One-Nearest-Neighbor Search is All You Need for Minimax Optimal Regression and Classification

• URL: http://arxiv.org/abs/2202.02464v1
• Date: Sat, 5 Feb 2022 01:59:09 GMT
• Title: One-Nearest-Neighbor Search is All You Need for Minimax Optimal Regression and Classification
• Authors: J. Jon Ryu and Young-Han Kim
• Abstract summary: This paper shows that the distributed algorithm with $k=1$ over a sufficiently large number of groups attains a minimax optimal error rate up to a multiplicative logarithmic factor under some regularity conditions. In the analysis, alternative rules with a refined aggregation method are proposed and shown to attain exact minimax optimal rates.
• Score: 22.98602552223454
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recently, Qiao, Duan, and Cheng~(2019) proposed a distributed nearest-neighbor classification method, in which a massive dataset is split into smaller groups, each processed with a $k$-nearest-neighbor classifier, and the final class label is predicted by a majority vote among these groupwise class labels. This paper shows that the distributed algorithm with $k=1$ over a sufficiently large number of groups attains a minimax optimal error rate up to a multiplicative logarithmic factor under some regularity conditions, for both regression and classification problems. Roughly speaking, distributed 1-nearest-neighbor rules with $M$ groups has a performance comparable to standard $\Theta(M)$-nearest-neighbor rules. In the analysis, alternative rules with a refined aggregation method are proposed and shown to attain exact minimax optimal rates.

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