$\alpha$-Information-theoretic Privacy Watchdog and Optimal Privatization Scheme
Ni Ding, Mohammad Amin Zarrabian, Parastoo Sadeghi · 2021
This paper proposes an$\alpha$-lift measure for data privacy and determines the optimal privatization scheme that minimizes the$\alpha$-lift in the watchdog method. To release useful data$X$that is correlated with sensitive data$S$, the ratio of the posterior belief to the prior belief on sensitive data with respect to the useful data is called ‘lift’, which quantifies privacy risk. The$\alpha$-lift denoted by$\ell_{\alpha}(x)$is proposed as the$L_{\alpha}$-norm of the lift for a given realization$x$. This is a tunable measure: when$\alpha < \infty$, each lift is weighted by its likelihood of appearing in the dataset (w.r.t. the marginal probability$p(s)$); for$\alpha=\infty,\ \alpha$-lift reduces to the existing maximum lift. To generate the sanitized data$Y$, we adopt the privacy watchdog method using$\alpha$-lift: obtain realizations of useful data such that the$\alpha$-lift is greater than a threshold$e^{\epsilon}$; apply a randomization mechanism to these ‘high-risk’ realizations, while all other realizations of$X$are published directly. For the resulting$\alpha$-lift denoted by$\ell_{\alpha}(y)$, it is shown that the Sibson mutual information$I_{\alpha}^{S}(S;Y)$is proportional to$\mathbb{E}[\ell_{\alpha}(y)]$. We further define a stronger privacy measure denoted$\overline{I}_{\alpha}^{S}(S;Y)$using the worst-case$\alpha$-lift:$\bar{I}_{\alpha}^{S}(S;Y)\propto\max olimits_{y}\ell_{\alpha}(y)$. We prove that the optimal watchdog randomization that minimizes both$I_{\alpha}^{S}(S;Y)$and$\overline{I}_{\alpha}^{S}(S;Y)$is$X$-invariant. Numerical experiments show that$\alpha$-lift can provide flexibility in the privacy-utility tradeoff.