Learning Conditional Preference Networks from Inconsistent Examples

The problem of learning conditional preference networks (CP-nets) from a set of examples has received great attention recently. However, because of the randomicity of the users´ behaviors and the observation errors, there is always some noise making the examples inconsistent, namely, there exists at least one outcome preferred over itself (by transferring) in examples. Existing CP-nets learning methods cannot handle inconsistent examples. In this work, we introduce the model of learning consistent CP-nets from inconsistent examples and present a method to solve this model. We do not learn the CP-nets directly. Instead, we first learn a preference graph from the inconsistent examples, because dominance testing and consistency testing in preference graphs are easier than those in CP-nets. The problem of learning preference graphs is translated into a 0-1 programming and is solved by the branch-and-bound search. Then, the obtained preference graph is transformed into a CP-net equivalently, which can entail a subset of examples with maximal sum of weight. Examples are given to show that our method can obtain consistent CP-nets over both binary and multivalued variables from inconsistent examples. The proposed method is verified on both simulated data and real data, and it is also compared with existing methods.