Large Scale Prediction with Decision Trees
It provides theoretical guarantees for widely used machine learning methods like CART, C4.5, and random forests, which is foundational for practitioners relying on these tools in high-dimensional data scenarios.
This paper tackles the problem of ensuring consistency in decision trees and random forests for regression and classification tasks under high-dimensional settings with sub-exponential growth of predictor variables and sparsity constraints, showing that these methods remain consistent across a wide range of models and data types.
This paper 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.