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Learning Concepts Described By Weight Aggregation Logic

Authors Steffen van Bergerem , Nicole Schweikardt

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Author Details

Steffen van Bergerem
  • RWTH Aachen University, Germany
Nicole Schweikardt
  • Humboldt-Universität zu Berlin, Germany

Funding

The second author has been partially supported by the ANR project EQUUS ANR-19-CE48-0019; funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 431183758 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 431183758).

Acknowledgements

We thank Martin Grohe and Sandra Kiefer for helpful discussions on the subject.

Cite AsGet BibTex

Steffen van Bergerem and Nicole Schweikardt. Learning Concepts Described By Weight Aggregation Logic. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CSL.2021.10

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Complexity theory and logic
  • Computing methodologies → Logical and relational learning
  • Computing methodologies → Supervised learning
Keywords
  • first-order definable concept learning
  • agnostic probably approximately correct learning
  • classification problems
  • locality
  • Feferman-Vaught decomposition
  • Gaifman normal form
  • first-order logic with counting
  • weight aggregation logic

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