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We consider a data corruption scenario in the classical $k$ Nearest Neighbors($k$-NN) algorithm, that is, the testing data are randomly perturbed. Undersuch a scenario, the impact of corruption level on the asymptotic regret iscarefully characterized. In particular, our theoretical analysis reveals aphase transition phenomenon that, when the corruption level $\omega$ is below acritical order (i.e., small-$\omega$ regime), the asymptotic regret remains thesame; when it is beyond that order (i.e., large-$\omega$ regime), theasymptotic regret deteriorates polynomially. Surprisingly, we obtain a negativeresult that the classical noise-injection approach will not help improve thetesting performance in the beginning stage of the large-$\omega$ regime, evenin the level of the multiplicative constant of asymptotic regret. As atechnical by-product, we prove that under different model assumptions, thepre-processed 1-NN proposed in \cite{xue2017achieving} will at most achieve asub-optimal rate when the data dimension $d>4$ even if $k$ is chosen optimallyin the pre-processing step.

Predictive Power of Nearest Neighbors Algorithm under Random Perturbation