A New Branch-and-Bound Pruning Framework for $\ell_0$-Regularized Problems

TL;DR

This paper introduces an efficient pruning framework for branch-and-bound algorithms tackling $\u2113_0$-regularized learning problems, significantly accelerating solution times in machine learning applications.

Contribution

It proposes a novel pruning test implementation that evaluates multiple regions simultaneously with minimal overhead, enhancing BnB efficiency for $\u2113_0$-regularized problems.

Findings

Pruning strategy reduces solving time by several orders of magnitude.

Method effectively handles typical machine-learning $\u2113_0$-regularized problems.

Numerical simulations validate the efficiency of the proposed approach.

Abstract

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.