arXiv 2005.018862020-05-04cs.LGmath.LOstat.ML

一个学习问题，其一致性等价于实值可测基数的不存在性

A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals

Instituto de Matemática e Estatística, Universidade Federal da Bahia（巴伊亚联邦大学数学与统计学院）；Department of Mathematics and Statistics, University of Ottawa（渥太华大学数学与统计系）

Vladimir G. Pestov

备注：: 16 pp., journal macros
发表信息：: Addendum was revised and published as a separate paper: On a result of K P. Hart about non-existence of measurable solutions to the discrete expectation maximization problem, Comment. Math. Univ. Carolin. 64 (2023), 353--358

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AI总结：

本文证明 k-最近邻学习规则的普遍一致性与实值可测基数的不存在性等价，揭示了机器学习一致性问题与集合论大基数假设之间的深刻联系。

摘要：

我们证明，在度量空间 $X$ 中，$k$-最近邻学习规则是普遍一致的，当且仅当它在 $X$ 的每个可分子空间中都是普遍一致的，并且 $X$ 的密度小于每个实值可测基数。特别地，$k$-NN 分类器在每个可分子空间在 Nagata 和 Preiss 意义下为 sigma-有限维的度量空间中都是普遍一致的，当且仅当不存在实值可测基数。后一假设与 ZFC 相对一致，但此类基数存在性的一致性无法在 ZFC 内证明。我们的结果受到 Cérou 和 Guyader 在 2006 年以直观严谨程度勾勒的一个例子的启发。

英文摘要：

We show that the $k$-nearest neighbour learning rule is universally consistent in a metric space $X$ if and only if it is universally consistent in every separable subspace of $X$ and the density of $X$ is less than every real-measurable cardinal. In particular, the $k$-NN classifier is universally consistent in every metric space whose separable subspaces are sigma-finite dimensional in the sense of Nagata and Preiss if and only if there are no real-valued measurable cardinals. The latter assumption is relatively consistent with ZFC, however the consistency of the existence of such cardinals cannot be proved within ZFC. Our results were inspired by an example sketched by Cérou and Guyader in 2006 at an intuitive level of rigour.

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