Binary Partitions with Approximate Minimum Impurity

The problem of splitting attributes is one of the main steps in the construction of decision trees. In order to decide the best split, impurity measures such as Entropy and Gini are widely used. In practice, decision-tree inducers use heuristics for finding splits with small impurity when they consider nominal attributes with a large number of distinct values. However, there are no known guarantees for the quality of the splits obtained by these heuristics. To fill this gap, we propose two new splitting procedures that provably achieve nearoptimal impurity. We also report experiments that provide evidence that the proposed methods are interesting candidates to be employed in splitting nominal attributes with many values during decision tree/random forest induction.