Contraction of Locally Differentially Private Mechanisms

Understand

We investigate the contraction properties of locally differentially private mechanisms.

• More specifically, we derive tight upper bounds on the divergence between $PK$ and $QK$ output distributions of an $\epsilon$-LDP mechanism $K$ in terms of a divergence between the corresponding input distributions $P$ and $Q$, respectively.
• Our first main technical result presents a sharp upper bound on the $\chi^2$-divergence $\chi^2(PK}\|QK)$ in terms of $\chi^2(P\|Q)$ and $\varepsilon$.
• We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance.