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.

翻译：Koch、Strassle 和 Tan [SODA 2023] 证明，在随机指数时间假设下，对于 $n$ 变量规模为 $s$ 的 DNF、规模为 $s$ 的决策树以及由 DNF 表示的 $\log s$-Junta 类（返回 DNF 假设），不存在运行时间为 $n^{\tilde O(\log\log s)}$ 的无分布 PAC 学习算法。基于集合覆盖难度的自然猜想，他们给出了 $n^{\Omega(\log s)}$ 的下界。这匹配了 $n$ 变量规模为 $s$ 的决策树和 $\log s$-Junta 类已知的最佳上界。本文中，我们给出了对于 $n$ 变量规模为 $s$ 的单调 DNF、规模为 $s$ 的单调决策树以及由 DNF 表示的单调 $\log s$-Junta 类进行 PAC 学习的相同下界。这解决了 Koch、Strassle 和 Tan 提出的开放问题，并涵盖了上述结果。即使学习器已知分布、能在多项式时间内根据该分布采样，且能在多项式时间内计算目标函数在分布支撑集上所有点的取值，该下界依然成立。