Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives

• URL: http://arxiv.org/abs/2104.07084v1
• Date: Wed, 14 Apr 2021 19:21:59 GMT
• Title: Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives
• Authors: Hussein Hazimeh, Rahul Mazumder, Peter Radchenko
• Abstract summary: We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. Our methodology covers both high-dimensional linear regression and non- additive modeling with sparse programming. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer (MIP) problem to certified optimality.
• Score: 9.593208961737572
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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