Skirting Additive Error Lower Bounds for Private Turnstile Streams

We study differentially private continual release of the number of distinct items in a stream, where items may be both inserted and deleted. In this turnstile setting, a recent work of Jain, Kalemaj, Raskhodnikova, Sivakumar, and Smith (NeurIPS '23) showed 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 *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 by polylogarithmic space streaming algorithms where some multiplicative error is necessary even without privacy. Lastly, we raise questions aimed at better understanding trade-offs between multiplicative and additive error in private continual estimation problems.