Superpolynomial Lower Bounds for Learning Monotone Classes

Author Nader H. Bshouty

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Nader H. Bshouty
  • Department of Computer Science, Technion, Haifa, Israel


I would like to express my sincere gratitude to the reviewers of RANDOM for their valuable comments on this research. Their feedback was greatly appreciated and helped improve the quality of this work.

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Nader H. Bshouty. Superpolynomial Lower Bounds for Learning Monotone Classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time n^Õ(log log s) for the classes of n-variable size-s DNF, size-s Decision Tree, and log s-Junta by DNF (that returns a DNF hypothesis). Assuming a natural conjecture on the hardness of set cover, they give the lower bound n^Ω(log s). This matches the best known upper bound for n-variable size-s Decision Tree, and log s-Junta. In this paper, we give the same lower bounds for PAC-learning of n-variable size-s Monotone DNF, size-s Monotone Decision Tree, and Monotone log s-Junta by DNF. This solves the open problem proposed by Koch, Strassle, and Tan and subsumes the above results. The lower bound holds, even if the learner knows the distribution, can draw a sample according to the distribution in polynomial time, and can compute the target function on all the points of the support of the distribution in polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • PAC Learning
  • Monotone DNF
  • Monotone Decision Tree
  • Monotone Junta
  • Lower Bound


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