  Multi-hop reasoning approaches over knowledge graphs infer a missing
relationship between entities with a multi-hop rule, which corresponds to a
chain of relationships. We extend existing works to consider a generalized form
of multi-hop rules, where each rule is a set of relation chains. To learn such
generalized rules efficiently, we propose a two-step approach that first
selects a small set of relation chains as a rule and then evaluates the
confidence of the target relationship by jointly scoring the selected chains. A
game-theoretical framework is proposed to this end to simultaneously optimize
the rule selection and prediction steps. Empirical results show that our
multi-chain multi-hop (MCMH) rules result in superior results compared to the
standard single-chain approaches, justifying both our formulation of
generalized rules and the effectiveness of the proposed learning framework.
