Minimax Rates of Estimating Approximate Differential Privacy

• DOI: 10.48550/arxiv.1905.10335
• Publication Date: 2019-01-01
• Authors: Liu, Xiyang; Oh, Sewoong
• Keywords: FOS: Computer and information sciences; Computer Science - Information Theory; Information Theory (cs.IT); 0202 electrical engineering, electronic engineering, information engineering; 02 engineering and technology

Differential privacy has become a widely accepted notion of privacy, leading to the introduction and deployment of numerous privatization mechanisms. However, ensuring the privacy guarantee is an error-prone process, both in designing mechanisms and in implementing those mechanisms. Both types of errors will be greatly reduced, if we have a data-driven approach to verify privacy guarantees, from a black-box access to a mechanism. We pose it as a property estimation problem, and study the fundamental trade-offs involved in the accuracy in estimated privacy guarantees and the number of samples required. We introduce a novel estimator that uses polynomial approximation of a carefully chosen degree to optimally trade-off bias and variance. With $n$ samples, we show that this estimator achieves performance of a straightforward plug-in estimator with $n \ln n$ samples, a phenomenon referred to as effective sample size amplification. The minimax optimality of the proposed estimator is proved by comparing it to a matching fundamental lower bound.