Universal Shuffle Asymptotics: Sharp Privacy Analysis in the Gaussian Regime

TL;DR

This paper provides a precise privacy analysis for shuffling mechanisms in the Gaussian regime, deriving exact formulas and asymptotic bounds for differential privacy using advanced statistical tools.

Contribution

It introduces a comprehensive, sharp theoretical framework for privacy amplification by shuffling, including exact likelihood identities, divergence expansions, and Gaussian DP equivalences.

Findings

Exact likelihood-ratio identities for shuffled histograms

Universal leading constant for Jensen-Shannon divergence in shuffling

Full limiting (epsilon, delta) privacy curve derived

Abstract

We develop a sharp, experiment-level privacy theory for amplification by shuffling in the Gaussian regime: a fixed finite-output local randomizer with full support and neighboring binary datasets differing in one user. We first prove exact likelihood-ratio identities for shuffled histograms and a complete conditional-expectation linearization theorem with explicit typical-set remainders. We then derive sharp Jensen-Shannon divergence expansions, identifying the universal leading constant I_pi/(8n) for proportional compositions and emphasizing the correct fixed-composition covariance Sigma_pi=(1-pi)Sigma_0+pi*Sigma_1. Next we establish equivalence to Gaussian Differential Privacy with Berry-Esseen bounds and obtain the full limiting (epsilon,delta) privacy curve. Finally, for unbundled multi-message shuffling we give an exact degree-m likelihood ratio, asymptotic GDP formulas, and a strict bundled-versus-unbundled comparison. Further results include Local Asymptotic Normality with quantitative Le Cam equivalence, exact finite-n privacy curves, a boundary Berry-Esseen theorem for randomized response, and a frequency-estimation application.