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Skirting Additive Error Lower Bounds for Private Turnstile Streams

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
Differential privacystreamingdistinctelements
Abstract:

We study differentially private continual release of the number of distinct items in a stream, where items may be both inserted and deleted. In this turnstile setting, a recent work of Jain, Kalemaj, Raskhodnikova, Sivakumar, and Smith (NeurIPS '23) showed that for streams of length TT, polynomial additive error of Ω(T1/4)\Omega(T^{1/4}) is necessary, even without any space restrictions. We show that this additive error lower bound can be circumvented if the algorithm is allowed to output estimates with multiplicative error. We give an algorithm for the continual release of the number of distinct elements with polylog(T)\text{polylog} (T) multiplicative and polylog(T)\text{polylog}(T) additive error. We also show a qualitatively similar phenomenon for estimating the F2F_2 moment of a turnstile stream, where we can obtain 1+o(1)1+o(1) multiplicative and polylog(T)\text{polylog} (T) additive error. Both results can be achieved by polylogarithmic space streaming algorithms where some multiplicative error is necessary even without privacy. Lastly, we raise questions aimed at better understanding trade-offs between multiplicative and additive error in private continual estimation problems.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper contributes algorithms for continual distinct element counting and F2 moment estimation in turnstile streams, achieving polylogarithmic multiplicative and additive error under differential privacy. It resides in the 'Distinct Element Counting' leaf of the taxonomy, which contains five papers total, including the original work. This leaf sits within 'Core Statistical Estimation Problems,' a moderately populated branch addressing fundamental counting and moment estimation tasks. The research direction is relatively sparse compared to broader streaming or privacy literature, reflecting the technical difficulty of combining turnstile dynamics with continual privacy guarantees.

The taxonomy reveals that 'Distinct Element Counting' is one of three sibling categories under 'Core Statistical Estimation Problems,' alongside 'Histogram and Column Sum Maintenance' (four papers) and 'Moment Estimation and Low-Rank Factorization' (two papers). These neighboring leaves address related but distinct challenges: histogram maintenance focuses on frequency queries rather than cardinality, while moment estimation explores higher-order statistics. The paper's F2 moment result bridges into the moment estimation category, but its primary contribution remains anchored in distinct counting. The taxonomy's scope notes clarify that cardinality-only estimation excludes histogram queries, positioning this work at the boundary between counting primitives and richer statistical summaries.

Among seven candidates examined via semantic search, the distinct elements contribution shows one refutable candidate out of four examined, suggesting some prior overlap in this specific direction. The F2 moment contribution examined two candidates with none refutable, indicating relatively less prior work in that area. The reduction from sublinear additive to near-constant multiplicative error examined only one candidate with no refutation, reflecting limited exploration of this transformation. The search scope is modest—seven total candidates across three contributions—so these statistics capture a snapshot of closely related work rather than exhaustive coverage. The multiplicative-versus-additive error trade-off appears less explored in prior continual release literature based on this limited sample.

Given the restricted search scale and the sparse taxonomy leaf, the work appears to occupy a relatively underexplored niche within differentially private streaming. The one refutable candidate for distinct elements suggests incremental refinement rather than entirely novel territory, yet the multiplicative error focus and the F2 moment extension show substantive technical departures. The analysis covers top semantic matches and does not claim comprehensive field coverage, so adjacent work in broader streaming or privacy venues may exist outside this scope.

Taxonomy

Core-task Taxonomy Papers
32
3
Claimed Contributions
7
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: differentially private continual release in turnstile streams. This field addresses the challenge of publishing privacy-preserving statistics over data streams where elements can be both added and removed (turnstile model), with results released continuously over time. The taxonomy reveals a rich structure organized around five main branches: Core Statistical Estimation Problems focuses on fundamental tasks like distinct element counting and histogram maintenance under privacy constraints; Graph and Network Analysis extends these ideas to dynamic graph structures; Synthetic Data Generation and Publication explores methods for releasing entire synthetic datasets that preserve temporal patterns; Specialized Application Domains targets privacy in contexts such as trajectory data, wearable sensors, and real-time event streams; and Algorithmic Frameworks and Theoretical Foundations develops general-purpose mechanisms, sketching techniques, and composition bounds that underpin the entire area. Representative works like Private Linear Sketches[2] and Private Continual Histograms[1] illustrate how core primitives enable broader applications. A particularly active line of work centers on distinct element counting in turnstile streams, where methods must handle negative updates while maintaining tight privacy-utility trade-offs. Private Turnstile Streams[0] sits squarely within this cluster, addressing the technical challenges of continual cardinality estimation when the underlying set can shrink as well as grow. It shares close ties with Private Distinct Counting[3] and Private Turnstile Cardinality[4], which similarly tackle cardinality queries but may differ in their algorithmic approach or the granularity of continual releases. Meanwhile, works like Continual Distinct Counting[23] and Private Cardinality Releases[6] explore related estimation problems, highlighting ongoing questions about optimal noise calibration, composition strategies, and the interplay between streaming constraints and privacy guarantees. The original paper's emphasis on turnstile dynamics positions it at the intersection of classical streaming algorithms and modern differential privacy, contributing to a small but growing body of work that reconciles these two demanding computational models.

Claimed Contributions

Polylogarithmic multiplicative and additive error algorithm for continual distinct elements estimation

The authors present a differentially private streaming algorithm that estimates the number of distinct elements in turnstile streams with both multiplicative and additive error bounded by polylogarithmic factors in the stream length T, circumventing the polynomial additive error lower bound of Ω(T^(1/4)) that applies when only additive error is allowed.

4 retrieved papers
Can Refute
Near-optimal multiplicative and additive error algorithm for continual F2 moment estimation

The authors develop a differentially private algorithm for estimating the F2 moment (sum of squared frequencies) in turnstile streams that achieves (1+η) multiplicative error and polylogarithmic additive error for any constant η, improving upon the Ω(T) additive error lower bound that applies to purely additive error algorithms.

2 retrieved papers
Reduction from sublinear additive error to near-constant multiplicative error for distinct elements

The authors establish a reduction showing that any differentially private algorithm achieving sublinear additive error in the domain size n for continual distinct elements can be converted to an algorithm with (1+η) multiplicative error and polylogarithmic additive error for any constant η, demonstrating a fundamental connection between additive and multiplicative error regimes.

1 retrieved paper

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Polylogarithmic multiplicative and additive error algorithm for continual distinct elements estimation

The authors present a differentially private streaming algorithm that estimates the number of distinct elements in turnstile streams with both multiplicative and additive error bounded by polylogarithmic factors in the stream length T, circumventing the polynomial additive error lower bound of Ω(T^(1/4)) that applies when only additive error is allowed.

Contribution

Near-optimal multiplicative and additive error algorithm for continual F2 moment estimation

The authors develop a differentially private algorithm for estimating the F2 moment (sum of squared frequencies) in turnstile streams that achieves (1+η) multiplicative error and polylogarithmic additive error for any constant η, improving upon the Ω(T) additive error lower bound that applies to purely additive error algorithms.

Contribution

Reduction from sublinear additive error to near-constant multiplicative error for distinct elements

The authors establish a reduction showing that any differentially private algorithm achieving sublinear additive error in the domain size n for continual distinct elements can be converted to an algorithm with (1+η) multiplicative error and polylogarithmic additive error for any constant η, demonstrating a fundamental connection between additive and multiplicative error regimes.

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