Differentially Private Chi-squared Test by Unit Circle Mechanism
Kazuya Kakizaki, Kazuto Fukuchi, Jun Sakuma
Abstract
This paper develops differentially private mechanisms for $\chi^2$ test of independence. While existing works put their effort into properly controlling the type-I error, in addition to that, we investigate the type-II error of differentially private mechanisms. Based on the analysis, we present unit circle mechanism: a novel differentially private mechanism based on the geometrical property of the test statistics. Compared to existing output perturbation mechanisms, our mechanism improves the dominated term of the type-II error from $O(1)$ to $O(\exp(-\sqrt{N}))$ where $N$ is the sample size. Furthermore, we introduce novel procedures for multiple $\chi^2$ tests by incorporating the unit circle mechanism into the sparse vector technique and the exponential mechanism. These procedures can control the family-wise error rate (FWER) properly, which has never been attained by existing mechanisms.
BibTeX
@InProceedings{pmlr-v70-kakizaki17a,
title = {Differentially Private Chi-squared Test by Unit Circle Mechanism},
author = {Kazuya Kakizaki and Kazuto Fukuchi and Jun Sakuma},
booktitle = {Proceedings of the 34th International Conference on Machine Learning},
pages = {1761--1770},
year = {2017},
editor = {Precup, Doina and Teh, Yee Whye},
volume = {70},
series = {Proceedings of Machine Learning Research},
month = {06--11 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v70/kakizaki17a/kakizaki17a.pdf},
url = {https://proceedings.mlr.press/v70/kakizaki17a.html},
abstract = {This paper develops differentially private mechanisms for $\chi^2$ test of independence. While existing works put their effort into properly controlling the type-I error, in addition to that, we investigate the type-II error of differentially private mechanisms. Based on the analysis, we present unit circle mechanism: a novel differentially private mechanism based on the geometrical property of the test statistics. Compared to existing output perturbation mechanisms, our mechanism improves the dominated term of the type-II error from $O(1)$ to $O(\exp(-\sqrt{N}))$ where $N$ is the sample size. Furthermore, we introduce novel procedures for multiple $\chi^2$ tests by incorporating the unit circle mechanism into the sparse vector technique and the exponential mechanism. These procedures can control the family-wise error rate (FWER) properly, which has never been attained by existing mechanisms.}
}