A New Branch-and-Bound Pruning Framework for $\ell_0$-Regularized Problems
This work addresses efficiency issues for researchers and practitioners using ℓ₀-regularization in machine learning, though it is incremental as it builds on existing BnB methods.
The paper tackles the computational bottleneck in Branch-and-Bound algorithms for ℓ₀-regularized problems by introducing a new pruning framework that simultaneously assesses multiple regions with negligible overhead, resulting in solving time improvements by several orders of magnitude in machine-learning applications.
We consider the resolution of learning problems involving $\ell_0$-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 $\ell_0$-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.