On Local Mutual-Information Privacy

TL;DR

This work analyzes local mutual-information privacy (LMIP) and its relationship to local differential privacy (LDP) and local information privacy (LIP). It derives explicit, closed-form conversion rules bounding the privacy parameters between LMIP and the other notions, clarifying that LMIP is a weak privacy notion even when combined with delta allowances. The authors prove that, under an average power constraint, uncorrelated Gaussian noise is the best-case perturbation for CI-LMIP, reinforcing the practical appeal of the Gaussian mechanism. The results provide guidance on when LMIP can be used for analysis rather than as a design objective and offer benchmarks via Gaussian mechanisms for strong privacy guarantees. Overall, the paper sharpens understanding of how LMIP relates to standard local privacy notions and establishes Gaussian noise as optimal under the considered power constraints.

Abstract

Local mutual-information privacy (LMIP) is a privacy notion that aims to quantify the reduction of uncertainty about the input data when the output of a privacy-preserving mechanism is revealed. We study the relation of LMIP with local differential privacy (LDP), the de facto standard notion of privacy in context-independent (CI) scenarios, and with local information privacy (LIP), the state-of-the-art notion for context-dependent settings. We establish explicit conversion rules, i.e., bounds on the privacy parameters for an LMIP mechanism to also satisfy LDP/LIP, and vice versa. We use our bounds to formally verify that LMIP is a weak privacy notion. We also show that uncorrelated Gaussian noise is the best-case noise in terms of CI-LMIP if both the input data and the noise are subject to an average power constraint.

Figures (4)

• Figure 1: The achievable curve $\delta^{\text{{}LDP}}_\mu(\epsilon)$ of a $\mu$-- mechanism, its lower bound $\bar{p}_\mu$, and the optimal curve $\bar{\delta}^{\text{{}LDP}}_G(\epsilon)$ of the Gaussian mechanism $G$ in \ref{['eq:Gaussian_mechanism']} with $d = 10$, $\Delta = 1$, and $\sigma^2$ chosen such that $\bar{\mu}^{\text{{}CI}}_G = \mu$.
• Figure 2: The achievable - parameter $\mu^{\text{{}CI}}_{\delta(\epsilon)}$ of a mechanism that satisfies $(\epsilon,\delta(\epsilon))$- with $\delta(\epsilon) = \bar{\delta}^{\text{{}LDP}}_G(\epsilon)$, and the optimal - parameter $\bar{\mu}^{\text{{}CI}}_G$ of the Gaussian mechanism \ref{['eq:Gaussian_mechanism']} with $\Delta = 1$.
• Figure 3: The achievable curve $\delta^{\text{{}LIP}}_\mu(\epsilon)$ given in \ref{['eq:delta_mu_LIP']} of a $\mu$-- mechanism, its lower bound $1-2^{-\mu}$, and the optimal curve $\bar{\delta}^{\text{{}LIP}}_G(\epsilon)$ of the Gaussian mechanism $G$ in \ref{['eq:Gaussian_mechanism']} with $d = 1$, $\Delta = 1$, $P_X$ being the Gaussian distribution $\mathcal{N}\xspace(0,1/4)$ truncated in a set of $21$ points placed evenly in $[-\Delta,\Delta]$, and $\sigma^2$ chosen such that $\bar{\mu}^{\text{{}CD}}_G = \mu$.
• Figure 4: The achievable - parameter $\mu^{\text{{}CD}}_{\delta(\epsilon)}$ of a mechanism that satisfies $(\epsilon,\delta(\epsilon))$- with $\delta(\epsilon) = \bar{\delta}^{\text{{}LIP}}_G(\epsilon)$, and the optimal - parameter $\bar{\mu}^{\text{{}CD}}_G$ of the Gaussian mechanism considered in Fig. \ref{['fig:LMIDP_LIP']}.

Theorems & Definitions (8)

• Definition 1: -
• Definition 2: -
• Definition 3: Kas11Duc13
• Definition 4: Bo21
• Theorem 1: - vs.
• Theorem 2: - vs.
• Theorem 3: Uncorrelated Gaussian is the best-case noise for -
• Lemma 1