[Paper Review] Subspace Differential Privacy

Jie Gao, Ruobin Gong+1 more|arXiv (Cornell University)|Aug 26, 2021

Privacy-Preserving Technologies in Data4 citations

TL;DR

This paper introduces subspace differential privacy, a framework that integrates mandatory data invariants—such as population totals or spatio-temporal constraints—into differentially private mechanisms from the outset. By projecting noise from standard mechanisms (e.g., Laplace or Gaussian) onto invariant subspaces, the method ensures unbiased, transparent, and post-processing-free outputs while preserving formal privacy guarantees.

ABSTRACT

Many data applications have certain invariant constraints due to practical needs. Data curators who employ differential privacy need to respect such constraints on the sanitized data product as a primary utility requirement. Invariants challenge the formulation, implementation, and interpretation of privacy guarantees. We propose subspace differential privacy, to honestly characterize the dependence of the sanitized output on confidential aspects of the data. We discuss two design frameworks that convert well-known differentially private mechanisms, such as the Gaussian and the Laplace mechanisms, to subspace differentially private ones that respect the invariants specified by the curator. For linear queries, we discuss the design of near-optimal mechanisms that minimize the mean squared error. Subspace differentially private mechanisms rid the need for post-processing due to invariants, preserve transparency and statistical intelligibility of the output, and can be suitable for distributed implementation. We showcase the proposed mechanisms on the 2020 Census Disclosure Avoidance demonstration data, and a spatio-temporal dataset of mobile access point connections on a large university campus.

Motivation & Objective

• To address the challenge of enforcing hard invariants—such as population totals or sum constraints—in differentially private data releases without introducing bias.
• To eliminate the need for post-processing, which often distorts results and obscures privacy guarantees.
• To develop mechanisms that are statistically transparent, unbiased, and suitable for distributed deployment.
• To extend differential privacy to settings where data must satisfy linear equality constraints, such as contingency tables or time-series aggregates.
• To demonstrate the framework’s utility and accuracy on real-world datasets, including 2020 Census DAS data and a university WiFi spatio-temporal dataset.

Proposed method

• Proposes a new privacy model—subspace differential privacy—that explicitly incorporates linear equality invariants into the privacy mechanism design.
• Designs two general frameworks: projection and extension, to transform standard differentially private mechanisms (e.g., Laplace, Gaussian) into subspace DP-compliant ones.
• Applies projection to noise variables drawn from unconstrained distributions, ensuring the final output lies within the invariant subspace.
• For linear queries, derives near-optimal mechanisms that minimize mean squared error under the subspace constraint.
• Uses the projected Laplace and projected Gaussian mechanisms as core building blocks, with theoretical guarantees on privacy and unbiasedness.
• Employs distance minimization in the design phase rather than post-processing, avoiding data-dependent bias.

Experimental results

Research questions

• RQ1 How can differential privacy be formally extended to respect hard invariants such as population totals or sum constraints?
• RQ2 What is the impact of post-processing on bias and privacy interpretation when invariants are enforced after noise injection?
• RQ3 Can differentially private mechanisms be designed to be unbiased while still satisfying linear equality constraints?
• RQ4 How does the error scale of the proposed mechanism compare to existing systems like the Census DAS?
• RQ5 What are the statistical and implementation advantages of embedding invariants into the mechanism design rather than applying them afterward?

Key findings

• The proposed subspace differentially private mechanisms are provably unbiased, unlike the TopDown algorithm used in the 2020 Census DAS, which exhibits a systematic negative bias between errors and true population size.
• In the 2020 Census demonstration data, 11 out of 48 state-level regressions showed statistically significant negative slopes between DAS errors and log population size, indicating a clear bias trend.
• The proposed projected Laplace mechanism applied to Illinois counties shows no such bias trend, with errors stemming from unbiased, projected random variables.
• On the spatio-temporal WiFi dataset, the projected Gaussian mechanism achieved a median standard deviation of 0.88 for elementwise errors, with 90% of quantiles between 0.86 and 0.91.
• Over 50 repetitions, the mechanism maintained consistent error scales and preserved statistical intelligibility without post-processing.
• The framework avoids privacy leakage concerns associated with post-processing, as the invariants are built into the mechanism from the start.