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.

We introduce an adaptive nearest neighbor classification rule.

This is indeed an omission, but it is not terribly complicated and we will add it to the paper.

After reading the author feedback, I am increasing my score as the authors have largely addressed my main concern of comparison/relating to existing locally adaptive NN estimators. The authors address this concern on the theory side. I still think additional numerical experiments to compare the proposed method with other adaptive NN approaches would strengthen the paper and be beneficial to practitioners (and could possibly lead to interesting theoretical questions if, for instance, there are some surprising performance gaps). The analysis is quite clean and elegant due to the newly introduced local notion of "advantage" as an alternative to the usual local Lipschitz conditions. The paper, however, does not discuss enough related work, especially adaptive nearest neighbor and kernel methods based on Lepski's method (for example, the adaptive k-NN method by Kpotufe (2011)).

The paper proposes a variant of the k-Nearest Neighbors algorithm (called adaptive knn) in which k is chosen for each example to classify, instead of being tuned as a global hyperparameter. To do so, the authors define a new notion that applied locally in the input space they call the advantage, instead of the local Lipschitz condition that is often used in such setting. An important contribution of the paper is the prove that the proposed algorithm is consistent and have pointwise convergence at the limit. The proposed notion of advantage is allers related to some error bounds for pointwise convergence. The experimental part is clearly sufficient for this type of paper, even if there is no comparison with other state-of-the-art algorithm.

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. 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.

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. 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.

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.