Online Consistency of the Nearest Neighbor Rule
This addresses the problem of ensuring vanishing mistake rates in online learning for a broad class of functions and spaces, making it foundational for theoretical machine learning.
The paper proves that the nearest neighbor rule achieves online consistency for all measurable functions in doubling metric spaces, under the mild assumption that instances are generated by a process uniformly absolutely continuous with respect to a finite, upper doubling measure, eliminating the need for strong i.i.d. or geometric assumptions.
In the realizable online setting, a learner is tasked with making predictions for a stream of instances, where the correct answer is revealed after each prediction. A learning rule is online consistent if its mistake rate eventually vanishes. The nearest neighbor rule (Fix and Hodges, 1951) is a fundamental prediction strategy, but it is only known to be consistent under strong statistical or geometric assumptions: the instances come i.i.d. or the label classes are well-separated. We prove online consistency for all measurable functions in doubling metric spaces under the mild assumption that the instances are generated by a process that is uniformly absolutely continuous with respect to a finite, upper doubling measure.