Gaussian Differential Privacy
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Differential privacy has seen remarkable success as a rigorous and practical formalization of data privacy in the past decade. But it also has some well known weaknesses: notably, it does not tightly handle composition. This weakness has inspired several recent relaxations of differential privacy based on Renyi divergences. We propose an alternative relaxation of differential privacy, which we term "f-differential privacy", which has a number of appealing properties and avoids some of the difficulties associated with divergence based relaxations. First, it preserves the hypothesis testing interpretation of differential privacy, which makes its guarantees easily interpretable. It allows for lossless reasoning about composition and post-processing, and notably, a direct way to import existing tools from differential privacy, including privacy amplification by subsampling. We define a canonical single parameter family of definitions within our class which we call "Gaussian Differential Privacy", defined based on the hypothesis testing of two shifted Gaussian distributions. We show that this family is focal by proving a central limit theorem, which shows that the privacy guarantees of any hypothesis-testing based definition of privacy (including differential privacy) converges to Gaussian differential privacy in the limit under composition. We also prove a finite (Berry-Esseen style) version of the central limit theorem, which gives a useful tool for tractably analyzing the exact composition of potentially complicated expressions. We demonstrate the use of the tools we develop by giving an improved analysis of the privacy guarantees of noisy stochastic gradient descent.
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