Analyzing Tree Architectures in Ensembles via Neural Tangent Kernel

TL;DR

This paper analyzes the Neural Tangent Kernel induced by soft tree ensembles, revealing that only the number of leaves at each depth influences training and generalization, regardless of the specific tree architecture.

Contribution

It introduces a theoretical framework for understanding soft tree ensembles via NTK, showing architecture invariance based on leaf counts at each depth.

Findings

Only leaf counts at each depth affect NTK and performance.

Asymmetric trees like decision lists do not degenerate with depth.

Binary trees' NTK degenerates as they become deeper.

Abstract

A soft tree is an actively studied variant of a decision tree that updates splitting rules using the gradient method. Although soft trees can take various architectures, their impact is not theoretically well known. In this paper, we formulate and analyze the Neural Tangent Kernel (NTK) induced by soft tree ensembles for arbitrary tree architectures. This kernel leads to the remarkable finding that only the number of leaves at each depth is relevant for the tree architecture in ensemble learning with an infinite number of trees. In other words, if the number of leaves at each depth is fixed, the training behavior in function space and the generalization performance are exactly the same across different tree architectures, even if they are not isomorphic. We also show that the NTK of asymmetric trees like decision lists does not degenerate when they get infinitely deep. This is in contrast to the perfect binary trees, whose NTK is known to degenerate and leads to worse generalization performance for deeper trees.