### Abstract

A binary classification problem is considered. The excess error probability of the k-nearest-neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation errors. Under a weak margin condition and under a modified Lipschitz condition or a local Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded. The concept of modified Lipschitz condition is applied for discrete distributions, too. As a consequence of both concepts, we present the rate of convergence of L_{2} error for the corresponding nearest neighbor regression estimate.

Original language | English |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Journal of Machine Learning Research |

Volume | 18 |

Publication status | Published - Jun 1 2018 |

### Fingerprint

### Keywords

- Classification
- Error probability
- K-nearest-neighbor rule
- Rate of convergence

### ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence

### Cite this

*Journal of Machine Learning Research*,

*18*, 1-16.

**Rate of convergence of k-nearest-neighbor classification rule.** / Döring, Maik; Györfi, L.; Walk, Harro.

Research output: Contribution to journal › Article

*Journal of Machine Learning Research*, vol. 18, pp. 1-16.

}

TY - JOUR

T1 - Rate of convergence of k-nearest-neighbor classification rule

AU - Döring, Maik

AU - Györfi, L.

AU - Walk, Harro

PY - 2018/6/1

Y1 - 2018/6/1

N2 - A binary classification problem is considered. The excess error probability of the k-nearest-neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation errors. Under a weak margin condition and under a modified Lipschitz condition or a local Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded. The concept of modified Lipschitz condition is applied for discrete distributions, too. As a consequence of both concepts, we present the rate of convergence of L2 error for the corresponding nearest neighbor regression estimate.

AB - A binary classification problem is considered. The excess error probability of the k-nearest-neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation errors. Under a weak margin condition and under a modified Lipschitz condition or a local Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded. The concept of modified Lipschitz condition is applied for discrete distributions, too. As a consequence of both concepts, we present the rate of convergence of L2 error for the corresponding nearest neighbor regression estimate.

KW - Classification

KW - Error probability

KW - K-nearest-neighbor rule

KW - Rate of convergence

UR - http://www.scopus.com/inward/record.url?scp=85052018548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052018548&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85052018548

VL - 18

SP - 1

EP - 16

JO - Journal of Machine Learning Research

JF - Journal of Machine Learning Research

SN - 1532-4435

ER -