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

We propose a simple extension of top-down decision tree learning heuristics such as ID3, C4.5, and CART. Our algorithm achieves provable guarantees for all target functions $f: {-1,1}ⁿ \to {-1,1}$ with respect to the uniform distribution, circumventing impossibility results showing that existing heuristics fare poorly even for simple target functions. The crux of our extension is a new splitting criterion that takes into account the correlations between $f$ and small subsets of its attributes. The splitting criteria of existing heuristics (e.g. Gini impurity and information gain), in contrast, are based 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 $s^{{\tilde{O}((\log} s)^{2/\epsilon2)}$} 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.