On statistical learning of graphs
This addresses theoretical graph learning problems for researchers in computational learning theory, providing incremental results on learnability conditions for infinite graphs.
The paper tackles the problem of learning hypothesis classes formed by copies of an infinite graph through vertex permutations, showing that PAC learnability of finite-support copies implies online learnability of the full isomorphism type and is equivalent to automorphic triviality. It also characterizes graphs where two-vertex permutations are not learnable and establishes a four-class partition of infinite graphs based on learnability equivalence for k-vertex permutations.
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.