Distribution Simulation Under Local Differential Privacy

We investigate the problem of distribution simulation under local differential privacy: Alice and Bob observe sequences X-n and Y-n respectively, where Y-n is generated by a non-interactive epsilon-locally 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 hypercontractivity coefficient of P-UV is strictly bigger than (e(epsilon) - 1/e(epsilon) + 1)(2). The proof of this result also leads to a new operational interpretation of LDP mechanisms: if Y is an output of an "-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 e(epsilon) - 1/e(epsilon) + 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.