  Classification and Regression Trees (CARTs) are off-the-shelf techniques in
modern Statistics and Machine Learning. CARTs are traditionally built by means
of a greedy procedure, sequentially deciding the splitting predictor
variable(s) and the associated threshold. This greedy approach trains trees
very fast, but, by its nature, their classification accuracy may not be
competitive against other state-of-the-art procedures. Moreover, controlling
critical issues, such as the misclassification rates in each of the classes, is
difficult. To address these shortcomings, optimal decision trees have been
recently proposed in the literature, which use discrete decision variables to
model the path each observation will follow in the tree. Instead, we propose a
new approach based on continuous optimization. Our classifier can be seen as a
randomized tree, since at each node of the decision tree a random decision is
made. The computational experience reported demonstrates the good performance
of our procedure.
