Skirting Additive Error Barriers for Private Turnstile Streams
Anders Aamand, Justin Y. Chen, Sandeep Silwal
TL;DR
This work shows that polynomial additive lower bounds for private continual release in turnstile streams can be circumvented by tolerating a small multiplicative error. By combining differential privacy with private continual counting as a primitive and employing MinHash and domain-reduction techniques, the authors obtain polylogarithmic additive and multiplicative errors for counting distinct elements in polylogarithmic space, and achieve 1+o(1) multiplicative accuracy with polylogarithmic additive error for the F2 moment estimation. The results extend to general turnstile streams and rely on Johnson–Lindenstrauss reductions to a small domain, enabling efficient private continual counting in reduced spaces. These advances improve the privacy-utility-space trade-offs for foundational streaming problems and raise open questions about optimal multiplicative/additive tradeoffs and extensions to other statistics like triangle counts.
Abstract
We study differentially private continual release of the number of distinct items in a turnstile stream, where items may be both inserted and deleted. A recent work of Jain, Kalemaj, Raskhodnikova, Sivakumar, and Smith (NeurIPS '23) shows that for streams of length $T$, polynomial additive error of $Ω(T^{1/4})$ is necessary, even without any space restrictions. We show that this additive error lower bound can be circumvented if the algorithm is allowed to output estimates with both additive \emph{and multiplicative} error. We give an algorithm for the continual release of the number of distinct elements with $\text{polylog} (T)$ multiplicative and $\text{polylog}(T)$ additive error. We also show a qualitatively similar phenomenon for estimating the $F_2$ moment of a turnstile stream, where we can obtain $1+o(1)$ multiplicative and $\text{polylog} (T)$ additive error. Both results can be achieved using polylogarithmic space whereas prior approaches use polynomial space. In the sublinear space regime, some multiplicative error is necessary even if privacy is not a consideration. We raise several open questions aimed at better understanding trade-offs between multiplicative and additive error in private continual release.