We study the computational cost of differential privacy in terms of memory efficiency. While the trade-off between accuracy and differential privacy is well-understood, the inherent cost of privacy regarding memory use remains largely unexplored. This paper establishes for the first time an unconditional space lower bound for user-level differential privacy by introducing a novel proof technique based on a multi-player communication game. Central to our approach, this game formally links the hardness of low-memory private algorithms to the necessity of ``contribution capping'' -- tracking and limiting the users who disproportionately impact the dataset. We demonstrate that winning this communication game requires transmitting information proportional to the number of over-active users, which translates directly to memory lower bounds. We apply this framework, as an example, to the fundamental problem of estimating the number of distinct elements in a stream and we prove that any private algorithm requires almost $\widetildeΩ(T^{1/3})$ space to achieve certain error rates in a promise variant of the problem. This resolves an open problem in the literature (by Jain et al. NeurIPS 2023 and Cummings et al. ICML 2025) and establishes the first exponential separation between the space complexity of private algorithms and their non-private $\widetilde{O}(1)$ counterparts for a natural statistical estimation task. Furthermore, we show that this communication-theoretic technique generalizes to broad classes of problems, yielding lower bounds for private medians, quantiles, and max-select.

翻译：本研究从内存效率角度探讨差分隐私的计算代价。尽管精度与差分隐私之间的权衡关系已得到充分理解，但隐私保护在内存使用方面的固有代价仍鲜有研究。本文通过引入基于多参与者通信博弈的新型证明技术，首次建立了用户级差分隐私的无条件空间下界。该博弈的核心在于形式化地建立了低内存私有算法的困难度与"贡献上限"必要性之间的关联——即追踪并限制对数据集产生过度影响的用户。我们证明赢得该通信博弈需要传输与过度活跃用户数量成正比的信息量，这直接转化为内存下界。作为应用示例，我们将该框架应用于流数据中不同元素数量估计这一基础问题，证明在该问题的承诺变体中，任何私有算法都需要至少 $\widetildeΩ(T^{1/3})$ 空间才能达到特定误差率。这解决了文献中的开放问题（Jain等人NeurIPS 2023与Cummings等人ICML 2025），并首次针对自然统计估计任务，建立了私有算法与其非私有 $\widetilde{O}(1)$ 对应算法之间空间复杂度的指数级分离。此外，我们证明该通信理论技术可推广至更广泛的问题类别，为私有中位数、分位数及最大值选择等问题推导出相应的空间下界。