Universal Bayes consistency in metric spaces

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 the first known to enjoy this property; by comparison, k-NN classifier its variants are not generally consistent, except under additional structural assumptions, as inner product, norm, finite dimension or Besicovitch-type property. The which universal consistency possible “essentially separable” ones—a notion we define, more general than standard separability. existence of essentially separable widely believed be independent ZFC axioms set theory. essential separability exactly characterizes learner for given space. In particular, yields impossibility result consistency. Taken together, results completely characterize strong weak spaces.