The Relative Gaussian Mechanism and its Application to Private Gradient Descent

Hadrien Hendrikx, Paul Mangold, Aurélien Bellet
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:3079-3087, 2024.

Abstract

The Gaussian Mechanism (GM), which consists in adding Gaussian noise to a vector-valued query before releasing it, is a standard privacy protection mechanism. In particular, given that the query respects some L2 sensitivity property (the L2 distance between outputs on any two neighboring inputs is bounded), GM guarantees Rényi Differential Privacy (RDP). Unfortunately, precisely bounding the L2 sensitivity can be hard, thus leading to loose privacy bounds. In this work, we consider a Relative L2 sensitivity assumption, in which the bound on the distance between two query outputs may also depend on their norm. Leveraging this assumption, we introduce the Relative Gaussian Mechanism (RGM), in which the variance of the noise depends on the norm of the output. We prove tight bounds on the RDP parameters under relative L2 sensitivity, and characterize the privacy loss incurred by using output-dependent noise. In particular, we show that RGM naturally adapts to a latent variable that would control the norm of the output. Finally, we instantiate our framework to show tight guarantees for Private Gradient Descent, a problem that naturally fits our relative L2 sensitivity assumption.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-hendrikx24a, title = {The Relative {G}aussian Mechanism and its Application to Private Gradient Descent}, author = {Hendrikx, Hadrien and Mangold, Paul and Bellet, Aur\'{e}lien}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {3079--3087}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/hendrikx24a/hendrikx24a.pdf}, url = {https://proceedings.mlr.press/v238/hendrikx24a.html}, abstract = {The Gaussian Mechanism (GM), which consists in adding Gaussian noise to a vector-valued query before releasing it, is a standard privacy protection mechanism. In particular, given that the query respects some L2 sensitivity property (the L2 distance between outputs on any two neighboring inputs is bounded), GM guarantees Rényi Differential Privacy (RDP). Unfortunately, precisely bounding the L2 sensitivity can be hard, thus leading to loose privacy bounds. In this work, we consider a Relative L2 sensitivity assumption, in which the bound on the distance between two query outputs may also depend on their norm. Leveraging this assumption, we introduce the Relative Gaussian Mechanism (RGM), in which the variance of the noise depends on the norm of the output. We prove tight bounds on the RDP parameters under relative L2 sensitivity, and characterize the privacy loss incurred by using output-dependent noise. In particular, we show that RGM naturally adapts to a latent variable that would control the norm of the output. Finally, we instantiate our framework to show tight guarantees for Private Gradient Descent, a problem that naturally fits our relative L2 sensitivity assumption.} }
Endnote
%0 Conference Paper %T The Relative Gaussian Mechanism and its Application to Private Gradient Descent %A Hadrien Hendrikx %A Paul Mangold %A Aurélien Bellet %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-hendrikx24a %I PMLR %P 3079--3087 %U https://proceedings.mlr.press/v238/hendrikx24a.html %V 238 %X The Gaussian Mechanism (GM), which consists in adding Gaussian noise to a vector-valued query before releasing it, is a standard privacy protection mechanism. In particular, given that the query respects some L2 sensitivity property (the L2 distance between outputs on any two neighboring inputs is bounded), GM guarantees Rényi Differential Privacy (RDP). Unfortunately, precisely bounding the L2 sensitivity can be hard, thus leading to loose privacy bounds. In this work, we consider a Relative L2 sensitivity assumption, in which the bound on the distance between two query outputs may also depend on their norm. Leveraging this assumption, we introduce the Relative Gaussian Mechanism (RGM), in which the variance of the noise depends on the norm of the output. We prove tight bounds on the RDP parameters under relative L2 sensitivity, and characterize the privacy loss incurred by using output-dependent noise. In particular, we show that RGM naturally adapts to a latent variable that would control the norm of the output. Finally, we instantiate our framework to show tight guarantees for Private Gradient Descent, a problem that naturally fits our relative L2 sensitivity assumption.
APA
Hendrikx, H., Mangold, P. & Bellet, A.. (2024). The Relative Gaussian Mechanism and its Application to Private Gradient Descent. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:3079-3087 Available from https://proceedings.mlr.press/v238/hendrikx24a.html.

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