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The Gaussian Mechanism (GM), which consists in adding Gaussian noise to avector-valued query before releasing it, is a standard privacy protectionmechanism. In particular, given that the query respects some L2 sensitivityproperty (the L2 distance between outputs on any two neighboring inputs isbounded), GM guarantees R\'enyi Differential Privacy (RDP). Unfortunately,precisely bounding the L2 sensitivity can be hard, thus leading to looseprivacy 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 ontheir norm. Leveraging this assumption, we introduce the Relative GaussianMechanism (RGM), in which the variance of the noise depends on the norm of theoutput. We prove tight bounds on the RDP parameters under relative L2sensitivity, and characterize the privacy loss incurred by usingoutput-dependent noise. In particular, we show that RGM naturally adapts to alatent variable that would control the norm of the output. Finally, weinstantiate our framework to show tight guarantees for Private GradientDescent, a problem that naturally fits our relative L2 sensitivity assumption.

The Relative Gaussian Mechanism and its Application to Private Gradient Descent