Hierarchical Gradient Smoothing for Probability Estimation Trees

Zhang, He; Petitjean, François; Buntine, Wray

Decision trees are still seeing use in online, non-stationary and embedded contexts, as well as for interpretability. For applications like ranking and cost-sensitive classification, probability estimation trees (PETs) are used. These are built using smoothing or calibration techniques. Older smoothing techniques used counts local to a leaf node, but a few more recent techniques consider the broader context of a node when doing estimation. We apply a recent advanced smoothing method called Hierarchical Dirichlet Process (HDP) to PETs, and then propose a novel hierarchical smoothing approach called Hierarchical Gradient Smoothing (HGS) as an alternative. HGS smooths leaf nodes up to all the ancestors, instead of recursively smoothing to the parent used by HDP. HGS is made faster by efficiently optimizing the Leave-One-Out Cross-Validation (LOOCV) loss measure using gradient descent, instead of sampling used in HDP. An extensive set of experiments are conducted on 143 datasets showing that our HGS estimates are not only more accurate but also do so within a fraction of HDP time. Besides, HGS makes a single tree almost as good as a Random Forest with 10 trees. For applications that require more interpretability and efficiency, a single decision tree plus HGS is more preferred.