Abstract
This article shows that decision trees constructed with Classification and Regression Trees (CART) and C4.5 methodology are consistent for regression and classification tasks, even when the number of predictor variables grows sub-exponentially with the sample size, under natural 0-norm and 1-norm sparsity constraints. The theory applies to a wide range of models, including (ordinary or logistic) additive regression models with component functions that are continuous, of bounded variation, or, more generally, Borel measurable. Consistency holds for arbitrary joint distributions of the predictor variables, thereby accommodating continuous, discrete, and/or dependent data. Finally, we show that these qualitative properties of individual trees are inherited by Breiman’s random forests. A key step in the analysis is the establishment of an oracle inequality, which allows for a precise characterization of the goodness of fit and complexity tradeoff for a mis-specified model. Supplementary materials for this article are available online.
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