We consider weighted structures, which extend ordinary relational structures by assigning weights, i.e. elements from a particular group or ring, to tuples present in the structure. We introduce an extension of first-order logic that allows to aggregate weights of tuples, compare such aggregates, and use them to build more complex formulas. We provide locality properties of fragments of this logic including Feferman-Vaught decompositions and a Gaifman normal form for a fragment called FOW1, as well as a localisation theorem for a larger fragment called FOWA1. This fragment can express concepts from various machine learning scenarios. Using the locality properties, we show that concepts definable in FOWA1 over a weighted background structure of at most polylogarithmic degree are agnostically PAC-learnable in polylogarithmic time after pseudo-linear time preprocessing.

中文翻译：

---

权重聚合逻辑描述的学习概念

我们考虑加权结构，它通过将权重（即来自特定组或环的元素）分配给结构中存在的元组来扩展普通关系结构。我们引入了一阶逻辑的扩展，允许聚合元组的权重，比较这些聚合，并使用它们来构建更复杂的公式。我们提供了该逻辑片段的局部性属性，包括 Feferman-Vaught 分解和名为 FOW1 的片段的 Gaifman 范式，以及名为 FOWA1 的更大片段的定位定理。这个片段可以表达来自各种机器学习场景的概念。使用局部属性，