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.

翻译：在可实现在线学习场景中，学习者需要对连续输入的实例流进行预测，每次预测后会立即揭示正确答案。若某学习规则的错误率最终趋近于零，则称其具有在线一致性。最近邻规则（Fix and Hodges, 1951）是一种基础预测策略，但此前仅知其在强统计或几何假设下具有一致性：要求实例独立同分布或标签类别具有良好分离性。本文证明，在加倍度量空间中，只要实例生成过程相对于某个有限上加倍测度满足一致绝对连续性这一温和假设，最近邻规则对所有可测函数均具有在线一致性。