More Flexible Differential Privacy: The Application of Piecewise Mixture Distributions in Query Release

There is an increasing demand to make data “open” to third parties, as data sharing has great benefits in data-driven decision making. However, with a wide variety of sensitive data collected, protecting privacy of individuals, communities and organizations, is an essential factor in making data “open”. The approaches currently adopted by industry in releasing private data are often ad hoc and prone to a number of attacks, including re-identification attacks, as they do not provide adequate privacy guarantees. While differential privacy has attracted significant interest from academia and industry by providing rigorous and reliable privacy guarantees, the reduced utility and inflexibility of current differentially private algorithms for data release is a barrier to their use in real-life. This paper aims to address these two challenges. First, we propose a novel mechanism to augment the conventional utility of differential privacy by fusing two Laplace or geometric distributions together. We derive closed form expressions for entropy, variance of added noise, and absolute expectation of noise for the proposed piecewise mixtures. Then the relevant distributions are utilised to theoretically prove the privacy and accuracy guarantees of the proposed mechanisms. Second, we show that our proposed mechanisms have greater flexibility, with three parameters to adjust, giving better utility in bounding noise, and mitigating larger inaccuracy, in comparison to typical one-parameter differentially private mechanisms. We then empirically evaluate the performance of piecewise mixture distributions with extensive simulations and with a real-world dataset for both linear count queries and histogram queries. The empirical results show an increase in all utility measures considered, while maintaining privacy, for the piecewise mixture mechanisms compared to standard Laplace or geometric mechanisms.

💡 Research Summary

The paper addresses two major practical limitations of differential privacy (DP): the loss of utility caused by the noise required for privacy guarantees, and the inflexibility of existing mechanisms that rely on a single privacy parameter ε. To overcome these issues, the authors propose a novel “piecewise mixture” mechanism that combines two Laplace (for real‑valued queries) or two symmetric geometric (discrete Laplace, for integer‑valued queries) distributions at a predefined break‑point c. The mechanism is parameterized by three values: the standard privacy budget ε, a scaling factor r > 0 that determines a second privacy budget r·ε for the tails of the distribution, and the break‑point c that separates the central region (|x| ≤ c) from the tails (|x| > c).

In the central region the noise follows a Laplace (or geometric) distribution with scale b₂ = Δf/ε, identical to the classic DP mechanism. In the tails the scale is reduced to b₁ = Δf/(r·ε), which yields smaller magnitude noise and therefore higher accuracy for outlier perturbations. By carefully choosing the normalizing constants a₁, a₂ and the mixing probabilities p₁, p₂, the authors ensure that the resulting probability density (or mass) function is continuous and integrates to one.

The paper provides closed‑form expressions for the entropy, variance, and absolute‑value expectation of the piecewise mixture distribution. These formulas reveal that increasing r (making the tail distribution tighter) reduces variance and absolute error at the cost of a modest entropy reduction, while the break‑point c controls how much probability mass is allocated to the low‑noise tail.

A rigorous privacy analysis shows that for any neighboring databases X¹ and X², the privacy loss Lₓ = ln(Pr