Optimal Generalized Decision Trees via Integer Programming
  Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if allowed to grow large, they lose interpretability. In this paper, we present a mixed integer programming formulation to construct optimal decision trees of a prespecified size. We take the special structure of categorical features into account and allow combinatorial decisions (based on subsets of values of features) at each node. Our approach can also handle numerical features via thresholding. We show that very good accuracy can be achieved with small trees using moderately-sized training sets. The optimization problems we solve are tractable with modern solvers. 
Oktay Gunluk, Jayant Kalagnanam, Minhan Li, Matt Menickelly, Katya   Scheinberg
cs.LG math.OC stat.ML