将理性选择理论嵌入神经网络，实现可微分的合理决策建模。
Tree-Preconditioned Differentiable Optimization and Axioms as Layers
- 用树状预条件器改进优化算法，解决高维约束下的数值不稳定性。
- 在稀疏数据下仍能保持模型理性，避免传统方法的结构过拟合。
- 适用于需要可解释决策逻辑的AI系统，如推荐与规划场景。
本文提出一种可微分框架，将随机效用模型（RUM）的公理结构直接嵌入深度神经网络。尽管将经验选择数据投影到RUM多面体在一般情况下是NP难问题，我们发现RUM一致性与布尔格上的流守恒存在同构关系。利用这一组合结构，我们推导出一种新型树状预条件共轭梯度求解器。通过利用约束图的生成树，该预条件器有效“白化”由内点法障碍项引起的病态海森谱，实现超线性收敛，并可扩展至此前被认为不可解的大规模问题。进一步地，我们通过隐函数定理将投影过程形式化为可微分层，使得精确雅可比在反向传播中传递几何约束。实验表明，这种‘公理即层’范式消除了基于惩罚项方法固有的结构过拟合，使模型具备联合训练、可证明合理性，且能在标准近似失效的稀疏数据场景中实现良好泛化。
原文摘要 · Abstract (English)
This paper introduces a differentiable framework that embeds the axiomatic structure of Random Utility Models (RUM) directly into deep neural networks. Although projecting empirical choice data onto the RUM polytope is NP-hard in general, we uncover an isomorphism between RUM consistency and flow conservation on the Boolean lattice. Leveraging this combinatorial structure, we derive a novel Tree-Preconditioned Conjugate Gradient solver. By exploiting the spanning tree of the constraint graph, our preconditioner effectively "whitens" the ill-conditioned Hessian spectrum induced by the Interior Point Method barrier, achieving superlinear convergence and scaling to problem sizes previously deemed unsolvable. We further formulate the projection as a differentiable layer via the Implicit Function Theorem, where the exact Jacobian propagates geometric constraints during backpropagation. Empirical results demonstrate that this "Axioms-as-Layers" paradigm eliminates the structural overfitting inherent in penalty-based methods, enabling models that are jointly trainable, provably rational, and capable of generalizing from sparse data regimes where standard approximations fail.
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