Fingerprint Codes Meet Geometry: Improved Lower Bounds for Private Query Publishing and Adaptive Data Analysis

Fingerprint codes are a crucial tool for demonstrating the lower bounds of differential privacy. They have been used to demonstrate strict lower bounds for several fundamental questions, especially in the “low precision” regime. However, unlike reconstruction/discrepancy approaches, they are more suitable for testing worst-case lower bounds, for query sets that arise naturally from the construction of fingerprint codes. In this work, we propose a general framework for testing lower bounds on fingerprint type, which allows us to adapt the technique to the geometry of the query set. Our approach allows us to test several new results.

First, we show that any precise algorithm (sample and population) to answer q<annotation encoding="application/x-tex”>qq arbitrary adaptive counting queries over a universe unknown<annotation encoding="application/x-tex”>\mathcal{x}unknown to the accuracy to<annotation encoding="application/x-tex”>\alphato needs Oh(record⁡∣unknown∣⋅record⁡qto3)<annotation encoding="application/x-tex”>\Omega(\frac{\sqrt{\log |\mathcal{x}|}\cdot \log Q}{\alpha^3})Oh(to3Iohgram∣unknown∣​⋅Iohgramq​) samples. This shows that differential privacy based approaches are optimal for this question and significantly improve previously known privacy lower bounds. record⁡qto2<annotation encoding="application/x-tex”>\frac{\log Q}{\alpha^2}to2Iohgramq​ and min.⁡(q,record⁡∣unknown∣)/to2<annotation encoding="application/x-tex”>\min(\sqrt{Q}, \sqrt{\log |\mathcal{x}|})/\alpha^2min.(q​,lookgram∣unknown∣​)/to2. Second, we show that any (my,d)<annotation encoding="application/x-tex”>(\varepsilon,\delta)(my,d)-DP algorithm to respond q<annotation encoding="application/x-tex”>qq count queries accurately to<annotation encoding="application/x-tex”>\alphato needs Oh(drecord⁡(1/d)record⁡qmyto2)<annotation encoding="application/x-tex”>\Omega\left( \frac{\sqrt{d \log(1/\delta)} \log Q}{\varepsilon \alpha^2} \right)Oh(myto2dIohgram(1/d)​Iohgramq​) samples. Our framework allows this limit to be directly tested and improved by record⁡(1/d)<annotation encoding="application/x-tex”>\sqrt{\log(1/\delta)}lookgram(1/d)​ the limit demonstrated by Bun, Ullman, and Vadhan (2013) using composition. Third, we characterize the sample complexity by answering a set of 0-1 random queries under approximate differential privacy. To achieve this, we provide new upper and lower bounds that, combined with the existing bounds, allow us to complete the picture.