Fast And Bayes-consistent Nearest Neighbors

Efremenko Klim, Kontorovich Aryeh, Noivirt Moshe. Arxiv 2019

[Paper]    
ARXIV Independent LSH

Research on nearest-neighbor methods tends to focus somewhat dichotomously either on the statistical or the computational aspects – either on, say, Bayes consistency and rates of convergence or on techniques for speeding up the proximity search. This paper aims at bridging these realms: to reap the advantages of fast evaluation time while maintaining Bayes consistency, and further without sacrificing too much in the risk decay rate. We combine the locality-sensitive hashing (LSH) technique with a novel missing-mass argument to obtain a fast and Bayes-consistent classifier. Our algorithm’s prediction runtime compares favorably against state of the art approximate NN methods, while maintaining Bayes-consistency and attaining rates comparable to minimax. On samples of size $n$ in ${\text{R}}^d$, our pre-processing phase has runtime $O(dnlogn)$, while the evaluation phase has runtime $O(dlogn)$ per query point.

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