Tight Bounds for Differentially Private Anonymized Histograms

TL;DR

This paper establishes tight bounds for differentially private algorithms computing anonymized histograms, providing nearly matching upper and lower bounds on error in both low and high privacy regimes.

Contribution

It introduces new bounds that precisely characterize the error for DP anonymized histograms, improving understanding of privacy-utility trade-offs.

Findings

Expected error bound of O(√n / e^ε) for ε ≥ 1

Lower bound of Ω(√(n log(1/ε))/ε) for ε < 1

Bounds asymptotically match previous results from Suresh, NeurIPS 2019

Abstract

In this note, we consider the problem of differentially privately (DP) computing an anonymized histogram, which is defined as the multiset of counts of the input dataset (without bucket labels). In the low-privacy regime $\epsilon \geq 1$, we give an $\epsilon$-DP algorithm with an expected $\ell_1$-error bound of $O(\sqrt{n} / e^\epsilon)$. In the high-privacy regime $\epsilon < 1$, we give an $\Omega(\sqrt{n \log(1/\epsilon) / \epsilon})$ lower bound on the expected $\ell_1$ error. In both cases, our bounds asymptotically match the previously known lower/upper bounds due to [Suresh, NeurIPS 2019].