Universal Bayes consistency in metric spaces

TL;DR

This paper introduces a new multiclass learning algorithm that is universally strongly Bayes-consistent across all essentially separable metric spaces, providing a complete characterization of universal Bayes consistency.

Contribution

It extends a 1-nearest-neighbor based algorithm and proves its universal strong Bayes-consistency in all essentially separable metric spaces, a first in the field.

Findings

The new algorithm is universally strongly Bayes-consistent in all essentially separable metric spaces.

Universal Bayes consistency is characterized by the property of essential separability.

The paper establishes the first impossibility result for universal Bayes consistency.

Abstract

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.