Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies

Differential private (DP) mechanisms protect individual-level information by introducing randomness into the statistical analysis procedure. While there are now many DP tools for various statistical problems, there is still a lack of general techniques to understand the sampling distribution of a DP estimator, which is crucial for uncertainty quantification in statistical inference. We analyze a DP bootstrap procedure that releases multiple private bootstrap estimates to infer the sampling distribution and construct confidence intervals. Our privacy analysis includes new results on the privacy cost of a single DP bootstrap estimate applicable to incorporate arbitrary DP mechanisms and identifies some misuses of the bootstrap in the existing literature. We show that the release of $B$ DP bootstrap estimates from mechanisms satisfying $(\mu/\sqrt{(2-2/\mathrm{e})B})$-Gaussian DP asymptotically satisfies $\mu$-Gaussian DP as $B$ goes to infinity. We also develop a statistical procedure based on the DP bootstrap estimates to correctly infer the sampling distribution using techniques related to the deconvolution of probability measures, an approach which is novel in analyzing DP procedures. From our density estimate, we construct confidence intervals and compare them to existing methods through simulations and real-world experiments using the 2016 Canada Census Public Use Microdata. The coverage of our private confidence intervals achieves the nominal confidence level, while other methods fail to meet this guarantee.