On the Convergence of CART under Sufficient Impurity Decrease Condition
Rahul Mazumder Haoyue Wang
The decision tree is a flexible machine learning model that finds its success
in numerous applications. It is usually fitted in a recursively greedy manner
using CART. In this paper, we investigate the convergence rate of CART under a
regression setting. First, we establish an upper bound on the prediction error
of CART under a sufficient impurity decrease (SID) condition
\cite{chi2022asymptotic} -- our result improves upon the known result by
\cite{chi2022asymptotic} under a similar assumption. Furthermore, we provide
examples that demonstrate the error bound cannot be further improved by more
than a constant or a logarithmic factor. Second, we introduce a set of easily
verifiable sufficient conditions for the SID condition. Specifically, we
demonstrate that the SID condition can be satisfied in the case of an additive
model, provided that the component functions adhere to a ``locally reverse
Poincar{\'e} inequality". We discuss several well-known function classes in
non-parametric estimation to illustrate the practical utility of this concept.