Strong Data Processing Inequalities for Locally Differentially Private Mechanisms

We investigate the strong data processing inequalities of locally differentially private mechanisms under a specific f -divergence, namely the E
γ
-divergence. More specifically, we characterize an upper bound on the E
γ
-divergence between PK and QK, the output distributions of an ε-LDP mechanism K, in terms of the E
γ
-divergence between the corresponding input distributions P and Q. Interestingly, the tightest such upper bound in the binary case turns out to have a non-multiplicative form. We then extend our results to derive a tight upper bound for general f-divergences. As an application of our main findings, we derive a lower bound on the locally private Bayesian estimation risk that is tighter than the available divergence-based bound in the literature.