最优分类树的精确联结树扩展形式
An Exact Junction-Tree Extended Formulation for Optimal Classification Trees
- The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种基于联结树的精确线性规划形式用于有界深度二元特征分类树，通过精确约简缩小模型，支持列生成与消息传递求解，显著提升可证明最优性的实例数并大幅降低运行时间。
AI中文摘要：
我们针对具有二元特征的有界深度分类树，利用联结树表示，开发了一种精确的线性规划（LP）形式。该形式是整数的，并支持递归子树优化。精确约简使模型更小，同时保留最优值以及最优树的恢复能力。约简后的模型支持两种求解方法：列生成和消息传递。列生成求解整数受限LP，并利用完整可行域上的界来证明最优性。消息传递递归地组合最优子树成本。两种方法都求解常见的子树问题，一旦先前的树决策被固定，这些问题可以独立且并行地评估。计算实验表明，精确约简显著减小了联结树形式的规模。由此产生的线性规划形式证明了在相同计算预算内，所测试的混合整数形式无法建立最优性的实例，而列生成和消息传递方法证明了更多实例，并相对于现有的最优分类树最先进精确方法，实现了几何平均运行时间一个数量级的减少。
英文摘要：
We develop an exact linear programming (LP) formulation for bounded-depth classification trees with binary features, using a junction-tree representation. The formulation is integral and supports recursive subtree optimization. Exact reductions make the model smaller while preserving the optimal value and recovery of an optimal tree. The reduced model supports two solution methods: column generation and message passing. Column generation solves integral restricted LPs and uses bounds over the full feasible domain to certify optimality. Message passing recursively combines optimal subtree costs. Both methods solve common subtree problems that, once the preceding tree decisions are fixed, can be evaluated independently and in parallel. Computational experiments show that the exact reductions substantially reduce the size of the junction-tree formulation. The resulting linear programming formulation certifies instances for which the tested mixed-integer formulation does not establish optimality within the same computational budget, while the column-generation and message-passing methods certify more instances and achieve an order-of-magnitude reduction in geometric-mean runtime relative to an existing state-of-the-art exact method for optimal classification trees.
