Recently, a number of approaches and techniques have been introduced for reporting software statistics with strong privacy guarantees. These range from abstract algorithms to comprehensive systems with varying assumptions and built upon local differential privacy mechanisms and anonymity. Based on the Encode-Shuffle-Analyze (ESA) framework, notable results formally clarified large improvements in privacy guarantees without loss of utility by making reports anonymous. However, these results either comprise of systems with seemingly disparate mechanisms and attack models, or formal statements with little guidance to practitioners. Addressing this, we provide a formal treatment and offer prescriptive guidelines for privacy-preserving reporting with anonymity. We revisit the ESA framework with a simple, abstract model of attackers as well as assumptions covering it and other proposed systems of anonymity. In light of new formal privacy bounds, we examine the limitations of sketch-based encodings and ESA mechanisms such as data-dependent crowds. We also demonstrate how the ESA notion of fragmentation (reporting data aspects in separate, unlinkable messages) improves privacy/utility tradeoffs both in terms of local and central differential-privacy guarantees. Finally, to help practitioners understand the applicability and limitations of privacy-preserving reporting, we report on a large number of empirical experiments. We use real-world datasets with heavy-tailed or near-flat distributions, which pose the greatest difficulty for our techniques; in particular, we focus on data drawn from images that can be easily visualized in a way that highlights reconstruction errors. Showing the promise of the approach, and of independent interest, we also report on experiments using anonymous, privacy-preserving reporting to train high-accuracy deep neural networks on standard tasks---MNIST and CIFAR-10.
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