[Paper Review] Constrained Differential Privacy for Count Data

Graham Cormode, Tejas Kulkarni+1 more|arXiv (Cornell University)|Oct 2, 2017

Privacy-Preserving Technologies in Data14 references3 citations

TL;DR

This paper proposes constrained differential privacy mechanisms for count queries that eliminate pathological behaviors—such as gaps (never reporting certain outputs) and spikes (over-reporting specific values)—in optimal mechanisms derived via linear programming. By enforcing structural properties like fairness and monotonicity, the authors show that only a few optimal mechanisms exist, including a novel one and the well-known geometric mechanism, with experiments demonstrating that constrained mechanisms outperform unconstrained ones, especially under high privacy budgets and non-extreme input distributions.

ABSTRACT

Concern about how to aggregate sensitive user data without compromising individual privacy is a major barrier to greater availability of data. The model of differential privacy has emerged as an accepted model to release sensitive information while giving a statistical guarantee for privacy. Many different algorithms are possible to address different target functions. We focus on the core problem of count queries, and seek to design mechanisms to release data associated with a group of n individuals. Prior work has focused on designing mechanisms by raw optimization of a loss function, without regard to the consequences on the results. This can leads to mechanisms with undesirable properties, such as never reporting some outputs (gaps), and overreporting others (spikes). We tame these pathological behaviors by introducing a set of desirable properties that mechanisms can obey. Any combination of these can be satisfied by solving a linear program (LP) which minimizes a cost function, with constraints enforcing the properties. We focus on a particular cost function, and provide explicit constructions that are optimal for certain combinations of properties, and show a closed form for their cost. In the end, there are only a handful of distinct optimal mechanisms to choose between: one is the well-known (truncated) geometric mechanism; the second a novel mechanism that we introduce here, and the remainder are found as the solution to particular LPs. These all avoid the bad behaviors we identify. We demonstrate in a set of experiments on real and synthetic data which is preferable in practice, for different combinations of data distributions, constraints, and privacy parameters.

Motivation & Objective

• To address the pathological behaviors—gaps and spikes—in existing differentially private mechanisms for count queries, which arise from unconstrained optimization of loss functions.
• To identify and formalize desirable structural properties (e.g., fairness, weak honesty, monotonicity) that ensure mechanisms are well-behaved and avoid extreme output biases.
• To demonstrate that any combination of these properties can be enforced via linear programming, yielding a small set of optimal mechanisms for the L0 loss function.
• To evaluate the practical performance of constrained mechanisms against standard ones (e.g., geometric, uniform) across varying input distributions, group sizes, and privacy parameters.
• To show that constrained mechanisms, particularly the novel mechanism and the constrained exponential mechanism, consistently outperform the geometric mechanism in real-world settings, especially under high privacy budgets.

Proposed method

• The authors define seven structural properties (e.g., fairness, weak honesty, monotonicity) to constrain mechanism design and prevent pathological output distributions.
• They extend the linear programming framework for differential privacy to incorporate these properties as constraints, minimizing the L0 error (probability of output deviating by more than d from the true input).
• The resulting optimal mechanisms are derived by solving a linear program with constraints enforcing the desired properties, yielding a small set of distinct optimal mechanisms.
• The paper introduces a novel mechanism that is optimal under fairness and monotonicity constraints, distinct from the geometric mechanism.
• Experiments use real and synthetic data to evaluate mechanisms under varying input distributions (parameterized by p), group sizes (n), and privacy parameters (α), measuring L0,d, L0,1, and RMSE error.
• The evaluation compares the constrained mechanisms (EM, WM) against the geometric mechanism (GM) and uniform mechanism (UM), with results averaged over 30 repetitions to ensure low variance.

Experimental results

Research questions

• RQ1 Can structural constraints be formally defined to eliminate pathological behaviors such as gaps and spikes in differentially private mechanisms for count queries?
• RQ2 Which combinations of structural properties lead to optimal mechanisms under the L0 loss function, and how many distinct optimal mechanisms exist?
• RQ3 How do constrained mechanisms compare to the widely used geometric mechanism in terms of utility across different input distributions and privacy budgets?
• RQ4 Does enforcing fairness or monotonicity improve robustness to input distribution skew, particularly under high privacy parameters?
• RQ5 Under what conditions does the geometric mechanism fail to outperform simpler mechanisms like uniform or constrained ones, especially in terms of RMSE and tail error?

Key findings

• The geometric mechanism often fails to report values near the true input, especially under high privacy budgets (α > 0.7), and can perform worse than random guessing in some cases.
• Constrained mechanisms that enforce fairness and monotonicity (e.g., EM and WM) consistently outperform the geometric mechanism when input distributions are spread across the input space (p near 0.5).
• For high privacy parameters (α = 0.91), the novel constrained exponential mechanism (EM) achieves lower RMSE than the geometric mechanism across all group sizes and input distributions.
• When input distributions are skewed (p near 0 or 1), the geometric mechanism performs better, but the constrained exponential mechanism still maintains strong performance, outperforming uniform and often the geometric mechanism.
• The constrained mechanism that enforces fairness (WM) reduces sensitivity to input distribution skew, making it more robust than the geometric mechanism in non-extreme cases.
• The L0,d error analysis shows that the constrained exponential mechanism (EM) significantly reduces tail probability mass, especially for larger d, and maintains this advantage even as d increases.