We give an $n^{O(\log\log n)}$-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over $\{\pm 1\}^n$. Even in the realizable setting, the previous fastest runtime was $n^{O(\log n)}$, a consequence of a classic algorithm of Ehrenfeucht and Haussler. Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of O'Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be "pruned" so that every variable in the resulting tree is influential.

中文翻译：

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在几乎多项式时间内正确学习决策树

我们给出了 $n^{O(\log\log n)}$-time 成员资格查询算法，用于在 $\{\pm 1\}^n$ 上的均匀分布下正确且不可知地学习决策树。即使在可实现的设置中，之前最快的运行时间也是 $n^{O(\log n)}$，这是 Ehrenfeucht 和 Haussler 的经典算法的结果。我们的算法与学习决策树的实用启发式方法有相似之处，我们增加了额外的想法来绕过这些启发式方法的已知下界。为了分析我们的算法，我们证明了一个新的决策树结构结果，它加强了 O'Donnell、Saks、Schramm 和 Servedio 的定理。虽然OSSS定理说每个决策树都有一个有影响的变量，但我们展示了如何“修剪”每个决策树，以便结果树中的每个变量都有影响。