Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on Sampled Gaussian Mechanism

Nikita P. Kalinin

TL;DR

The 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.

Abstract

In these notes, we prove a recent conjecture posed in the paper by Räisä, 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, $σ_{\text{eff}} = \frac{σ(q)}{q}$, 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.

Notes on Sampled Gaussian Mechanism

TL;DR

The 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.

Abstract

In these notes, we prove a recent conjecture posed in the paper by Räisä, 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, , decreases as a function of the subsampling rate . 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.
Paper Structure (3 sections, 2 theorems, 18 equations)

This paper contains 3 sections, 2 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Sampled Gaussian Mechanism
  3. Proof of Conjecture

Key Result

Lemma 1

For any $\epsilon > 0$ and $q\in(0,1]$, $\Psi_{\epsilon,q}(\sigma)$ is an invertable and strictly monotonically decreasing function with respect to $\sigma\in\mathbb{R}_+$.

Theorems & Definitions (5)

  • Conjecture
  • Lemma 1: Monotonicity of $\Psi$
  • proof
  • Lemma 2
  • proof