Superpolynomial Lower Bounds for Decision Tree Learning and Testing
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We establish new hardness results for decision tree optimization problems, adding to a line of work that dates back to Hyafil and Rivest in 1976. We prove, under randomized ETH, superpolynomial lower bounds for two basic problems: given an explicit representation of a function f and a generator for a distribution 𝒟, construct a small decision tree approximator for f under 𝒟, and decide if there is a small decision tree approximator for f under 𝒟. Our results imply new lower bounds for distribution-free PAC learning and testing of decision trees, settings in which the algorithm only has restricted access to f and 𝒟. Specifically, we show: n-variable size-s decision trees cannot be properly PAC learned in time n^Õ(loglog s), and depth-d decision trees cannot be tested in time exp(d^ O(1)). For learning, the previous best lower bound only ruled out poly(n)-time algorithms (Alekhnovich, Braverman, Feldman, Klivans, and Pitassi, 2009). For testing, recent work gives similar though incomparable bounds in the setting where f is random and 𝒟 is nonexplicit (Blais, Ferreira Pinto Jr., and Harms, 2021). Assuming a plausible conjecture on the hardness of Set-Cover, we show our lower bound for learning decision trees can be improved to n^Ω(log s), matching the best known upper bound of n^O(log s) due to Ehrenfeucht and Haussler (1989). We obtain our results within a unified framework that leverages recent progress in two lines of work: the inapproximability of Set-Cover and XOR lemmas for query complexity. Our framework is versatile and yields results for related concept classes such as juntas and DNF formulas.
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