Improved Lower Bounds for Privacy under Continual Release

TL;DR

The paper investigates the privacy-utility tradeoffs in continual data release under differential privacy, focusing on event-level versus item-level privacy for evolving graphs and simultaneous norm estimation. It introduces gadget-based inner-product reductions to establish polynomial additive-error lower bounds for event-level insertions-only problems (e.g., maximum matching, degree histogram, k-core) and extends these techniques to Simultaneous Norm Estimation, showing limitations even when multiple norms are queried. It also demonstrates that allowing small multiplicative error enables polylog additive error for monotone symmetric norms, while item-level settings incur strong polynomial lower-bound products between multiplicative and additive errors, reinforcing a dichotomy between event-level and item-level privacy in continual release. The results are complemented by improved incremental upper bounds, tightness results, and a general lower-bound framework that deepens our understanding of what private continual release can achieve for graph problems and norm estimation.

Abstract

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.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5