An Adaptive Nearest Neighbor Rule For Classification

Abstract

We introduce a variant of the \(k\)-nearest neighbor classifier in which \(k\) is chosen adaptively for each query, rather than 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 derive bounds on the convergence rates of our classifier that depend on a local quantity we call the `advantage' which is significantly weaker than the Lipschitz conditions used in previous convergence rate proofs. These generalization bounds hinge on a variant of the seminal Uniform Convergence Theorem due to Vapnik and Chervonenkis; this variant concerns conditional probabilities and may be of independent interest.

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