Early Stopping for Regression Trees

We develop early stopping rules for growing regression tree estimators. The fully data-driven stopping rule is based on monitoring the global residual norm. The best-first search and the breadth-first search algorithms together with linear interpolation give rise to generalized projection or regularization flows. A general theory of early stopping is established. Oracle inequalities for the early-stopped regression tree are derived without any smoothness assumption on the regression function, assuming the original CART splitting rule, yet with a much broader scope. The remainder terms are of smaller order than the best achievable rates for Lipschitz functions in dimension $d\ge 2$. In real and synthetic data the early stopping regression tree estimators attain the statistical performance of cost-complexity pruning while significantly reducing computational costs.