Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA

• Activity Number: 133 - Statistical Methods for Functional Data
• Type: Contributed
• Date/Time: Monday, July 29, 2019 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Jordan Awan* and Ana Kenney and Matthew Reimherr and Aleksandra Slavkovic
• Companies: Penn State University and Pennsylvania State University and Penn State University and Penn State University
• Keywords: Differential Privacy; Statistical Disclosure Control; Functional Data Analysis; Hilbert Space; Principal Component Analysis; Central Limit Theorem

Abstract

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that one can design the mechanism with respect to a specific base measure over the output space, such as a Guassian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide an apparent negative result, showing that the magnitude of the noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an $\ep$-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces. We demonstrate its performance via simulations and applications to two datasets.