Domain-Liftability of Relational Marginal Polytopes

TL;DR

This paper investigates the computational properties of relational marginal polytopes in statistical relational learning, establishing conditions under which their construction and related problems are domain-liftable, with implications for learning Markov logic networks.

Contribution

It extends domain-liftability results from partition functions of MLNs to relational marginal polytopes and analyzes the complexity of polytope containment problems.

Findings

Domain-liftability results carry over to relational marginal polytope construction.

Weight learning of MLNs is domain-liftable if the partition function is domain-liftable.

Relational marginal polytope containment problem is computationally hard under certain assumptions.

Abstract

We study computational aspects of relational marginal polytopes which are statistical relational learning counterparts of marginal polytopes, well-known from probabilistic graphical models. Here, given some first-order logic formula, we can define its relational marginal statistic to be the fraction of groundings that make this formula true in a given possible world. For a list of first-order logic formulas, the relational marginal polytope is the set of all points that correspond to the expected values of the relational marginal statistics that are realizable. In this paper, we study the following two problems: (i) Do domain-liftability results for the partition functions of Markov logic networks (MLNs) carry over to the problem of relational marginal polytope construction? (ii) Is the relational marginal polytope containment problem hard under some plausible complexity-theoretic assumptions? Our positive results have consequences for lifted weight learning of MLNs. In particular, we show that weight learning of MLNs is domain-liftable whenever the computation of the partition function of the respective MLNs is domain-liftable (this result has not been rigorously proven before).