$\alpha$-Information-theoretic Privacy Watchdog and Optimal Privatization Scheme

This paper proposes an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift measure for data privacy and determines the optimal privatization scheme that minimizes the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift in the watchdog method. To release useful data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> that is correlated with sensitive data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{\alpha}(x)$</tex> is proposed as the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{\alpha}$</tex> -norm of the lift for a given realization <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex> . This is a tunable measure: when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha &lt; \infty$</tex> , each lift is weighted by its likelihood of appearing in the dataset (w.r.t. the marginal probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p(s)$</tex> ); for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha=\infty,\ \alpha$</tex> -lift reduces to the existing maximum lift. To generate the sanitized data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Y$</tex> , we adopt the privacy watchdog method using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift: obtain realizations of useful data such that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift is greater than a threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$e^{\epsilon}$</tex> ; apply a randomization mechanism to these ‘high-risk’ realizations, while all other realizations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> are published directly. For the resulting <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{\alpha}(y)$</tex> , it is shown that the Sibson mutual information <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$I_{\alpha}^{S}(S;Y)$</tex> is proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{E}[\ell_{\alpha}(y)]$</tex> . We further define a stronger privacy measure denoted <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\overline{I}_{\alpha}^{S}(S;Y)$</tex> using the worst-case <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\bar{I}_{\alpha}^{S}(S;Y)\propto\max\nolimits_{y}\ell_{\alpha}(y)$</tex> . We prove that the optimal watchdog randomization that minimizes both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$I_{\alpha}^{S}(S;Y)$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\overline{I}_{\alpha}^{S}(S;Y)$</tex> is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> -invariant. Numerical experiments show that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> -lift can provide flexibility in the privacy-utility tradeoff.