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We develop new abstractions for reasoning about three relaxations of differential privacy: Re&#x0301;nyi differential privacy, zero-concentrated differential privacy, and truncated concentrated differential privacy, which express bounds on statistical divergences between two output probability distributions. In order to reason about such properties compositionally, we introduce approximate span-lifting, a novel construction extending the approximate relational lifting approaches previously developed for standard differential privacy to a more general class of divergences, and also to continuous distributions. As an application, we develop a program logic based on approximate span-liftings capable of proving relaxations of differential privacy and other statistical divergence properties.

Approximate Span Liftings: Compositional Semantics for Relaxations of Differential Privacy