Local Differential Privacy Is Equivalent to Contraction of $E_gamma$-Divergence

We investigate the local differential privacy (LDP) guarantees of a randomized privacy mechanism via its contraction properties. We first show that LDP constraints can be equivalently cast in terms of the contraction coefficient of the $E_gamma$-divergence. We then use this equivalent formula to express LDP guarantees of privacy mechanisms in terms of contraction coefficients of arbitrary $f$-divergences. When combined with standard estimation-theoretic tools (such as Le Cams and Fanos converse methods), this result allows us to study the trade-off between privacy and utility in several testing and minimax and Bayesian estimation problems.