$α$-Information-theoretic Privacy Watchdog and Optimal Privatization Scheme

Abstract

This paper proposes an $α$-lift measure for data privacy and determines the optimal privatization scheme that minimizes the $α$-lift in the watchdog method. To release data $X$ that is correlated with sensitive information $S$, the ratio $l(s,x) = \frac{p(s|x)}{p(s)} $ denotes the `lift' of the posterior belief on $S$ and quantifies data privacy. The $α$-lift is proposed as the $L_α$-norm of the lift: $\ell_α(x) = \| (\cdot,x) \|_α = (E[l(S,x)^α])^{1/α}$. This is a tunable measure: When $α< \infty$, each lift is weighted by its likelihood of appearing in the dataset (w.r.t. the marginal probability $p(s)$); For $α= \infty$, $α$-lift reduces to the existing maximum lift. To generate the sanitized data $Y$, we adopt the privacy watchdog method using $α$-lift: Obtain $\mathcal{X}_ε$ containing all $x$'s such that $\ell_α(x) > e^ε$; Apply the randomization $r(y|x)$ to all $x \in \mathcal{X}_ε$, while all other $x \in \mathcal{X} \setminus \mathcal{X}_ε$ are published directly. For the resulting $α$-lift $\ell_α(y)$, it is shown that the Sibson mutual information $I_α^{S}(S;Y)$ is proportional to $E[ \ell_α(y)]$. We further define a stronger measure $\bar{I}_α^{S}(S;Y)$ using the worst-case $α$-lift: $\max_{y} \ell_α(y)$. We prove that the optimal randomization $r^*(y|x)$ that minimizes both $I_α^{S}(S;Y)$ and $\bar{I}_α^{S}(S;Y)$ is $X$-invariant, i.e., $r^*(y|x) = R(y), \forall x\in \mathcal{X}_ε$ for any probability distribution $R$ over $y \in \mathcal{X}_ε$. Numerical experiments show that $α$-lift can provide flexibility in the privacy-utility tradeoff.