Total Variation Meets Differential Privacy

The framework of approximate differential privacy is considered, and augmented by introducing the notion of "the total variation of a (privacy-preserving) mechanism" (denoted by $\eta$-TV). With this refinement, an exact composition result is derived, and shown to be significantly tighter than the optimal bounds for differential privacy (which do not consider the total variation). Furthermore, it is shown that $(\varepsilon,\delta)$-DP with $\eta$-TV is closed under subsampling. The induced total variation of commonly used mechanisms are computed. Moreover, the notion of total variation of a mechanism is extended to the local privacy setting and privacy-utility tradeoffs are investigated. In particular, total variation distance and KL divergence are considered as utility functions and upper bounds are derived. Finally, the results are compared and connected to the (purely) locally differentially private setting.