Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA
Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA
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发表时间：
2019-01
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通讯作者：
Jordan Awan;Ana M. Kenney;M. Reimherr;A. Slavkovic
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作者：
Jordan Awan;Ana M. Kenney;M. Reimherr;A. Slavkovic
指数机制由于其强大的隐私保证和灵活性而成为差分隐私（DP）的基本工具。我们研究了它的扩展设置与摘要的基础上无限维输出，如功能数据分析，形状分析，和非参数统计。我们表明，可以设计一个特定的基础措施的输出空间，如高斯过程的机制。我们提供了一个积极的结果，建立了一个中心极限定理的指数机制相当广泛。我们还提供了一个明显的负面结果，表明为隐私引入的噪声的大小是渐近不可忽略的相对于统计估计误差。我们开发了一个\ep-DP机制的功能主成分分析，适用于可分离的希尔伯特空间。我们证明了它的性能，通过模拟和应用程序的两个数据集。
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.