[Paper Review] An Uncertainty Principle is a Price of Privacy-Preserving Microdata

John M. Abowd, Robert Ashmead+8 more|arXiv (Cornell University)|Oct 25, 2021

Privacy-Preserving Technologies in Data32 references4 citations

TL;DR

This paper establishes an uncertainty principle in differential privacy that quantifies the inherent trade-off between accuracy for aggregate (sum) queries and sub-population (point) queries when publishing privacy-preserving microdata. It proves that requiring microdata output forces a logarithmic or polynomial degradation in error bounds—up to a log²d or d² factor—compared to non-microdata systems, even under pure, approximate, or concentrated differential privacy, with empirical validation on PUMS datasets.

ABSTRACT

Privacy-protected microdata are often the desired output of a differentially private algorithm since microdata is familiar and convenient for downstream users. However, there is a statistical price for this kind of convenience. We show that an uncertainty principle governs the trade-off between accuracy for a population of interest ("sum query") vs. accuracy for its component sub-populations ("point queries"). Compared to differentially private query answering systems that are not required to produce microdata, accuracy can degrade by a logarithmic factor. For example, in the case of pure differential privacy, without the microdata requirement, one can provide noisy answers to the sum query and all point queries while guaranteeing that each answer has squared error $O(1/ε^2)$. With the microdata requirement, one must choose between allowing an additional $\log^2(d)$ factor ($d$ is the number of point queries) for some point queries or allowing an extra $O(d^2)$ factor for the sum query. We present lower bounds for pure, approximate, and concentrated differential privacy. We propose mitigation strategies and create a collection of benchmark datasets that can be used for public study of this problem.

Motivation & Objective

• To identify and formalize the statistical cost of requiring differentially private algorithms to output microdata instead of just noisy query answers.
• To demonstrate that this requirement introduces an unavoidable trade-off between accuracy on population-level sum queries and sub-population point queries.
• To establish theoretical lower bounds on error degradation under pure, approximate, and concentrated differential privacy when microdata is mandated.
• To validate the theoretical findings with empirical benchmarks on real-world PUMS datasets using various noise mechanisms.
• To propose mitigation strategies and release public benchmark datasets for studying this trade-off.

Proposed method

• Formalizing the trade-off using per-query expected squared error as the primary metric, distinguishing it from simultaneous/outlier error used in prior work.
• Defining a set of d disjoint point queries and one sum query that aggregates them, modeling real-world use cases like census data or redistricting.
• Deriving theoretical lower bounds on error for both sum and point queries under pure, approximate, and concentrated differential privacy when microdata must be published.
• Using the Gauss mechanism with ρ = 0.005 to generate privacy-preserving microdata and comparing query errors across multiple algorithms (olsalg, nnlsalg, maxalg, seqalg, weightalg).
• Creating a public benchmark suite of PUMS datasets with computed squared errors and standard deviations for each query type and algorithm.
• Analyzing discrepancies between original noisy query answers and those derived from synthesized microdata to quantify error inflation.

Experimental results

Research questions

• RQ1 What is the fundamental accuracy cost of requiring differentially private algorithms to produce microdata instead of just noisy query answers?
• RQ2 How does the requirement to publish microdata affect the error bounds for sum queries versus their constituent point queries under differential privacy?
• RQ3 Can theoretical lower bounds be established for error degradation in pure, approximate, and concentrated differential privacy when microdata is required?
• RQ4 To what extent do real-world microdata synthesis methods amplify error compared to direct noisy query publishing?
• RQ5 What mitigation strategies can reduce the error trade-off between aggregate and sub-population accuracy in privacy-preserving microdata?

Key findings

• Requiring microdata output forces a logarithmic or polynomial degradation in error bounds: under pure differential privacy, sum query error can increase by a factor of O(d²), while point query errors can increase by log²d.
• Without the microdata requirement, each query (sum and point) can achieve O(1/ε²) squared error; with microdata, this guarantee is lost unless one query type pays a higher error penalty.
• Empirical results on PUMS datasets show that microdata synthesis leads to significantly higher query errors than the original noisy measurements, with squared errors ranging from ~300 to over 7,000 depending on the query and algorithm.
• The Gauss mechanism with ρ = 0.005 produced sum query squared errors between 323.4 and 1,661.8, with standard deviations from ±12.2 to ±283.4, indicating substantial variability across datasets and algorithms.
• The discrepancy between original measurement errors and derived microdata query errors confirms that microdata synthesis introduces systematic error inflation, especially for point queries.
• The benchmark datasets released in the study show that error inflation is not random but follows a predictable pattern tied to the query structure and privacy mechanism used.