Superpolynomial Lower Bounds for Learning Monotone Classes

TL;DR

This paper establishes superpolynomial lower bounds for PAC-learning monotone classes such as monotone DNF and monotone decision trees, under standard complexity assumptions, extending previous non-monotone results.

Contribution

It proves the first superpolynomial lower bounds for PAC-learning monotone classes, resolving an open problem and generalizing prior non-monotone lower bounds.

Findings

Lower bounds hold even with distribution knowledge and polynomial-time sampling.

Results match upper bounds for certain classes, indicating tight complexity.

Extends non-monotone lower bounds to monotone classes, broadening understanding.

Abstract

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 $n^{\tilde O(\log\log s)}$ 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 $n^{\Omega(\log s)}$. 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.