[论文] Exactness at Inference: A Representational Criterion for Out-of-Distri...

论文概要

研究领域: ML 作者: Filipe Marinho Rocha, Inês Dutra, Vítor Santos Costa, Luís Paulo Reis 发布时间: 2026-09-21 arXiv: 2609.24942

中文摘要

模型只有在计算出的表示与生成机制结构等价（而非拟合近似）时，才能在训练分布之外泛化。这种等价是分布内外精确性的必要条件，外推受推理时精确性的支配，无论其实现方式如何。张量逻辑（Tensor Logic）展示了这一点：零温收缩等价于离散逻辑，就地演绎而不提取任何中间产物，其张量为布尔值、嵌入为正交归一，仅算术是连续的。由于缺乏无限递归，它达到 Datalog 而非 Prolog，虽然在封闭域上是精确的，但需要外部记忆来绑定新实体。该标准既不需要离散表示也不需要提取的表达式，它约束的是推理而非训练：精确的 [0,1] 边际通过，而阈值化为硬标签的神经网络不通过。逻辑张量网络不通过该标准，而可微 ILP 和零温张量逻辑通过。分段线性外推发散和无法绑定新实体，是精确可表示性不足的两种表现。对于混合架构，一条传播规则随之而来：输出继承其路径上每个拟合估计器的界限，这解释了等变模型中哪些轴会失败以及 ARC-AGI 中归纳/转导的分裂。只有精确的假设类才能认证训练数据未能确定的结论：在定律导出的划分上，它找到了 56.3% 可回答的遥远查询——集成方法以虚假信心应对，距离度量则反向排序。常见的归纳偏置（从对称性到记忆）之所以能达到精确性，仅因为人类注入了它们——这论证了应当诱导精确表示，而非拟合其残差即使在训练算术底线上也会发散并在组合下复合的代理模型。

原文摘要

A model generalizes outside its training distribution only when it computes a representation structurally equivalent to the generating mechanism, not an approximation fitted to it. Such equivalence is necessary for exactness in and out of distribution, and extrapolation is governed by this exactness at inference, whatever its realization. Tensor Logic shows this: a zero-temperature contraction is equivalent to discrete logic, deducing in place with no artefact extracted, its tensors Boolean, its embeddings orthonormal, only its arithmetic continuous. Lacking infinite recursion it reaches Datalog, not Prolog, and though exact over closed domains it needs external memory to bind a novel entity. The criterion needs neither a discrete representation nor an extracted expression, and constrains ...

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