Consistency of Random Forests and Other Averaging Classifiers

In the last years of his life, Leo Breiman promoted random forests for use in classification. He suggested using averaging as a means of obtaining good discrimination rules. The base classifiers used for averaging are simple and randomized, often based on random samples from the data. He left a few questions unanswered regarding the consistency of such rules. In this paper, we give a number of theorems that establish the universal consistency of averaging rules. We also show that some popular classifiers, including one suggested by Breiman, are not universally consistent.