Mining Optimal Decision Trees from Itemset Lattices

We present DL8, an exact algorithm for finding a decision tree that optimizes a ranking function under size, depth, accuracy and leaf constraints. Because the discovery of optimal trees has high theoretical complexity, until now few efforts have been made to compute such trees for real-world datasets. An exact algorithm is of both scientific and practical interest. Front a scientific point of view, it can be used as a gold standard to evaluate the performance of heuristic constraint-based decision tree learners and to gain new insight in traditional decision tree learners. From the application point of view, it can be used to discover trees that cannot be found by heuristic decision tree learners. The key idea behind our algorithm is that there is a relation between constraints on decision trees and constraints on itemsets. We show that optimal decision trees can be extracted from lattices of itemsets in linear time. We give several strategies to efficiently build these lattices. Experiments show that under the same constraints, DL8 obtains better results than C4.5, which confirms that exhaustive search does not always imply overfitting. The results also show that DL8 is a useful and interesting tool to learn decision trees under constraints.