UNIVERSAL BAYES CONSISTENCY IN METRIC SPACES

• 成果类型: Article
• 署名作者: Hanneke, Steve; Kontorovich, Aryeh; Sabato, Sivan; Weiss, Roi
• 署名单位: Toyota Technological Institute - Chicago; Ben-Gurion University of the Negev; Ariel University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS2029
• 发表日期: 2021
• 页码: 2129-2150
• 关键词: nearest; CLASSIFICATION; compression; CONVERGENCE; regression; bounds; error; rates

摘要

We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes consistent in all metric spaces admitting any such learner, making it an optimistically universal Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, the k-NN classifier and its variants are not generally universally Bayes consistent, except under additional structural assumptions, such as an inner product, a norm, finite dimension or a Besicovitch-type property. The metric spaces in which universal Bayes consistency is possible are the essentially separable ones-a notion that we define, which is more general than standard separability. The existence of metric spaces that are not essentially separable is widely believed to be independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal Bayes-consistent learner for the given metric space. In particular, this yields the first impossibility result for universal Bayes consistency. Taken together, our results completely characterize strong and weak universal Bayes consistency in metric spaces.