On the exact learnability of graph parameters: The case of partition functions

TL;DR

This paper investigates the exact learnability of graph parameters represented as partition functions counting weighted homomorphisms, demonstrating that a broad class of these functions can be learned efficiently using a query-based model.

Contribution

It introduces a model for learning graph partition functions via membership and equivalence queries, and proves that rigid partition functions are learnable in polynomial time.

Findings

Rigid partition functions can be learned efficiently.

The learning algorithm operates in polynomial time in the Blum-Shub-Smale model.

The model uses connection matrices and counterexamples for hypothesis refinement.

Abstract

We study the exact learnability of real valued graph parameters $f$ which are known to be representable as partition functions which count the number of weighted homomorphisms into a graph $H$ with vertex weights $\alpha$ and edge weights $\beta$. M. Freedman, L. Lov\'asz and A. Schrijver have given a characterization of these graph parameters in terms of the $k$-connection matrices $C(f,k)$ of $f$. Our model of learnability is based on D. Angluin's model of exact learning using membership and equivalence queries. Given such a graph parameter $f$, the learner can ask for the values of $f$ for graphs of their choice, and they can formulate hypotheses in terms of the connection matrices $C(f,k)$ of $f$. The teacher can accept the hypothesis as correct, or provide a counterexample consisting of a graph. Our main result shows that in this scenario, a very large class of partition functions, the rigid partition functions, can be learned in time polynomial in the size of $H$ and the size of the largest counterexample in the Blum-Shub-Smale model of computation over the reals with unit cost.