What Functions Does XGBoost Learn?

TL;DR

This paper provides a rigorous theoretical characterization of the function class learned by XGBoost, connecting it to classical variation measures and establishing near-optimal convergence rates.

Contribution

It introduces a new infinite-dimensional function class and complexity measure that explain XGBoost's implicit regularization and theoretical properties.

Findings

XGBoost optimizers are equivalent to penalized regression over a new function class.

The function class relates to Hardy--Krause variation, linking to classical mathematical concepts.

The estimator over this class achieves near-minimax convergence rates.

Abstract

This paper establishes a rigorous theoretical foundation for the function class implicitly learned by XGBoost, bridging the gap between its empirical success and our theoretical understanding. We introduce an infinite-dimensional function class $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ that extends finite ensembles of bounded-depth regression trees, together with a complexity measure $V^{d, s}_{\infty-\text{XGB}}(\cdot)$ that generalizes the $L^1$ regularization penalty used in XGBoost. We show that every optimizer of the XGBoost objective is also an optimizer of an equivalent penalized regression problem over $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ with penalty $V^{d, s}_{\infty-\text{XGB}}(\cdot)$, providing an interpretation of XGBoost as implicitly targeting a broader function class. We also develop a smoothness-based interpretation of $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ and $V^{d, s}_{\infty-\text{XGB}}(\cdot)$ in terms of Hardy--Krause variation. We prove that the least squares estimator over $\{f \in \mathcal{F}^{d, s}_{\infty-\text{ST}}: V^{d, s}_{\infty-\text{XGB}}(f) \le V\}$ achieves a nearly minimax-optimal rate of convergence $n^{-2/3} (\log n)^{4(\min(s, d) - 1)/3}$, thereby avoiding the curse of dimensionality. Our results provide the first rigorous characterization of the function space underlying XGBoost, clarify its connection to classical notions of variation, and identify an important open problem: whether the XGBoost algorithm itself achieves minimax optimality over this class.