  Decision trees are popular Classification and Regression tools and, when
small-sized, easy to interpret. Traditionally, a greedy approach has been used
to build the trees, yielding a very fast training process; however, controlling
sparsity (a proxy for interpretability) is challenging. In recent studies,
optimal decision trees, where all decisions are optimized simultaneously, have
shown a better learning performance, especially when oblique cuts are
implemented. In this paper, we propose a continuous optimization approach to
build sparse optimal classification trees, based on oblique cuts, with the aim
of using fewer predictor variables in the cuts as well as along the whole tree.
Both types of sparsity, namely local and global, are modeled by means of
regularizations with polyhedral norms. The computational experience reported
supports the usefulness of our methodology. In all our data sets, local and
global sparsity can be improved without harming classification accuracy. Unlike
greedy approaches, our ability to easily trade in some of our classification
accuracy for a gain in global sparsity is shown.
