推理时的精确性：面向分布外泛化的表示准则
Exactness at Inference: A Representational Criterion for Out-of-Distribution Generalization
- Faculdade de Ciências da Universidade do Porto（波尔图大学理学院）
- INESC TEC
- Faculdade de Engenharia da Universidade do Porto（波尔图大学工程学院）
- LIACC
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出表示与生成机制结构等价是分布外泛化精确性的准则，约束推理而非训练，并论证应归纳精确表示而非拟合替代品。
AI中文摘要：
一个模型只有在计算出的表示与生成机制在结构上等价（而非仅是对其的近似拟合）时，才能在训练分布之外进行泛化。这种等价性对于分布内和分布外的精确性都是必要的，而外推由推理时的这种精确性所支配，无论其实现方式如何。张量逻辑证明了这一点：零温度收缩等价于离散逻辑，在原位进行演绎而不提取任何人工产物，其张量是布尔型的，其嵌入是正交的，仅其算术运算是连续的。由于缺乏无限递归，它达到的是Datalog而非Prolog，尽管在封闭域上是精确的，但它需要外部记忆来绑定新实体。该准则既不需要离散表示，也不需要提取的表达式，并且约束的是推理而非训练：一个精确的边际在$[0,1]$内通过，而一个被阈值化为硬标签的神经网络则不通过。逻辑张量网络无法满足该准则，而可微ILP和$T=0$时的张量逻辑则通过。分段仿射外推发散和无法绑定新实体是精确可表示性不足的两个方面。对于混合架构，存在一条传播规则：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失败以及ARC-AGI归纳/转导分裂。只有精确的假设类才能证明训练数据未确定的内容：在基于定律的划分上，它找到了$56.3\%$的可回答的远距离查询，而集成方法以虚假置信度应对，距离度量则反向排序。从对称性到记忆的常见归纳偏置之所以能达到精确性，只是因为人类注入了它们，这论证了应归纳精确表示而非拟合替代品，因为替代品的残差即使在训练中的算术下限处，也会在数据之外发散并在组合下复合。
英文摘要：
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 inference, not training: an exact marginal in $[0,1]$ passes, a Neural Network thresholded to a hard label does not. Logic Tensor Networks fail it, while differentiable ILP and Tensor Logic at $T=0$ pass. Piecewise-affine extrapolation divergence and an inability to bind novel entities are two faces of a shortfall in exact representability. For hybrid architectures, a propagation rule follows: the output inherits the bounds of every fitted estimator on its path, explaining which axes fail in equivariant models and the ARC-AGI induction/transduction split. Only an exact hypothesis class certifies what the training data leave underdetermined: on a law-derived partition it finds the $56.3\%$ of distant queries that are answerable, which ensembles meet with false confidence and distance metrics rank backwards. Common inductive biases, from symmetries to memory, reach exactness only because humans inject them, an argument for inducing exact representations rather than fitting surrogates whose residuals, even at the arithmetic floor in training, diverge outside the data and compound under composition.
