Notes on Sampled Gaussian Mechanism

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, _eff = (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.

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