Learning Concepts Described by Weight Aggregation Logic

TL;DR

This paper introduces an extension of first-order logic for weighted structures, enabling aggregation and comparison of weights, and demonstrates its applicability to machine learning concepts with locality properties and learnability results.

Contribution

It presents a novel logical framework for weighted structures with aggregation, locality, and normal form results, and shows its relevance to machine learning concept definability and learnability.

Findings

Logic extension supports weight aggregation and comparison.

Locality properties enable efficient learning algorithms.

Weighted concepts are PAC-learnable in polylogarithmic time.

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 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.