ZFC中具有非PAC一致学习器的VC维为1的Borel概念类

中文摘要 AI 辅助

统计学习基本定理指出，在适当的可测性假设下，有限Vapnik–Chervonenkis（VC）维保证每个恰当一致学习规则都是概率近似正确（PAC）的。Blumer、Ehrenfeucht、Haussler和Warmuth假设连续统假设成立，证明了概念类的“良态”条件不可省略：他们构造了一个VC维为1的Borel集概念类，该类存在一个非PAC的一致学习规则。我们证明连续统假设并非必要条件，仅在带有选择公理的策梅洛-弗兰克尔集合论（ZFC）框架下，即可构造[0,1]上的VC维为1的Borel集概念类，以及一个非PAC的恰当一致学习规则。更准确地说，对于合适的Borel概率测度和目标概念，该规则在所有样本量下，在一个外概率为1的样本集合上的真实风险为1。因此，单个概念的有限VC维和Borel可测性，不足以保证每个恰当一致学习规则都是PAC的。该结果表明，无需额外的集合论假设，基本定理中的额外正则性条件通常不可省略。

英文摘要

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

发表机构

• Instituto de Matemática, Estatística e Ciência da Computação, Universidade de São Paulo（圣保罗大学数学与统计及计算机科学学院）
• Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算机科学学院）

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