The Relative Gaussian Mechanism and its Application to Private Gradient Descent

TL;DR

This paper introduces the Relative Gaussian Mechanism (RGM), which adapts noise variance based on output norms under a relative L2 sensitivity assumption, providing tighter privacy guarantees especially for private gradient descent.

Contribution

The paper proposes RGM, a novel privacy mechanism that uses output-dependent noise based on relative L2 sensitivity, with tight RDP bounds and applications to private gradient descent.

Findings

RGM achieves tighter privacy bounds than traditional Gaussian Mechanism.

The framework adapts to latent variables controlling output norms.

Application to private gradient descent yields improved privacy guarantees.

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 R\'enyi 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.