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.

翻译：本文研究了在旋转门流中不同项目数量的差分隐私持续发布问题，其中项目可能被插入也可能被删除。Jain、Kalemaj、Raskhodnikova、Sivakumar和Smith（NeurIPS '23）的最新工作表明，对于长度为$T$的流，即使没有任何空间限制，$Ω(T^{1/4})$的多项式加法误差也是不可避免的。我们证明，如果允许算法输出同时具有加法误差和乘法误差的估计值，则可以规避这一加法误差下界。我们提出了一种算法，用于持续发布不同元素的数量，其误差为$\text{polylog} (T)$乘法误差和$\text{polylog}(T)$加法误差。我们还展示了在估计旋转门流的$F_2$矩时存在类似的现象，我们可以获得$1+o(1)$乘法误差和$\text{polylog} (T)$加法误差。这两个结果都可以在多项式对数空间内实现，而先前的方法需要多项式空间。在亚线性空间机制下，即使不考虑隐私问题，一定的乘法误差也是必要的。我们提出了几个开放性问题，旨在更好地理解私有持续发布中乘法误差与加法误差之间的权衡关系。