Learning Concepts Described by Weight Aggregation Logic

Abstract

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 FOW 1 , as well as a localisation theorem for a larger fragment called FOWA 1 . This fragment can express concepts from various machine learning scenarios. Using the locality properties, we show that concepts definable in FOWA 1 over a weighted background structure of at most polylogarithmic degree are agnostically PAC-learnable in polylogarithmic time after pseudo-linear time preprocessing.