k-NN Regression Adapts to Local Intrinsic Dimension

Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that $k$-NN regression is also adaptive to intrinsic dimension. In particular our rates are local to a query $x$ and depend only on the way masses of balls centered at $x$ vary with radius. Furthermore, we show a simple way to choose $k = k(x)$ locally at any $x$ so as to nearly achieve the minimax rate at $x$ in terms of the unknown intrinsic dimension in the vicinity of $x$. We also establish that the minimax rate does not depend on a particular choice of metric space or distribution, but rather that this minimax rate holds for any metric space and doubling measure.