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Top-down induction of decision trees: rigorous guarantees and inherent limitations

Consider the following heuristic for building a decision tree for a function $f : \{0,1\}^n \to \{\pm 1\}$. Place the most influential variable $x_i$ of $f$ at the root, and recurse on the subfunctions $f_{x_i=0}$ and $f_{x_i=1}$ on the left and right subtrees respectively; terminate once the tree is an $\varepsilon$-approximation of $f$. We analyze the quality of this heuristic, obtaining near-matching upper and lower bounds: $\circ$ Upper bound: For every $f$ with decision tree size $s$ and every $\varepsilon \in (0,\frac1{2})$, this heuristic builds a decision tree of size at most $s^{O(\log(s/\varepsilon)\log(1/\varepsilon))}$. $\circ$ Lower bound: For every $\varepsilon \in (0,\frac1{2})$ and $s \le 2^{\tilde{O}(\sqrt{n})}$, there is an $f$ with decision tree size $s$ such that this heuristic builds a decision tree of size $s^{\tilde{\Omega}(\log s)}$. We also obtain upper and lower bounds for monotone functions: $s^{O(\sqrt{\log s}/\varepsilon)}$ and $s^{\tilde{\Omega}(\sqrt[4]{\log s } )}$ respectively. The lower bound disproves conjectures of Fiat and Pechyony (2004) and Lee (2009). Our upper bounds yield new algorithms for properly learning decision trees under the uniform distribution. We show that these algorithms---which are motivated by widely employed and empirically successful top-down decision tree learning heuristics such as ID3, C4.5, and CART---achieve provable guarantees that compare favorably with those of the current fastest algorithm (Ehrenfeucht and Haussler, 1989). Our lower bounds shed new light on the limitations of these heuristics. Finally, we revisit the classic work of Ehrenfeucht and Haussler. We extend it to give the first uniform-distribution proper learning algorithm that achieves polynomial sample and memory complexity, while matching its state-of-the-art quasipolynomial runtime.

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