{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/learning-sparse-classifiers-continuous-and","title":"Learning Sparse Classifiers: Continuous and Mixed Integer Optimization Perspectives","arxiv_id":"2001.06471","date":"2020-01-17","proceeding":null,"authors":["Antoine Dedieu","Hussein Hazimeh","Rahul Mazumder"],"abstract":"We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to bridge this gap in computation times by developing new MIP-based algorithms for $\\ell_0$-regularized classification. We propose two classes of scalable algorithms: an exact algorithm that can handle $p\\approx 50,000$ features in a few minutes, and approximate algorithms that can address instances with $p\\approx 10^6$ in times comparable to the fast $\\ell_1$-based algorithms. Our exact algorithm is based on the novel idea of \\textsl{integrality generation}, which solves the original problem (with $p$ binary variables) via a sequence of mixed integer programs that involve a small number of binary variables. Our approximate algorithms are based on coordinate descent and local combinatorial search. In addition, we present new estimation error bounds for a class of $\\ell_0$-regularized estimators. Experiments on real and synthetic data demonstrate that our approach leads to models with considerably improved statistical performance (especially, variable selection) when compared to competing methods.","url_abs":"https://arxiv.org/abs/2001.06471v2","url_pdf":"https://arxiv.org/pdf/2001.06471v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"learning-sparse-classifiers-continuous-and","repo_url":"https://github.com/hazimehh/L0Learn","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[{"task_slug":"variable-selection","task_name":"Variable Selection"},{"task_slug":"global-optimization","task_name":"global-optimization"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2001.06471","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}