Learning Concepts Described By Weight Aggregation Logic

arXiv (Cornell University)

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.

Authors 2

1. Steffen van Bergerem Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Germany

   Rheinisch-Westfälische Technische Hochschule Aachen University

2. Nicole Schweikardt

   Humboldt-Universität zu Berlin

   Affiliation as printed

   Humboldt-Universität zu Berlin, Germany

   Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin