On the exact learnability of graph parameters: The case of partition functions
This addresses a theoretical problem in computational learning theory for graph parameters, providing an efficient algorithm for a specific class, but it is incremental as it builds on prior characterizations and learning models.
The paper tackles the problem of exactly learning graph parameters representable as partition functions, showing that a large class called rigid partition functions can be learned in polynomial time in the size of the underlying graph and the largest counterexample, within the Blum-Shub-Smale model over reals with unit cost.
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 $α$ and edge weights $β$. M. Freedman, L. Lovász 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.