Optimum Noise Mechanism for Differentially Private Queries in Discrete Finite Sets

TL;DR

This paper introduces a novel framework for designing an optimal noise probability mass function (PMF) tailored to discrete and finite query sets, and demonstrates that the optimal PMF can be obtained through solving a mixed-integer linear program.

Abstract

The Differential Privacy (DP) literature often centers on meeting privacy constraints by introducing noise to the query, typically using a pre-specified parametric distribution model with one or two degrees of freedom. However, this emphasis tends to neglect the crucial considerations of response accuracy and utility, especially in the context of categorical or discrete numerical database queries, where the parameters defining the noise distribution are finite and could be chosen optimally. This paper addresses this gap by introducing a novel framework for designing an optimal noise Probability Mass Function (PMF) tailored to discrete and finite query sets. Our approach considers the modulo summation of random noise as the DP mechanism, aiming to present a tractable solution that not only satisfies privacy constraints but also minimizes query distortion. Unlike existing approaches focused solely on meeting privacy constraints, our framework seeks to optimize the noise distribution under an arbitrary $(ε, δ)$ constraint, thereby enhancing the accuracy and utility of the response. We demonstrate that the optimal PMF can be obtained through solving a Mixed-Integer Linear Program (MILP). Additionally, closed-form solutions for the optimal PMF are provided, minimizing the probability of error for two specific cases. Numerical experiments highlight the superior performance of our proposed optimal mechanisms compared to state-of-the-art methods. This paper contributes to the DP literature by presenting a clear and systematic approach to designing noise mechanisms that not only satisfy privacy requirements but also optimize query distortion. The framework introduced here opens avenues for improved privacy-preserving database queries, offering significant enhancements in response accuracy and utility.

Figures (14)

• Figure 1: In these examples $f^\star{(\hat{\mu})}$ is single distance, $\hat{\mu}$, away from $f^\star{(0)}$ hence it is assigned $e^{-\epsilon}f^\star{(0)}$, next $f^\star{(2\hat{\mu})}$ is assigned $e^{-\epsilon}f^\star{(\hat{\mu})}$ since it is $\hat{\mu}$ away from $f^\star{(\hat{\mu})}$ and so on and so forth. So the order of assignment of values for plot (a) example is: $0,3,6,1,4,7,2,5$ and the order of assignment of values for plot (b) example is: $0, 2, 4, 6$. Since the values at $1, 3, 5, 7$ are not $\hat{\mu}$ away from $f^\star{(2k\hat{\mu})}, k \in [N_{\hat{\mu}}-1=3]$ they are assigned $0$ value to have a higher $f^\star{(0)}$.
• Figure 2: The variation of $\rho$ as a function of $\delta$ for SD neighborhood showing the alternate flat and linear regions.
• Figure 3: The PMF of the optimal noise mechanism for the BD neighborhood follows a staircase pattern. In the flat region, it has $b+1$ steps and $i^{th}$ step height is $\phi^k_i$. Similarly, in the linear region, the PMF has $b+2$ steps and $i^{th}$ step height is $\psi^k_i(\delta)$.
• Figure 4: The variation of $f^{\star}(0)$ as a function of $\delta$ for BD neighborhood showing $b$ segments with the alternate flat and linear regions.
• Figure 5: Comparison of the proposed optimal mechanism in terms of the expected distortion costs with those proposed in Ghosh_Universalcanonne2021discretesadeghi2020differentiallyMcSherry2007MechRavi2022Diff. In plot (a), the optimal noise mechanism is compared in terms of MSE, $\rho^{MSE}$, with the discrete geometric mechanism, discrete Gaussian mechanism, and Gumbel mechanism for a fixed a value of $\delta=0.3$ and $n=8$. In plot (b), the optimal noise mechanism is compared in terms of error rate, $\rho^{ER}$, with the discrete geometric mechanism, discrete count mechanism, and data independent mechanism for a fixed a value of $\delta=0.5$ and $n=7$.

Theorems & Definitions (30)

• Definition 1: $(\epsilon,\delta)$-Differential Privacy (DP) Calib_Dwork
• Definition 2: $(\epsilon, \delta)$-Probabilistic DP Privacy_Machanavajjhala
• Theorem 1: PDP implies DP mcclure2015relaxations
• Proposition 1
• proof
• Definition 3: Error Rate
• Definition 4: Mean Squared Error
• Remark 1
• Definition 5: Single Distance (SD)
• Definition 6: Bounded Difference (BD)