Generalized Rainbow Differential Privacy

TL;DR

This work introduces rainbow differential privacy, where datasets are nodes in a neighbor graph and each dataset has a preferred output ordering (rainbow) over a finite output space. Under a boundary homogeneous condition, the authors prove the existence and uniqueness of an optimal $(\epsilon,\delta)$-DP mechanism and provide a closed-form construction that depends only on the boundary distributions, enabling a pullback from a boundary rainbow graph and reducing the problem to line graphs. The core technical device is the $T_{\epsilon,\delta}$ operator, which propagate boundary probabilities inward along line graphs to yield distributions that dominate all $(\epsilon,\delta)$-close competitors, establishing optimality. They also analyze the special case $\delta=0$ and the general case $\delta>0$, providing explicit recursions and phase-transition behavior, plus numerical results, and they discuss limitations under non-homogeneous boundaries, dataset-dependence, and connections to lexicographic ordering and exponential mechanisms. The results extend prior two- and three-color rainbow DP work to arbitrary numbers of outputs with a unified, rigorous approach, with potential implications for dataset-adaptive DP mechanism design and broader ordering-based privacy frameworks.

Abstract

We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $(ε,δ)$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $(ε,δ)$-DP, improving upon previous results that only apply to at most three colors and for $ε$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism.

Figures (5)

• Figure 1: A rainbow graph and its corresponding boundary graph. A vertex represents a dataset and its neighboring datasets are connected by an edge. The function output space is represented by three colors blue, red, and green. Each dataset has a color preference, represented by the ordering inside the vertex. For instance, vertex $d_1$ prefers blue to red and red to green. We call each such color ordering a rainbow. A DP mechanism is then a probability distribution over colors for every vertex. In (B), we show the boundary rainbow graph of the rainbow graph shown in (A), as described in Definition \ref{['def:boundarymorph']} below. In Theorem \ref{['thm:reduce-to-line']} we show how, for homogeneous boundary conditions (defined in Definition \ref{['def:boundaryhomogeneity']} below), optimal $(\epsilon,\delta)$-DP mechanisms on (A) can be retrieved from optimal ones on (B). For example, for rainbow $c=(\texttt{red}, \texttt{green},\texttt{blue})$, the vertex $(c,0)$ in the boundary rainbow graph corresponds to datasets $d_4,d_9$ in the original rainbow graph because they are on the boundary of $B^c$ in the original graph. There is an edge between $(c=(\texttt{red}, \texttt{green},\texttt{blue}),0)$ and $(c'=(\texttt{red},\texttt{blue},\texttt{green}),0)$ in the boundary rainbow graph, because there is an edge $(d_9,d_{13})$ in the original rainbow graph, with $i\in B^c$, $m\in B^{c'}$.
• Figure 2: Illustration of the rainbow graph in Example \ref{['eg:no-optimal']}. We use blue, red, green to represent the choices $1,2,3$ respectively.
• Figure 3: The optimal $(\log (1.2),0)$-DP mechanism with homogeneous boundary condition $\vec{m} =(0.0005,0.0081,0.1364, 0.2727, 0.5822)$. We have $\tau_1= 38, \tau_2= 22, \tau_3= 7, \tau_4= 1,$ and $\tau_5=0$.
• Figure 4: The optimal $(\log(1.2),10^{-3})$-DP mechanism with homogeneous boundary condition $\vec{m} =(0.0005,0.0081,0.1364, 0.2727, 0.5822)$. We have $\tau_1= 25, \tau_2= 20, \tau_3= 7, \tau_4= 1,$ and $\tau_5=0$.
• Figure 5: The optimal $(\log (1.2),0.01)$-DP mechanism with the homogeneous boundary condition $\vec{m} =(0.0005,0.0081,0.1364, 0.2727, 0.5822)$. We have $\tau_1= 13, \tau_2= 12, \tau_3= 6, \tau_4= 1,$ and $\tau_5=0.$

Theorems & Definitions (20)

• Definition 1
• Definition 2: DworkBook
• Definition 3
• Definition 4: RafnoDP2colorPaper
• Theorem 1: RafnoDP2colorPaper
• Definition 5
• Example 1
• Definition 6
• Theorem 2
• Definition 7