Risk Bounds for Classification Trees under a Margin Condition

Risk bounds for Classification and Regression Trees (CART, Breiman et. al. 1984) classifiers are obtained under a margin condition in the binary supervised classification framework. These risk bounds are obtained conditionally on the construction of the maximal deep binary tree and permit to prove that the linear penalty used in the CART pruning algorithm is valid under a margin condition. It is also shown that, conditionally on the construction of the maximal tree, the final selection by test sample does not alter dramatically the estimation accuracy of the Bayes classifier. In the two-class classification framework, the risk bounds that are proved, obtained by using penalized model selection, validate the CART algorithm which is used in many data mining applications such as Biology, Medicine or Image Coding.