optimization-based framework for grouped variable selection that achieves scalable exact solutions and demonstrates superior statistical performance over existing methods.

Contribution

The paper develops a new algorithmic framework combining approximate and exact methods for $ ext{l}_0$-regularized grouped variable selection, enabling solutions for large-scale problems previously infeasible.

Findings

Exact algorithms solve problems with 5 million features in minutes to hours.

Proposed estimators outperform popular group-sparse methods statistically.

Open-source implementation is provided for practical use.

Abstract

We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the $\ell_0$-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with $5 \times 10^6$ features and $10^3$ observations in minutes to hours -- over $1000$ times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the $\ell_0$-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open-source implementation of our proposed framework.