决策树的信息复杂度

中文摘要 AI 辅助

我们定义并研究了一种用于随机决策树的信息复杂度度量。我们证明了关于该复杂度度量的三个主要结果：信息等于摊销规模复杂度。我们证明了随机决策树的信息复杂度等于计算函数 f 的摊销最坏情况随机树规模复杂度的对数。也就是说，当在 n 个输入上计算 f 时，随机树规模的对数恰好等于计算该函数所需的信息量。信息允许树规模压缩。我们证明了即使在单个输入上计算 f 时，如果允许成功概率有少量损失，信息复杂度也可用于压缩树的大小。结合 Chattopadhyay、Dahiya、Mande、Radhakrishnan 和 Sanyal（2023）最近的刻画，该结果表明 AND-OR 树的深度也可以根据信息复杂度进行压缩。直接乘积定理。我们证明了信息复杂度的成功条件变体满足一个完美的直接乘积定理。该结果给出了 Ben-David 和 Blais（2025）关于成功条件随机查询复杂度的直接乘积定理的信息复杂度类比。

英文摘要

We define and study a measure of information complexity for randomized decision trees. We prove three main results about this complexity measure: Information equals amortized size complexity. We show that the information complexity of randomized decision tree is equal to the logarithm of the amortized worst-case randomized tree size complexity of computing a function f. That is, when computing f on n inputs, the logarithm of the randomized tree size is exactly equal to the amount of information needed to compute the function. Information allows for tree size compression. We show that even when computing f on a single input, the information complexity can be used to compress the size of a tree, if we allow a small loss in success probability. With the recent characterization of Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (2023), this result shows that the depth of AND-OR trees can also be compressed in terms of information complexity. Direct Product Theorems. We show that the success-conditioned variant of information complexity satisfies a perfect direct product theorem. This result gives an information complexity analogue of the direct product theorem for success-conditioned randomized query complexity by Ben-David and Blais (2025).

发表机构

• Institute for Quantum Computing（量子计算研究所）
• University of Waterloo（滑铁卢大学）

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