决策树学习和测试的超多项式下界
Superpolynomial Lower Bounds for Decision Tree Learning and Testing
摘要：
我们建立了新的硬度结果决策树优化问题，添加到一行的工作，可以追溯到Hyafil和Rivest在1976年。我们证明，在随机ETH，超多项式下界的两个基本问题：给定一个明确的表示的功能$f$和生成器的分布$\mathcal{D}$，构造一个小的决策树逼近$f$下$\mathcal{D}$，并决定是否有一个小的决策树逼近$f$下$\mathcal{D}$。我们的研究结果意味着新的下限分布免费PAC学习和测试的决策树，设置中，该算法只有限制访问$f$和$\mathcal{D}$。具体而言，我们显示：$n$ -可变大小- $s$决策树不能在时间$n^{\tilde{O}(\log\log s)}$中正确地PAC学习，并且深度- $d$决策树不能在时间$\exp(d^{\,O(1)})$中测试。对于学习，以前的最佳下界只排除了$\text{poly}(n)$ -时间算法（Alekhnovich，Braverman，Feldman，Klivans和Pitassi，2009）。为了测试，最近的工作在$f$是随机的并且$\mathcal{D}$是非显式的设置中给出了类似但不可比较的界限（Blais，Ferreira Pinto Jr.，Harms，2021）。假设一个合理的猜想上的硬度集覆盖，我们证明我们的下限学习决策树可以提高到$n^{Ω(\log s)}$，匹配最好的已知的上限$n^{O(\log s)}$由于Escherifeucht和Haussler（1989）。我们得到我们的结果在一个统一的框架内，利用最近的进展，在两条工作线：集合覆盖和XOR引理查询复杂性的不可近似性。我们的框架是通用的，并产生相关的概念类，如juntas和DNF公式的结果。
英文摘要：
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 $\mathcal{D}$, construct a small decision tree approximator for $f$ under $\mathcal{D}$, and decide if there is a small decision tree approximator for $f$ under $\mathcal{D}$. 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 $\mathcal{D}$. Specifically, we show: $n$-variable size-$s$ decision trees cannot be properly PAC learned in time $n^{\tilde{O}(\log\log s)}$, and depth-$d$ decision trees cannot be tested in time $\exp(d^{\,O(1)})$. For learning, the previous best lower bound only ruled out $\text{poly}(n)$-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 $\mathcal{D}$ 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.
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。
