  Nearest neighbor methods are a popular class of nonparametric estimators with
several desirable properties, such as adaptivity to different distance scales
in different regions of space. Prior work on convergence rates for nearest
neighbor classification has not fully reflected these subtle properties. We
analyze the behavior of these estimators in metric spaces and provide
finite-sample, distribution-dependent rates of convergence under minimal
assumptions. As a by-product, we are able to establish the universal
consistency of nearest neighbor in a broader range of data spaces than was
previously known. We illustrate our upper and lower bounds by introducing
smoothness classes that are customized for nearest neighbor classification.
