Benefit of Interpolation in Nearest Neighbor Algorithms

• URL: http://arxiv.org/abs/2202.11817v1
• Date: Wed, 23 Feb 2022 22:47:18 GMT
• Title: Benefit of Interpolation in Nearest Neighbor Algorithms
• Authors: Yue Xing, Qifan Song, Guang Cheng
• Abstract summary: In some studies, it is observed that over-parametrized deep neural networks achieve a small testing error even when the training error is almost zero. We turn into another way to enforce zero training error (without over-parametrization) through a data mechanism.
• Score: 21.79888306754263
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In some studies \citep[e.g.,][]{zhang2016understanding} of deep learning, it is observed that over-parametrized deep neural networks achieve a small testing error even when the training error is almost zero. Despite numerous works towards understanding this so-called "double descent" phenomenon \citep[e.g.,][]{belkin2018reconciling,belkin2019two}, in this paper, we turn into another way to enforce zero training error (without over-parametrization) through a data interpolation mechanism. Specifically, we consider a class of interpolated weighting schemes in the nearest neighbors (NN) algorithms. By carefully characterizing the multiplicative constant in the statistical risk, we reveal a U-shaped performance curve for the level of data interpolation in both classification and regression setups. This sharpens the existing result \citep{belkin2018does} that zero training error does not necessarily jeopardize predictive performances and claims a counter-intuitive result that a mild degree of data interpolation actually {\em strictly} improve the prediction performance and statistical stability over those of the (un-interpolated) $k$-NN algorithm. In the end, the universality of our results, such as change of distance measure and corrupted testing data, will also be discussed.

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