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In some studies \citep[e.g.,][]{zhang2016understanding} of deep learning, itis observed that over-parametrized deep neural networks achieve a small testingerror even when the training error is almost zero. Despite numerous workstowards understanding this so-called "double descent" phenomenon\citep[e.g.,][]{belkin2018reconciling,belkin2019two}, in this paper, we turninto another way to enforce zero training error (without over-parametrization)through a data interpolation mechanism. Specifically, we consider a class ofinterpolated weighting schemes in the nearest neighbors (NN) algorithms. Bycarefully characterizing the multiplicative constant in the statistical risk,we reveal a U-shaped performance curve for the level of data interpolation inboth classification and regression setups. This sharpens the existing result\citep{belkin2018does} that zero training error does not necessarily jeopardizepredictive performances and claims a counter-intuitive result that a milddegree of data interpolation actually {\em strictly} improve the predictionperformance and statistical stability over those of the (un-interpolated)$k$-NN algorithm. In the end, the universality of our results, such as changeof distance measure and corrupted testing data, will also be discussed.

Benefit of Interpolation in Nearest Neighbor Algorithms