Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA

Jordan Awan, Ana Kenney, Matthew Reimherr, Aleksandra Slavković
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:374-384, 2019.

Abstract

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that the mechanism must be designed with respect to a specific base measure over the output space, such as a Gaussian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide a negative result, showing that the magnitude of noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an $\ep$-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces, and demonstrate its performance via simulations and applications to two datasets.

Cite this Paper


BibTeX
@InProceedings{pmlr-v97-awan19a, title = {Benefits and Pitfalls of the Exponential Mechanism with Applications to {H}ilbert Spaces and Functional {PCA}}, author = {Awan, Jordan and Kenney, Ana and Reimherr, Matthew and Slavkovi{\'c}, Aleksandra}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {374--384}, year = {2019}, editor = {Kamalika Chaudhuri and Ruslan Salakhutdinov}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/awan19a/awan19a.pdf}, url = { http://proceedings.mlr.press/v97/awan19a.html }, abstract = {The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that the mechanism must be designed with respect to a specific base measure over the output space, such as a Gaussian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide a negative result, showing that the magnitude of noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an $\ep$-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces, and demonstrate its performance via simulations and applications to two datasets.} }
Endnote
%0 Conference Paper %T Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA %A Jordan Awan %A Ana Kenney %A Matthew Reimherr %A Aleksandra Slavković %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-awan19a %I PMLR %P 374--384 %U http://proceedings.mlr.press/v97/awan19a.html %V 97 %X The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that the mechanism must be designed with respect to a specific base measure over the output space, such as a Gaussian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide a negative result, showing that the magnitude of noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an $\ep$-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces, and demonstrate its performance via simulations and applications to two datasets.
APA
Awan, J., Kenney, A., Reimherr, M. & Slavković, A.. (2019). Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:374-384 Available from http://proceedings.mlr.press/v97/awan19a.html .

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