Infinitely divisible privacy and beyond I: resolution of the $s^2=2k$ conjecture
Aaradhya Pandey, Arian Maleki, Sanjeev Kulkarni
TL;DR
The paper develops an infinitely divisible privacy (IDP) framework for f-DP, showing that the limit of composing many nearly perfectly private mechanisms corresponds to an infinitely divisible log-likelihood ratio and a baseline trade-off f_infinity determined by a Lévy–Khintchine triplet with an Esscher tilt. This unifies GDP with non-Gaussian limits (notably Poisson) and provides explicit mechanisms achieving Poisson differential privacy for count statistics, while resolving the s^2=2k conjecture under contiguity. It also explores how the number of private operations can be random, potentially yielding limits outside the IDD class, and discusses practical implications for graph-structured data and Bayesian viewpoints on information coarsening. Together, these results broaden the scope of baseline privacy curves and offer a principled route to optimal DP mechanisms aligned with the underlying limit laws. The framework thus connects hypothesis-testing privacy, infinite divisibility, and composition to deliver a richer, more flexible privacy theory with concrete mechanisms and applications.
Abstract
Differential privacy is increasingly formalized through the lens of hypothesis testing via the robust and interpretable $f$-DP framework, where privacy guarantees are encoded by a baseline Blackwell trade-off function $f_{\infty} = T(P_{\infty}, Q_{\infty})$ involving a pair of distributions $(P_{\infty}, Q_{\infty})$. The problem of choosing the right privacy metric in practice leads to a central question: what is a statistically appropriate baseline $f_{\infty}$ given some prior modeling assumptions? The special case of Gaussian differential privacy (GDP) showed that, under compositions of nearly perfect mechanisms, these trade-off functions exhibit a central limit behavior with a Gaussian limit experiment. Inspired by Le Cam's theory of limits of statistical experiments, we answer this question in full generality in an infinitely divisible setting. We show that suitable composition experiments $(P_n^{\otimes n}, Q_n^{\otimes n})$ converge to a binary limit experiment $(P_{\infty}, Q_{\infty})$ whose log-likelihood ratio $L = \log(dQ_{\infty} / dP_{\infty})$ is infinitely divisible under $P_{\infty}$. Thus any limiting trade-off function $f_{\infty}$ is determined by an infinitely divisible law $P_{\infty}$, characterized by its Levy--Khintchine triplet, and its Esscher tilt defined by $dQ_{\infty}(x) = e^{x} dP_{\infty}(x)$. This characterizes all limiting baseline trade-off functions $f_{\infty}$ arising from compositions of nearly perfect differentially private mechanisms. Our framework recovers GDP as the purely Gaussian case and yields explicit non-Gaussian limits, including Poisson examples. It also positively resolves the empirical $s^2 = 2k$ phenomenon observed in the GDP paper and provides an optimal mechanism for count statistics achieving asymmetric Poisson differential privacy.