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We propose a simple extension of top-down decision tree learning heuristicssuch as ID3, C4.5, and CART. Our algorithm achieves provable guarantees for alltarget functions $f: \{-1,1\}^n \to \{-1,1\}$ with respect to the uniformdistribution, circumventing impossibility results showing that existingheuristics fare poorly even for simple target functions. The crux of ourextension is a new splitting criterion that takes into account the correlationsbetween $f$ and small subsets of its attributes. The splitting criteria ofexisting heuristics (e.g. Gini impurity and information gain), in contrast, arebased solely on the correlations between $f$ and its individual attributes.  Our algorithm satisfies the following guarantee: for all target functions $f: \{-1,1\}^n \to \{-1,1\}$, sizes $s\in \mathbb{N}$, and error parameters$\epsilon$, it constructs a decision tree of size $s^{\tilde{O}((\logs)^2/\epsilon^2)}$ that achieves error $\le O(\mathsf{opt}_s) + \epsilon$,where $\mathsf{opt}_s$ denotes the error of the optimal size $s$ decision tree.A key technical notion that drives our analysis is the noise stability of $f$,a well-studied smoothness measure.

Universal guarantees for decision tree induction via a higher-order splitting criterion