A Linear Reduction Method for Local Differential Privacy and Log-lift

This paper considers the problem of publishing data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> while protecting the correlated sensitive information <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> . We propose a linear method 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> with the same alphabet <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{Y}=\mathcal{X}$</tex> that attains local differential privacy (LDP) and log-lift at the same time. It is revealed that both LDP and log-lift are inversely proportional to the statistical distance between conditional probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y\vert S}(x\vert s)$</tex> and marginal probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y}(x)$</tex> : the closer the two probabilities are, the more private <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Y$</tex> is. Specifying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y\vert S}(x\vert s)$</tex> that linearly reduces this distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\vert P_{Y\vert S}(x\vert s)-P_{Y}(x)\vert =(1-\alpha)\vert P_{X\vert S}(x\vert s)-P_{X}(x)\vert, \forall s, x$</tex> for some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha\in(0,1]$</tex> , we study the problem of how to generate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{Y}$</tex> from the original data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> . The Markov randomization/sanitization scheme <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y\vert X}(x\vert x^{\prime})=P_{Y\vert S,X}(x\vert s,x^{\prime})$</tex> is obtained by solving linear equations. The optimal non-Markov sanitization, the transition probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y\vert S,X}(x\vert s,x^{\prime})$</tex> that depends on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> ,, can be determined by maximizing the data utility subject to linear equality constraints on data privacy. We compute the solution for two linear utility function: the expected distance and total variance distance. It is shown that the non-Markov randomization significantly improves data utility and the marginal probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{X}(x)$</tex> remains the same after the linear sanitization method: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{Y}(x)=P_{X}(x),\forall x\in \mathcal{X}$</tex> .