Distribution Simulation Under Local Differential Privacy

We investigate the problem of distribution simu-lation under local differential privacy: Alice and Bob observe sequences $X^{n}$ and $Y^{n}$ respectively, where $Y^{n}$ is generated by a non-interactive $\varepsilon$ -Iocally differentially private (LDP) mechanism from $X^{n}$. The goal is for Alice and Bob to output $U$ and $V$ from a joint distribution that is close in total variation distance to a target distribution $P_{UV}$. As the main result, we show that such task is impossible if the hynercontractivity coefficient of $P_{UV}$ is strictly bigger than $\left(\frac{e^{\varepsilon}-1}{e^{\varepsilon}+1}\right)^{2}$ . The proof of this result also leads to a new operational interpretation of LDP mechanisms: if $Y$ is an output of an $\varepsilon$ -LDP mechanism with input $X$, then the probability of correctly guessing $f(X)$ given $Y$ is bigger than the probability of blind guessing only by $\frac{e^{\varepsilon}-1}{e^{\varepsilon}+1}$, for any deterministic finitely-supported function $f$. If $f(X)$ is continuous, then a similar result holds for the minimum mean-squared error in estimating $f(X)$ given $Y$.