决策树学习的超常数不可近似性
Superconstant Inapproximability of Decision Tree Learning
AI总结：
本文证明带查询的决策树恰当 PAC 学习在允许任意常数因子近似最优假设树时仍为 NP 难，并通过新证明和新的 XOR 引理实现了困难性假设与不可近似性因子之间的平滑权衡，同时为决策树最小化问题提供了新推论。
摘要：
我们考虑在带查询条件下对决策树进行恰当 PAC 学习的任务。Koch、Strassle 和 Tan 的近期工作表明，该任务的最严格版本——要求假设树 $T$ 达到最优最小规模——是 NP 难的。他们的工作留下了一个未解问题：如果仅要求 $T$ 接近最优，例如在 2 倍因子以内，而不是精确最优，该任务是否仍然难以处理。我们对此给出肯定回答，并证明即使允许 $T$ 处于最优规模的任意常数因子之内，该任务依然保持 NP 难。更一般地，我们的结果允许在困难性假设与不可近似性因子之间进行平滑权衡。由于 Koch 等人的技术似乎不适用于这种强化，我们首先用一个更简单的新证明重新得到他们的结果，并将其与一个新的决策树 XOR 引理相结合。虽然关于决策树 XOR 引理已有大量工作，但我们的设置要求参数极其精确，而已有的 XOR 引理尚不知道能否达到这些参数。我们的工作还对相关的决策树最小化问题带来了新的推论。
英文摘要：
We consider the task of properly PAC learning decision trees with queries. Recent work of Koch, Strassle, and Tan showed that the strictest version of this task, where the hypothesis tree $T$ is required to be optimally small, is NP-hard. Their work leaves open the question of whether the task remains intractable if $T$ is only required to be close to optimal, say within a factor of 2, rather than exactly optimal. We answer this affirmatively and show that the task indeed remains NP-hard even if $T$ is allowed to be within any constant factor of optimal. More generally, our result allows for a smooth tradeoff between the hardness assumption and the inapproximability factor. As Koch et al.'s techniques do not appear to be amenable to such a strengthening, we first recover their result with a new and simpler proof, which we couple with a new XOR lemma for decision trees. While there is a large body of work on XOR lemmas for decision trees, our setting necessitates parameters that are extremely sharp, and are not known to be attainable by existing XOR lemmas. Our work also carries new implications for the related problem of Decision Tree Minimization.
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