{"ID":2833300,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2601.06036","arxiv_id":"2601.06036","title":"Tree-Preconditioned Differentiable Optimization and Axioms as Layers","abstract":"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.","short_abstract":"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 Bo...","url_abs":"https://arxiv.org/abs/2601.06036","url_pdf":"https://arxiv.org/pdf/2601.06036v1","authors":"[\"Yuexin Liao\"]","published":"2025-12-03T04:47:37Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\",\"math.OC\"]","methods":"[]","has_code":false}