Outer Bounds on the CEO Problem with Privacy Constraints

TL;DR

This work studies the rate-distortion-leakage region for the two-encoder CEO problem in the presence of a passive eavesdropper, introducing a novel outer bound for general distortion via a dedicated privacy-leakage lemma. It specialized the analysis to log-loss distortion, obtaining a tight bound for discrete and Gaussian sources when Eve has no side information, and deriving outer bounds for the case where Eve has side information, with the two bounds differing only by a small leakage term that vanishes in high-distortion regimes. The results connect privacy leakage to classic rate-distortion theory and extend prior work by replacing equivocation constraints with privacy leakage constraints, offering exact results in several key scenarios and near-tight bounds in more general ones. These findings have implications for secure distributed sensing and privacy-aware multiterminal source coding, particularly in settings with soft-decoding distortions and public-rate links.

Abstract

We investigate the rate-distortion-leakage region of the Chief Executive Officer (CEO) problem, considering the presence of a passive eavesdropper and privacy constraints. We start by examining the region where a general distortion measure quantifies the distortion. While the inner bound of the region is derived from previous work, this paper newly develops an outer bound. To derive the outer bound, we introduce a new lemma tailored for analyzing privacy constraints. Next, as a specific instance of the general distortion measure, we demonstrate that the tight bound for discrete and Gaussian sources is obtained when the eavesdropper has no side information, and the distortion is quantified by the log-loss distortion measure. We further investigate the rate-distortion-leakage region for a scenario where the eavesdropper has side information, and the distortion is quantified by the log-loss distortion measure and provide an outer bound for this case. The derived outer bound differs from the inner bound by only a minor quantity that appears in the constraints associated with the privacy-leakage rates, and these bounds match when the distortion is large.

Figures (4)

• Figure 1: The CEO problem with two encoders where $Y^n_1$ and $Y^n_2$ denote measurements of source sequence $X^n$ observed through channels $P_{Y_1|X}$ and $P_{Y_2|X}$, respectively. The compressed messages $J_1$ and $J_2$ are transmitted to the central decoder over noiseless channels. The decoder utilizes these messages to reconstruct $\hat{X}^n$, a reproduced sequence of the source $X^n$.
• Figure 2: System model of the CEO problem in the presence of a passive eavesdropper; (a) when Eve has no side information of the hidden source and (b) when Eve has side information of the hidden source.
• Figure 3: A relation of privacy-leakage rate versus minimum distortion of the Gaussian CEO problem under log-loss distortion measures for the model without SI at Eve. In the calculation for each graph, we set $\sigma^2_{N_1}=\sigma^2_{N_2} = 1$.
• Figure 4: An illustration on the behavior of $L_1 + L_2$ and $D$ of the inner bound (Proposition \ref{['th-in-ll']}) and the outer bound (Proposition \ref{['th-out-ll']}) is shown using $*$ and $\square$, respectively. The relation of \ref{['k1k2-ll1']} and \ref{['d-ll1']} in Theorem \ref{['th4']} is indicated by $\star$. The right-upper parts of each graph represent the achievable regions.

Theorems & Definitions (31)

• Definition 1
• Remark 1
• Definition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Definition 3