Optimization by gradient boosting

TL;DR

This paper provides a comprehensive analysis of gradient boosting algorithms, demonstrating their convergence and consistency within a functional optimization framework, emphasizing the role of strong convexity and regularization.

Contribution

It introduces a general framework for analyzing gradient boosting, proving convergence and consistency without early stopping, and highlights the importance of strong convexity and regularization.

Findings

Proves convergence of gradient boosting algorithms as iterations increase.

Establishes conditions for statistical consistency of boosting predictors.

Highlights the role of strong convexity and regularization in boosting performance.

Abstract

Gradient boosting is a state-of-the-art prediction technique that sequentially produces a model in the form of linear combinations of simple predictors---typically decision trees---by solving an infinite-dimensional convex optimization problem. We provide in the present paper a thorough analysis of two widespread versions of gradient boosting, and introduce a general framework for studying these algorithms from the point of view of functional optimization. We prove their convergence as the number of iterations tends to infinity and highlight the importance of having a strongly convex risk functional to minimize. We also present a reasonable statistical context ensuring consistency properties of the boosting predictors as the sample size grows. In our approach, the optimization procedures are run forever (that is, without resorting to an early stopping strategy), and statistical regularization is basically achieved via an appropriate $L^2$ penalization of the loss and strong convexity arguments.