Boost Like a (Var)Pro: Trust-Region Gradient Boosting via Variable Projection

TL;DR

This paper introduces VPBoost, a novel gradient boosting algorithm for smooth models that combines variable projection and trust-region methods, leading to improved convergence and performance in various machine learning tasks.

Contribution

VPBoost is the first gradient boosting method for separable smooth models that integrates variable projection with trust-region theory, providing convergence guarantees and competitive results.

Findings

VPBoost converges to a stationary point under mild conditions.

VPBoost achieves superlinear convergence under stronger assumptions.

VPBoost outperforms gradient descent boosting in experiments.

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.