A Two-Stage Active Learning Algorithm for $k$-Nearest Neighbors

We introduce a simple and intuitive two-stage active learning algorithm for the training of $k$-nearest neighbors classifiers. We provide consistency guarantees for a modified $k$-nearest neighbors classifier trained on samples acquired via our scheme, and show that when the conditional probability function $\mathbb{P}(Y=y|X=x)$ is sufficiently smooth and the Tsybakov noise condition holds, our actively trained classifiers converge to the Bayes optimal classifier at a faster asymptotic rate than passively trained $k$-nearest neighbor classifiers.