dorsal/arxiv
View SchemaTree-Preconditioned Differentiable Optimization and Axioms as Layers
| Authors | Yuexin Liao |
|---|---|
| Categories | |
| ArXiv ID | 2601.06036vv1 |
| URL | https://arxiv.org/abs/2601.06036 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
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"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.",
"arxiv_id": "2601.06036",
"authors": [
"Yuexin Liao"
],
"categories": [
"cs.LG",
"cs.AI",
"math.OC"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Tree-Preconditioned Differentiable Optimization and Axioms as Layers",
"url": "https://arxiv.org/abs/2601.06036",
"version": "v1"
},
"schema_id": "dorsal/arxiv",
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"variant": "snapshot-2026-01-17",
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