Improved Lower Bounds for Privacy under Continual Release

2025-12-17

AI Summary

This paper investigates fundamental error lower bounds for continual release of statistics over dynamic data streams under differential privacy, focusing on insertion-only graph problems—including maximum matching, $k$-core decomposition, and degree histogram—as well as monotone symmetric norm estimation. Methodologically, it integrates combinatorial lower bound constructions, reductions to 1-way marginals, differential privacy analysis, and dynamic graph algorithm theory. In the event-level setting, it establishes the first polynomial additive error lower bound, refuting the conventional intuition that insertion-only updates inherently admit smaller error than fully dynamic updates. In the item-level setting, it provides the first tight multiplicative-plus-additive error lower bound. Furthermore, the paper proposes the first continual mechanism achieving $(1+zeta)$-multiplicative approximation with polylogarithmic additive error, attaining optimal asymptotic accuracy for simultaneous norm estimation.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Establishes polynomial lower bounds for differential privacy in continual release of graph statistics.
• Demonstrates polynomial additive error is unavoidable for insertions-only dynamic graph problems.
• Shows multiplicative approximations reduce additive error to polylogarithmic for certain graph problems.

Innovation

• Polynomial lower bounds for insertions-only graph problems
• Polylogarithmic error with multiplicative approximations allowed
• Lower bounds on multiplicative-additive error product for item-level