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Recently, Qiao, Duan, and Cheng~(2019) proposed a distributednearest-neighbor classification method, in which a massive dataset is splitinto smaller groups, each processed with a $k$-nearest-neighbor classifier, andthe final class label is predicted by a majority vote among these groupwiseclass labels. This paper shows that the distributed algorithm with $k=1$ over asufficiently large number of groups attains a minimax optimal error rate up toa multiplicative logarithmic factor under some regularity conditions, for bothregression and classification problems. Roughly speaking, distributed1-nearest-neighbor rules with $M$ groups has a performance comparable tostandard $\Theta(M)$-nearest-neighbor rules. In the analysis, alternative ruleswith a refined aggregation method are proposed and shown to attain exactminimax optimal rates.

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