A decomposable branch-and-price formulation for optimal classification trees

In this study, we focus on finding exact binary optimal classification trees (OCTs) via mixed-integer (linear) programs (MIP). To alleviate the scalability issues of existing MIP-based OCT formulations, we propose a novel decomposable model that treats each leaf of the classification tree as a decision rule. The model exploits the order-independent nature of decision rules to reduce the problem size and the solution space, and lends itself to a Dantzig–Wolfe (extended) reformulation. We then implement a tailored branch-and-price solution with pricing subproblems designated for each leaf solved in parallel. We identify the conditions under which the extended formulation performs better in order to form a hybrid algorithm, OrderIndependentOCT, that decides when to invoke branch-and-price. On 18 benchmark datasets, OrderIndependentOCT proves optimality on $$71.1%$$ 71.1 % of instances within a one-hour time limit, compared to $$60.0%$$ 60.0 % for the well-known Benders decomposition approach (BendersOCT). In cases where neither method attains optimality, OrderIndependentOCT yields higher (or the same) training accuracy than BendersOCT in $$50.68%$$ 50.68 % (resp. $$71.23%$$ 71.23 % ) of the instances. Similarly, in $$86.3%$$ 86.3 % of all instances, OrderIndependentOCT matches or exceeds BendersOCT’s out-of-sample accuracy. These findings illustrate the efficiency of the proposed decomposition approach for the optimal classification tree training problem compared to the state-of-the-art.