On statistical learning of graphs

TL;DR

This paper investigates the learnability of infinite graphs under permutations, establishing conditions for PAC and online learnability, and classifying graphs based on their permutation learnability complexity.

Contribution

It provides a characterization of when classes of permuted graphs are PAC and online learnable, linking automorphic triviality to learnability and classifying graphs by permutation complexity.

Findings

PAC learnability implies online learnability for certain graph classes.

Characterization of graphs where swapping two vertices affects learnability.

Equivalence of learnability for k-vertex and 2-vertex permutations for all graphs.

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.