Fingerprinting Codes Meet Geometry: Improved Lower Bounds for Private Query Release and Adaptive Data Analysis

Fingerprinting codes are a crucial tool for proving lower bounds in differential privacy. They have been used to prove tight lower bounds for several fundamental questions, especially in the “low accuracy” regime. Unlike reconstruction/discrepancy approaches however, they are more suited for proving worst-case lower bounds, for query sets that arise naturally from the fingerprinting codes construction. In this work, we propose a general framework for proving fingerprinting type lower bounds, that allows us to tailor the technique to the geometry of the query set.
Our approach allows us to prove several new results.

First, we show that any (sample- and population-)accurate algorithm for answering $Q$ arbitrary adaptive counting queries over a universe $\mathcal{X}$ to accuracy $\alpha$ needs $\Omega(\frac{\sqrt{\log |\mathcal{X}|}\cdot \log Q}{\alpha^3})$

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