Adaptive Concentration of Regression Trees, with Application to Random Forests

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.