An adaptive nearest neighbor rule for classification

We introduce a variant of the k-nearest neighbor classifier in which k is chosen adaptively for each query, rather than being supplied as a parameter. The choice of k depends on properties of each neighborhood, and therefore may significantly vary between different points. For example, the algorithm will use larger k for predicting the labels of points in noisy regions. We provide theory and experiments that demonstrate that the algorithm performs comparably to, and sometimes better than, k-NN with an optimal choice of k. In particular, we bound the convergence rate of our classifier in terms of a local quantity we call the "advantage", giving results that are both more general and more accurate than the smoothness-based bounds of earlier nearest neighbor work. Our analysis uses a variant of the uniform convergence theorem of Vapnik-Chervonenkis that is for empirical estimates of conditional probabilities and may be of independent interest.