A New Branch-and-Bound Pruning Framework for $\ell_0$-Regularized Problems (2406.03504v1)

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.

• The paper introduces an alternative pruning strategy using dual functions to bypass costly convex optimization tests in ℓ0-regularized problems.
• The paper establishes a robust theoretical framework with broad applicability and provides a complexity analysis that demonstrates its efficiency.
• Empirical results show the method reduces pruning test time by several orders of magnitude in both synthetic and real-world machine learning scenarios.

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

This paper addresses the resolution of learning problems involving $\ell_0$-regularization through Branch-and-Bound (BnB) algorithms. It proposes a novel pruning framework that aims to improve the computational efficiency of these algorithms by offering an alternative method to perform pruning tests.

Summary of Key Points

The primary challenge in BnB algorithms for $\ell_0$-regularized problems is the computational overhead involved in solving convex optimization problems at each pruning step. Traditional pruning tests necessitate solving these problems to evaluate whether a region of the feasible space contains any solutions. The paper introduces a more computationally efficient method to implement these pruning tests by leveraging Fenchel-Rockafellar duality.

Contributions

1. Alternative Pruning Strategy:
   • The paper proposes a method to evaluate pruning tests that does not rely on solving a convex optimization problem. This is achieved by using dual functions associated with BnB subproblems.
   • This new strategy allows for the simultaneous assessment of multiple regions at a negligible computational overhead, significantly reducing the evaluating time for pruning tests.
2. Theoretical Foundations:
   • A general framework is established under broad hypotheses, making the method applicable to a wide range of $\ell_0$-regularized problems in machine learning.
   • Theoretically, the paper demonstrates that the proposed pruning method can be embedded in standard BnB implementations and provides a complexity analysis indicating its efficiency.
3. Empirical Validation:
   • The effectiveness of the proposed method is validated through numerical simulations. These simulations show that the new pruning strategy can reduce the solving time of BnB procedures by several orders of magnitude for typical machine learning problems.

Technical Insights

• The approach exploits the properties of the $\ell_0$-norm to promote sparsity, and the regularization function $h$ that enforces additional application-specific properties.
• The use of dual functions in evaluating the pruning tests is a critical innovation. By constructing lower bounds through dual functions, this method avoids the computational bottlenecks associated with traditional convex optimization-based pruning tests.
• The complexity of the proposed pruning method scales with $O(m+n)$, where $m$ is the number of data points and $n$ is the number of features, which is a substantial improvement over the traditional $O(\ddimn \kappa)$ complexity.

Numerical Results

Numerical experiments, both on synthetic data and real-world datasets, underscore the practical benefits of the new pruning framework:

• Synthetic Data: The new method consistently solved a larger percentage of instances within a given time budget compared to other state-of-the-art solvers.
• Real-World Data: Across multiple datasets related to various application domains, the proposed method showed significant time savings, sometimes achieving up to four orders of magnitude improvement over existing methods.

Implications and Future Directions

The methods presented in this paper suggest several theoretical and practical implications:

• Scalability: The reduced computational overhead implies that more complex and larger-scale $\ell_0$-regularized problems can be tackled more efficiently.
• Versatility: The broad applicability of the proposed method makes it suitable for a wide variety of machine learning tasks, including feature selection, compressive sensing, and neural network pruning.
• Future Developments: The framework opens avenues for further refinements. Future work could explore adaptive strategies for selecting dual points or integrate more sophisticated heuristics to further enhance the efficiency of BnB algorithms.

Conclusion

The paper presents a significant contribution to the optimization community, particularly in the field of BnB algorithms for $\ell_0$-regularized problems. By proposing an efficient alternative to evaluate pruning tests, it paves the way for more scalable and faster algorithms. This advancement is especially pertinent for large-scale machine learning applications, where solving time is a critical factor. The comprehensive theoretical background, coupled with robust empirical results, showcases the potential of this novel pruning framework in enhancing optimization procedures in varied applications.