# A New Branch-and-Bound Pruning Framework for ℓ0-Regularized Problems

arXiv: https://arxiv.org/abs/2406.03504 · HTML: https://mlsanity.com/paper/2406.03504/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization](https://mlsanity.com/paper/2004.06152/) (2020)
- [Sparse High-Dimensional Regression: Exact Scalable Algorithms and Phase Transitions](https://mlsanity.com/paper/1709.10029/) (2017)
- [Learning Sparse Classifiers: Continuous and Mixed Integer Optimization Perspectives](https://mlsanity.com/paper/2001.06471/) (2020)
- [A unified approach to mixed-integer optimization problems with logical constraints](https://mlsanity.com/paper/1907.02109/) (2019)
- [Cardinality Minimization, Constraints, and Regularization: A Survey](https://mlsanity.com/paper/2106.09606/) (2021)
- [Bayesian Regularization: From Tikhonov to Horseshoe](https://mlsanity.com/paper/1902.06269/) (2019)
- [Compressed Sensing: A Discrete Optimization Approach](https://mlsanity.com/paper/2306.04647/) (2023)

## Built on by (most cited first, 5 of 5)

- [On the Efficiency of ERM in Feature Learning](https://mlsanity.com/paper/2411.12029/) (2024)
- [Scalable First-order Method for Certifying Optimal k-Sparse GLMs](https://mlsanity.com/paper/2502.09502/) (2025)
- [A Generic Branch-and-Bound Algorithm for ℓ0-Penalized Problems with Supplementary Material](https://mlsanity.com/paper/2506.03974/) (2025)
- [El0ps: An Exact L0-regularized Problems Solver](https://mlsanity.com/paper/2506.06373/) (2025)
- [From Sequential Nodes to GPU Batches: Parallel Branch and Bound for Optimal k-Sparse GLMs](https://mlsanity.com/paper/2605.22188/) (2026)

## Abstract (arXiv)

We consider the resolution of learning problems involving ℓ0-regularization via Branch-and-Bound (BnB) algorithms. These methods explore regions of the feasible space of the problem and check whether they do not contain solutions through "pruning tests". In standard implementations, evaluating a pruning test requires to solve a convex optimization problem, which may result in computational bottlenecks. In this paper, we present an alternative to implement pruning tests for some generic family of ℓ0-regularized problems. Our proposed procedure allows the simultaneous assessment of several regions and can be embedded in standard BnB implementations with a negligible computational overhead. We show through numerical simulations that our pruning strategy can improve the solving time of BnB procedures by several orders of magnitude for typical problems encountered in machine-learning applications.