单调类学习的超多项式下界
Superpolynomial Lower Bounds for Learning Monotone Classes
摘要：
Koch、Strassle和Tan [SODA 2023]表明，在随机指数时间假设下，对于$n$ -可变大小- $s$ DNF、大小- $s$决策树和$\log s$ -Junta by DNF（返回DNF假设）的类，不存在在时间$n^{\tilde O(\log\log s)}$中运行的无分布PAC学习算法。假设一个关于集合覆盖硬度的自然猜想，给出了集合覆盖的下界$n^{Ω(\log s)}$。这与$n$ -可变大小- $s$决策树和$\log s$ -Junta的最佳已知上限相匹配。本文通过~DNF给出了$n$ -可变大小- $s$单调DNF、大小- $s$单调决策树和单调$\log s$ -Junta的PAC学习的相同下界。这解决了由Koch，Strassle和Tan提出的开放问题，并包含了上述结果。下界成立，即使学习者知道分布，也能在多项式时间内根据分布绘制样本，并能在多项式时间内在分布的所有支撑点上计算目标函数。
英文摘要：
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 $n^{Ω(\log s)}$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
