Differentially Private Chi-squared Test by Unit Circle Mechanism

Kazuya Kakizaki, Kazuto Fukuchi, Jun Sakuma
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1761-1770, 2017.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Differentially Private Chi-squared Test by Unit Circle Mechanism %A Kazuya Kakizaki %A Kazuto Fukuchi %A Jun Sakuma %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-kakizaki17a %I PMLR %P 1761--1770 %U https://proceedings.mlr.press/v70/kakizaki17a.html %V 70 %X 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.
APA
Kakizaki, K., Fukuchi, K. & Sakuma, J.. (2017). Differentially Private Chi-squared Test by Unit Circle Mechanism. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:1761-1770 Available from https://proceedings.mlr.press/v70/kakizaki17a.html.

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