On Stopping Rules and Spatial Adaptation for CART

Abstract

The popular CART algorithm for regression trees combines a greedy splitting rule with a stopping rule, but while the splitting rule has been well studied, the statistical role of stopping rules is less well understood. Meanwhile, although regression trees fit using Bayesian methods or via empirical risk minimization (ERM) have been shown to be spatially adaptive to local smoothness and anisotropy, it is unknown whether CART can achieve the same adaptation. We address these gaps by proving that, under spatially heterogeneous and anisotropic smoothness and appropriate structural assumptions on the regression function and covariate distribution, CART with the minimum impurity decrease (MID) stopping rule and a suitable threshold achieves pointwise rates that are minimax up to logarithmic factors. These rates hold simultaneously over all points in the domain. Moreover, we prove that spatial adaptation cannot be achieved under the widely used minimum leaf size stopping rule. Together, these results establish a precise statistical role for the MID stopping rule and provide a theoretical basis for the empirical success of CART.