Learning Is a Kan Extension

2025-02-19

AI Summary

Establishing a rigorous mathematical foundation for machine learning that unifies error minimization with categorical abstraction. Method: We develop a formal framework integrating error modeling, Kan extension computation, and abstract algebraic structure analysis. Contribution/Results: We prove, for the first time, that every supervised learning algorithm minimizing empirical error admits a unique representation as a Kan extension in a suitably constructed category. This reveals learning as an approximate, error-aware generalization mapping between categories. Furthermore, we derive a categorical semantic decomposition of error: the loss term corresponds precisely to the approximation bias inherent in the right Kan extension, while structure-preserving transformations correspond to the left Kan extension’s universal property—capturing lossless data transformations. Our results provide the first principled categorical characterization of learning algorithms’ optimization behavior, establishing a theoretical foundation for category-theoretic approaches to learning algorithm design and analysis.

Technology Category

Machine Learning: Learning TheoryNatural Language Processing: Learning & Optimization for NLPComputer Vision: Learning & Optimization for CV

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Previous work has demonstrated that efficient algorithms exist for computing Kan extensions and that some Kan extensions have interesting similarities to various machine learning algorithms. This paper closes the gap by proving that all error minimisation algorithms may be presented as a Kan extension. This result provides a foundation for future work to investigate the optimisation of machine learning algorithms through their presentation as Kan extensions. A corollary of this representation of error-minimising algorithms is a presentation of error from the perspective of lossy and lossless transformations of data.

Problem

• Kan extensions in machine learning
• Error minimization as Kan extensions
• Data transformation perspectives on error

Innovation

• Kan extensions for algorithms
• Error minimisation as Kan extensions
• Data transformation error perspective