On Optimal Rule Mining: A Framework and a Necessary and Sufficient Condition of Antimonotonicity

Many studies have shown the limits of support/confidence framework used in Apriori-like algorithms to mine association rules. There are a lot of efficient implementations based on the antimonotony property of the support but candidate set generation is still costly. In addition many rules are uninteresting or redundant and one can miss interesting rules like nuggets. One solution is to get rid of frequent itemset mining and to focus as soon as possible on interesting rules. For that purpose algorithmic properties were first studied, especially for the confidence. They allow all confidence rules to be found without a preliminary support pruning. More recently, in the case of class association rules, the concept of optimal rules gave a pruning strategy compatible with more measures. However, all these properties have been demonstrated for a limited number of interestingness measures. We present a new formal framework which allows us to make the link between analytic and algorithmic properties of the measures. We apply this framework to optimal rules, and we demonstrate a necessary and sufficient condition of existence for this pruning strategy, which can be applied to any measure.