Universal Bayes consistency in metric spaces
该研究修改了一种基于1-最近邻的多类学习算法，证明其在所有允许此类学习器的度量空间中具有强通用贝叶斯一致性，并刻画了这种一致性存在的充要条件。
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.