Improved Lower Bounds for Privacy under Continual Release

We study the problem of continually releasing statistics of an evolving dataset under differential privacy. In the event-level setting, we show the first polynomial lower bounds on the additive error for insertions-only graph problems such as maximum matching, degree histogram and $k$-core. This is an exponential improvement on the polylogarithmic lower bounds of Fichtenberger et al.[ESA 2021] for the former two problems, and are the first continual release lower bounds for the latter. Our results run counter to the intuition that the difference between insertions-only vs fully dynamic updates causes the gap between polylogarithmic and polynomial additive error. We show that for maximum matching and $k$-core, allowing small multiplicative approximations is what brings the additive error down to polylogarithmic. Beyond graph problems, our techniques also show that polynomial additive error is unavoidable for Simultaneous Norm Estimation in the insertions-only setting. When multiplicative approximations are allowed, we circumvent this lower bound by giving the first continual mechanism with polylogarithmic additive error under $(1+ζ)$ multiplicative approximations, for $ζ>0$, for estimating all monotone symmetric norms simultaneously. In the item-level setting, we show polynomial lower bounds on the product of the multiplicative and the additive error of continual mechanisms for a large range of graph problems. To the best of our knowledge, these are the first lower bounds for any differentially private continual release mechanism with multiplicative error. To obtain this, we prove a new lower bound on the product of multiplicative and additive error for 1-Way-Marginals, from which we reduce to continual graph problems. This generalizes the lower bounds of Hardt and Talwar[STOC 2010] and Bun et al.[STOC 2014] on the additive error for mechanisms with no multiplicative error.