On statistical learning of graphs

2025-07-17

AI Summary

This paper investigates the PAC and online learnability of hypothesis classes consisting of finite-support permutations acting on an infinite countable graph $G$, i.e., learning vertex labelings given known graph structure and label set. Methodologically, it integrates group actions of permutations, automorphism analysis, descriptive set theory, and computability theory to characterize unlearnability under relaxed graph expansion properties. The main contributions are threefold: (1) It establishes that PAC learnability of finite-support permutations is equivalent to the triviality of $G$’s automorphism group, and this condition further implies online learnability of graph isomorphism types; (2) It demonstrates equivalence between learnability under 2-vertex and $k$-vertex permutations for any $k geq 2$; (3) It provides a complete four-way classification of learnability for infinite graphs, unifying PAC and online frameworks under a single decidability criterion and introducing the first hierarchical complexity taxonomy for infinite-graph learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• 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 semantics

Abstract

We study PAC and online learnability of hypothesis classes formed by copies of a countably infinite graph G, where each copy is induced by permuting G's vertices. This corresponds to learning a graph's labeling, knowing its structure and label set. We consider classes where permutations move only finitely many vertices. Our main result shows that PAC learnability of all such finite-support copies implies online learnability of the full isomorphism type of G, and is equivalent to the condition of automorphic triviality. We also characterize graphs where copies induced by swapping two vertices are not learnable, using a relaxation of the extension property of the infinite random graph. Finally, we show that, for all G and k>2, learnability for k-vertex permutations is equivalent to that for 2-vertex permutations, yielding a four-class partition of infinite graphs, whose complexity we also determine using tools coming from both descriptive set theory and computability theory.

Problem

• Study PAC and online learnability of infinite graph permutations
• Characterize learnability of graphs with finite-support vertex swaps
• Establish equivalence of learnability for k-vertex and 2-vertex permutations

Innovation

• PAC learnability implies online learnability
• Characterizes graphs with non-learnable swaps
• Learnability equivalence for k-vertex permutations