Keeping a Secret Requires a Good Memory: Space Lower-Bounds for Private Algorithms

2026-02-12

AI Summary

This work investigates the memory cost of user-level differentially private algorithms in the streaming model and establishes the first unconditional space complexity lower bound. By modeling the problem through a multi-player communication game, the authors link the inherent difficulty of low-memory private algorithms to the “contribution truncation” mechanism and introduce a novel proof technique based on communication complexity. For fundamental tasks such as distinct element counting, they prove that any user-level differentially private algorithm requires at least $\tilde{\Omega}(T^{1/3})$ space, revealing an exponential gap between private and non-private algorithms in space complexity. This result resolves open questions posed at NeurIPS 2023 and ICML 2025 and extends to other problems including median, quantile estimation, and maximum selection.

Technology Category

• Machine Learning: Privacy
• Data Mining & Knowledge Management: Data Stream Mining
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 $\widetilde{\Omega}(T^{1/3})$ 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.

Problem

• differential privacy
• space lower bounds
• memory efficiency
• user-level privacy
• private algorithms

Innovation

• differential privacy
• space lower bounds
• communication complexity
• contribution capping
• streaming algorithms