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Several recent publications report advances in training optimal decisiontrees (ODT) using mixed-integer programs (MIP), due to algorithmic advances ininteger programming and a growing interest in addressing the inherentsuboptimality of heuristic approaches such as CART. In this paper, we propose anovel MIP formulation, based on a 1-norm support vector machine model, to traina multivariate ODT for classification problems. We provide cutting planetechniques that tighten the linear relaxation of the MIP formulation, in orderto improve run times to reach optimality. Using 36 data-sets from theUniversity of California Irvine Machine Learning Repository, we demonstratethat our formulation outperforms its counterparts in the literature by anaverage of about 10% in terms of mean out-of-sample testing accuracy across thedata-sets. We provide a scalable framework to train multivariate ODT on largedata-sets by introducing a novel linear programming (LP) based data selectionmethod to choose a subset of the data for training. Our method is able toroutinely handle large data-sets with more than 7,000 sample points andoutperform heuristics methods and other MIP based techniques. We presentresults on data-sets containing up to 245,000 samples. Existing MIP-basedmethods do not scale well on training data-sets beyond 5,500 samples.

A Scalable MIP-based Method for Learning Optimal Multivariate Decision Trees