Provable guarantees for decision tree induction: the agnostic setting

We give strengthened provable guarantees on the performance of widely employed and empirically successful {\sl top-down decision tree learning heuristics}. While prior works have focused on the realizable setting, we consider the more realistic and challenging {\sl agnostic} setting. We show that for all monotone functions~$f$ and parameters $s\in \mathbb{N}$, these heuristics construct a decision tree of size $s^{{\tilde{O}((\log} s)/\varepsilon^2)}$ that achieves error $\le \mathsf{opt}s + \varepsilon$, where $\mathsf{opt}s$ denotes the error of the optimal size-$s$ decision tree for $f$. Previously, such a guarantee was not known to be achievable by any algorithm, even one that is not based on top-down heuristics. We complement our algorithmic guarantee with a near-matching $s^{{\tilde{\Omega}(\log} s)}$ lower bound.