Optimal choice of $k$ for $k$-nearest neighbor regression

The $k$-nearest neighbor algorithm ($k$-NN) is a widely used non-parametric method for classification and regression. We study the mean squared error of the $k$-NN estimator when $k$ is chosen by leave-one-out cross-validation (LOOCV). Although it was known that this choice of $k$ is asymptotically consistent, it was not known previously that it is an optimal $k$. We show, with high probability, the mean squared error of this estimator is close to the minimum mean squared error using the $k$-NN estimate, where the minimum is over all choices of $k$.