Decision tree heuristics can fail, even in the smoothed setting
This addresses a foundational problem in machine learning theory by disproving conjectures about the robustness of widely used heuristics, with implications for algorithm design and analysis.
The paper tackles the theoretical justification of greedy decision tree learning heuristics by constructing counterexamples showing they can fail even in the smoothed setting, building trees of depth 2^{Ω(k)} for depth-k decision tree targets, and also demonstrating failure in the agnostic setting for targets close to k-juntas.
Greedy decision tree learning heuristics are mainstays of machine learning practice, but theoretical justification for their empirical success remains elusive. In fact, it has long been known that there are simple target functions for which they fail badly (Kearns and Mansour, STOC 1996). Recent work of Brutzkus, Daniely, and Malach (COLT 2020) considered the smoothed analysis model as a possible avenue towards resolving this disconnect. Within the smoothed setting and for targets $f$ that are $k$-juntas, they showed that these heuristics successfully learn $f$ with depth-$k$ decision tree hypotheses. They conjectured that the same guarantee holds more generally for targets that are depth-$k$ decision trees. We provide a counterexample to this conjecture: we construct targets that are depth-$k$ decision trees and show that even in the smoothed setting, these heuristics build trees of depth $2^{Ω(k)}$ before achieving high accuracy. We also show that the guarantees of Brutzkus et al. cannot extend to the agnostic setting: there are targets that are very close to $k$-juntas, for which these heuristics build trees of depth $2^{Ω(k)}$ before achieving high accuracy.