Tamas Linder

Shahab Asoodeh, Mario Diaz, Fady Alajaji et al.

We investigate the problem of estimating a random variable $Y\in \mathcal{Y}$ under a privacy constraint dictated by another random variable $X\in \mathcal{X}$, where estimation efficiency and privacy are assessed in terms of two different loss functions. In the discrete case, we use the Hamming loss function and express the corresponding utility-privacy tradeoff in terms of the privacy-constrained guessing probability $h(P_{XY}, ε)$, the maximum probability $\mathsf{P}_\mathsf{c}(Y|Z)$ of correctly guessing $Y$ given an auxiliary random variable $Z\in \mathcal{Z}$, where the maximization is taken over all $P_{Z|Y}$ ensuring that $\mathsf{P}_\mathsf{c}(X|Z)\leq ε$ for a given privacy threshold $ε\geq 0$. We prove that $h(P_{XY}, \cdot)$ is concave and piecewise linear, which allows us to derive its expression in closed form for any $ε$ when $X$ and $Y$ are binary. In the non-binary case, we derive $h(P_{XY}, ε)$ in the high utility regime (i.e., for sufficiently large values of $ε$) under the assumption that $Z$ takes values in $\mathcal{Y}$. We also analyze the privacy-constrained guessing probability for two binary vector scenarios. When $X$ and $Y$ are continuous random variables, we use the squared-error loss function and express the corresponding utility-privacy tradeoff in terms of $\mathsf{sENSR}(P_{XY}, ε)$, which is the smallest normalized minimum mean squared-error (mmse) incurred in estimating $Y$ from its Gaussian perturbation $Z$, such that the mmse of $f(X)$ given $Z$ is within $ε$ of the variance of $f(X)$ for any non-constant real-valued function $f$. We derive tight upper and lower bounds for $\mathsf{sENSR}$ when $Y$ is Gaussian. We also obtain a tight lower bound for $\mathsf{sENSR}(P_{XY}, ε)$ for general absolutely continuous random variables when $ε$ is sufficiently small.

Shahab Asoodeh, Fady Alajaji, Tamas Linder

We study the maximal mutual information about a random variable $Y$ (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only $ε$ bits of information is leaked about a random variable $X$ (representing private information) that is correlated with $Y$. Denoting this quantity by $g_ε(X,Y)$, we show that for perfect privacy, i.e., $ε=0$, one has $g_0(X,Y)=0$ for any pair of absolutely continuous random variables $(X,Y)$ and then derive a second-order approximation for $g_ε(X,Y)$ for small $ε$. This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution $P_{XY}$. Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimation-theoretic quantities and obtain explicit bounds for this tradeoff when $ε$ is sufficiently small using the approximation formula derived for $g_ε(X,Y)$.