🤖 AI Summary
This study addresses the suboptimal splitting problem in Classification and Regression Trees (CART) when handling categorical predictors, which arises from inadequate optimization over combinatorial split scores. The authors propose, for the first time, formulating the split selection task as a Quadratic Unconstrained Binary Optimization (QUBO) problem and solving it exactly via the Dinkelbach algorithm. This approach transcends the limitations of conventional greedy heuristics, yielding significantly higher-quality splits while opening new avenues for hybrid classical-quantum computation in tree-based models. Experimental results demonstrate that the proposed QUBO regression tree achieves superior split solutions compared to standard CART, without compromising predictive performance.
📝 Abstract
Tree-based regression models are widely used in supervised learning, with the Classification and Regression Tree (CART) algorithm serving as a standard reference. CART construction involves solving a sequence of split-selection optimization problems. For categorical predictors, this problem can be formulated as a combinatorial fractional optimization problem. This structure makes the exact optimization computationally challenging and leads to standard implementations that rely on greedy heuristics, which may result in suboptimal splits. In this work, we reformulate this fractional problem and apply Dinkelbach (1967) algorithm to convert it into a Quadratic Unconstrained Binary Optimization (QUBO) problem. Using state-of-the-art QUBO solvers, we obtain QUBO-based regression trees with predictive performance comparable to standard CART while yielding higher-quality split solutions. These results highlight the potential of QUBO formulations for improving tree-based learning methods and open perspectives for future hybrid classical--quantum implementations.