Notes on Sampled Gaussian Mechanism

TL;DR

This paper rigorously proves a conjecture about the Sampled Gaussian Mechanism, showing that increasing subsampling rates improves privacy-utility trade-offs by reducing effective noise levels.

Contribution

It provides a rigorous proof of a conjecture related to the effective noise level in the Sampled Gaussian Mechanism, completing the proof of a key theorem.

Findings

Effective noise level decreases with higher subsampling rates

Larger subsampling rates lead to better privacy-utility trade-offs

Completes the proof of a significant theorem in differential privacy

Abstract

In these notes, we prove a recent conjecture posed in the paper by R\"ais\"a, O. et al. [Subsampling is not Magic: Why Large Batch Sizes Work for Differentially Private Stochastic Optimization (2024)]. Theorem 6.2 of the paper asserts that for the Sampled Gaussian Mechanism - a composition of subsampling and additive Gaussian noise, the effective noise level, $\sigma_{\text{eff}} = \frac{\sigma(q)}{q}$, decreases as a function of the subsampling rate $q$. Consequently, larger subsampling rates are preferred for better privacy-utility trade-offs. Our notes provide a rigorous proof of Conjecture 6.3, which was left unresolved in the original paper, thereby completing the proof of Theorem 6.2.