Adaptive Concentration of Regression Trees, with Application to Random Forests

TL;DR

This paper introduces a new notion of adaptive concentration for regression trees, providing theoretical guarantees for their convergence and enabling high-dimensional consistency and valid inference in random forests.

Contribution

It proposes a novel adaptive concentration framework that separates tree training into split selection and model fitting, with theoretical bounds and practical implications.

Findings

Fitted regression trees concentrate around the optimal predictor with high probability.

The bounds depend on feature dimension, sample size, and leaf size, and are rate-matching.

Results enable consistency proofs and valid post-selection inference for high-dimensional forests.

Abstract

We study the convergence of the predictive surface of regression trees and forests. To support our analysis we introduce a notion of adaptive concentration for regression trees. This approach breaks tree training into a model selection phase in which we pick the tree splits, followed by a model fitting phase where we find the best regression model consistent with these splits. We then show that the fitted regression tree concentrates around the optimal predictor with the same splits: as d and n get large, the discrepancy is with high probability bounded on the order of sqrt(log(d) log(n)/k) uniformly over the whole regression surface, where d is the dimension of the feature space, n is the number of training examples, and k is the minimum leaf size for each tree. We also provide rate-matching lower bounds for this adaptive concentration statement. From a practical perspective, our result enables us to prove consistency results for adaptively grown forests in high dimensions, and to carry out valid post-selection inference in the sense of Berk et al. [2013] for subgroups defined by tree leaves.