  The over-parameterized models attract much attention in the era of data
science and deep learning. It is empirically observed that although these
models, e.g. deep neural networks, over-fit the training data, they can still
achieve small testing error, and sometimes even {\em outperform} traditional
algorithms which are designed to avoid over-fitting. The major goal of this
work is to sharply quantify the benefit of data interpolation in the context of
nearest neighbors (NN) algorithm. Specifically, we consider a class of
interpolated weighting schemes and then carefully characterize their asymptotic
performances. Our analysis reveals a U-shaped performance curve with respect to
the level of data interpolation, and proves that a mild degree of data
interpolation {\em strictly} improves the prediction accuracy and statistical
stability over those of the (un-interpolated) optimal $k$NN algorithm. This
theoretically justifies (predicts) the existence of the second U-shaped curve
in the recently discovered double descent phenomenon. Note that our goal in
this study is not to promote the use of interpolated-NN method, but to obtain
theoretical insights on data interpolation inspired by the aforementioned
phenomenon.
