Privacy Profiles Under Tradeoff Composition

Abstract

Privacy profiles and tradeoff functions are two frameworks for comparing differential privacy guarantees of alternative privacy mechanisms. We study connections between these frameworks. We show that the composition of tradeoff functions corresponds to a binary operation on privacy profiles we call their T-convolution. Composition of tradeoff functions characterizes group privacy guarantees, so the T-convolution provides a bridge for translating group privacy properties from one framework to the other. Composition of tradeoff functions has also been used to characterize mechanisms with log-concave additive noise; we derive a corresponding property based on privacy profiles. We also derive new bounds on privacy profiles for log-concave mechanisms based on new convexity properties. In developing these ideas, we characterize regular privacy profiles, which are privacy profiles for mutually absolutely continuous probability measures.