Random forests are a type of ensemble method which makes predictions by combining the results of several independent trees. However, the theory of random forests has long been outpaced by their application. In this paper, we propose a novel random forests algorithm based on cooperative game theory. Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. Unlike the previously used information gain rate of information theory, which simply chooses the most informative feature, the Banzhaf power index can be considered as a metric of the importance of each feature on the dependency among a group of features. More importantly, we have proved the consistency of the proposed algorithm, named Banzhaf random forests (BRF). This theoretical analysis takes a step towards narrowing the gap between the theory and practice of random forests for classification problems. Experiments on several UCI benchmark data sets show that BRF is competitive with state-of-the-art classifiers and dramatically outperforms previous consistent random forests. Particularly, it is much more efficient than previous consistent random forests.
In theory, there are two main properties of theoretical interests related to random forests. One is the consistency of the models, that whether it can converge to an optimal solution as the data set grows infinitely large. The other is the rate of convergence. Our paper mainly focuses on consistency, which @cite_2 has proved that Breiman's random forests cannot guarantee.
@cite_2
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Random forests are a scheme proposed by Leo Breiman in the 2000's for building a predictor ensemble with a set of decision trees that grow in randomly selected subspaces of data. Despite growing interest and practical use, there has been little exploration of the statistical properties of random forests, and little is known about the mathematical forces driving the algorithm. In this paper, we offer an in-depth analysis of a random forests model suggested by Breiman (2004), which is very close to the original algorithm. We show in particular that the procedure is consistent and adapts to sparsity, in the sense that its rate of convergence depends only on the number of strong features and not on how many noise variables are present.