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.

翻译：基于树的回归模型在监督学习中应用广泛，其中分类与回归树（CART）算法作为标准参照。CART的构建涉及一系列分裂选择优化问题的求解。对于分类预测变量，该问题可表述为组合分式优化问题。这种结构使得精确优化在计算上具有挑战性，导致标准实现依赖贪婪启发式方法，可能产生次优分裂。在本工作中，我们重新表述该分式问题，并应用Dinkelbach（1967）算法将其转换为二次无约束二元优化（QUBO）问题。利用最先进的QUBO求解器，我们获得了预测性能与标准CART相当的基于QUBO的回归树，同时能产生更高质量的分裂解。这些结果凸显了QUBO公式在改进基于树的学习方法方面的潜力，并为未来混合经典-量子实现开辟了前景。