Predictive Power of Nearest Neighbors Algorithm under Random Perturbation

TL;DR

This paper analyzes how random perturbations in test data affect the asymptotic regret of the k-NN algorithm, revealing a phase transition and limitations of noise-injection methods in high-dimensional settings.

Contribution

It characterizes the impact of data corruption on k-NN's regret, identifies a phase transition phenomenon, and shows the limited effectiveness of noise-injection in high dimensions.

Findings

Asymptotic regret remains stable for small corruption levels.

Regret deteriorates polynomially when corruption exceeds a critical level.

Noise-injection does not improve performance in the large corruption regime.

Abstract

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