Skirting Additive Error Barriers for Private Turnstile Streams

TL;DR

This paper introduces algorithms for differentially private continual release of stream statistics that achieve polylogarithmic error bounds by allowing both additive and multiplicative errors, overcoming previous additive error barriers.

Contribution

It presents novel algorithms that break the additive error lower bounds for private turnstile streams by incorporating multiplicative error, using only polylogarithmic space.

Findings

Achieves polylogarithmic multiplicative and additive error for distinct count estimation.

Provides similar results for $F_2$ moment estimation with near-constant multiplicative error.

Uses polylogarithmic space, improving over prior polynomial space methods.

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 $\Omega(T^{1/4})$ 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.