Boost Like a (Var)Pro: Trust-Region Gradient Boosting via Variable Projection
arXiv:2603.23658v1 Announce Type: new Abstract: Gradient boosting, a method of building additive ensembles from weak learners, has established itself as a practical and theoretically-motivated approach to approximate functions, especially using decision tree weak learners. Comparable methods for smooth parametric learners, such as neural networks, remain less developed in both training methodology and theory. To this end, we introduce \texttt{VPBoost} ({\bf V}ariable {\bf P}rojection {\bf Boost}ing), a gradient boosting algorithm for separable smooth approximators, i.e., models with a smooth nonlinear featurizer followed by a final linear mapping. \texttt{VPBoost} fuses variable projection, a training paradigm for separable models that enforces optimality of the linear weights, with a second-order weak learning strategy. The combination of second-order boosting, separable models, and variable projection give rise to a closed-form solution for the optimal linear weights and a natural i
arXiv:2603.23658v1 Announce Type: new Abstract: Gradient boosting, a method of building additive ensembles from weak learners, has established itself as a practical and theoretically-motivated approach to approximate functions, especially using decision tree weak learners. Comparable methods for smooth parametric learners, such as neural networks, remain less developed in both training methodology and theory. To this end, we introduce \texttt{VPBoost} ({\bf V}ariable {\bf P}rojection {\bf Boost}ing), a gradient boosting algorithm for separable smooth approximators, i.e., models with a smooth nonlinear featurizer followed by a final linear mapping. \texttt{VPBoost} fuses variable projection, a training paradigm for separable models that enforces optimality of the linear weights, with a second-order weak learning strategy. The combination of second-order boosting, separable models, and variable projection give rise to a closed-form solution for the optimal linear weights and a natural interpretation of \VPBoost as a functional trust-region method. We thereby leverage trust-region theory to prove \VPBoost converges to a stationary point under mild geometric conditions and, under stronger assumptions, achieves a superlinear convergence rate. Comprehensive numerical experiments on synthetic data, image recognition, and scientific machine learning benchmarks demonstrate that \VPBoost learns an ensemble with improved evaluation metrics in comparison to gradient-descent-based boosting and attains competitive performance relative to an industry-standard decision tree boosting algorithm.
Executive Summary
The authors propose a novel gradient boosting algorithm, VPBoost, for separable smooth approximators. By combining variable projection with second-order weak learning, VPBoost achieves a closed-form solution for optimal linear weights and exhibits superlinear convergence rate under mild geometric conditions. Comprehensive numerical experiments demonstrate improved evaluation metrics compared to gradient-descent-based boosting and competitive performance relative to industry-standard decision tree boosting. The proposed method leverages trust-region theory to ensure convergence to a stationary point, providing a theoretically-motivated approach to function approximation. The results have significant implications for practical applications, particularly in machine learning and scientific computing, where accurate and efficient function approximation is critical.
Key Points
- ▸ VPBoost combines variable projection with second-order weak learning to achieve a closed-form solution for optimal linear weights.
- ▸ The proposed method exhibits superlinear convergence rate under mild geometric conditions.
- ▸ Comprehensive numerical experiments demonstrate improved evaluation metrics compared to gradient-descent-based boosting.
Merits
Strength in Theoretical Foundation
VPBoost's convergence to a stationary point and superlinear convergence rate are theoretically justified using trust-region theory, providing a robust foundation for the method.
Improved Practical Performance
Numerical experiments demonstrate that VPBoost achieves improved evaluation metrics compared to gradient-descent-based boosting, making it a promising approach for practical applications.
Demerits
Limited Model Scope
VPBoost is specifically designed for separable smooth approximators, limiting its applicability to other types of models.
Computational Complexity
The proposed method may require additional computational resources due to the variable projection step, which could be a limitation in certain applications.
Expert Commentary
The proposed VPBoost algorithm represents a significant advancement in the field of gradient boosting, leveraging novel theoretical insights to achieve improved practical performance. The authors' use of trust-region theory to ensure convergence to a stationary point is a particularly noteworthy innovation, demonstrating the power of interdisciplinary research in solving complex problems. While VPBoost's limitations, such as its focus on separable smooth approximators, are acknowledged, the method's potential to improve function approximation accuracy and efficiency makes it an exciting development in the field of machine learning.
Recommendations
- ✓ Future research should focus on extending VPBoost to other types of models and exploring its applications in various machine learning and scientific computing domains.
- ✓ The development of more efficient computational methods for variable projection could further enhance the practical performance of VPBoost.
Sources
Original: arXiv - cs.LG
