Privacy Amplification via Shuffling: Unified, Simplified, and Tightened

Abstract: The shuffle model of differential privacy provides promising privacy-utility balances in decentralized, privacy-preserving data analysis. However, the current analyses of privacy amplification via shuffling lack both tightness and generality. To address this issue, we propose the \emph{variation-ratio reduction} as a comprehensive framework for privacy amplification in both single-message and multi-message shuffle protocols. It leverages two new parameterizations: the total variation bounds of local messages and the probability ratio bounds of blanket messages, to determine indistinguishability levels. Our theoretical results demonstrate that our framework provides tighter bounds, especially for local randomizers with extremal probability design, where our bounds are exactly tight. Additionally, variation-ratio reduction complements parallel composition in the shuffle model, yielding enhanced privacy accounting for popular sampling-based randomizers employed in statistical queries (e.g., range queries, marginal queries, and frequent itemset mining). Empirical findings demonstrate that our numerical amplification bounds surpass existing ones, conserving up to $30\%$ of the budget for single-message protocols, $75\%$ for multi-message ones, and a striking $75\%$-$95\%$ for parallel composition. Our bounds also result in a remarkably efficient $\tilde{O}(n)$ algorithm that numerically amplifies privacy in less than $10$ seconds for $n=10^8$ users.