aiqbench analysis
This research provides sharp asymptotic characterizations for the privacy loss in differentially private Principal Component Analysis (PCA) within high-dimensional limits. By reducing the detection problem to distinguishing between two Gaussians, the study offers precise utility metrics that complement existing conservative non-asymptotic bounds. These findings are critical for understanding the fundamental trade-offs between data utility and privacy guarantees in large-scale spectral learning tasks.
High-Dimensional Asymptotics of Differentially Private PCA
arXiv:2511.07270v4 Announce Type: replace-cross Abstract: In differential privacy, random noise is introduced to privatize summary statistics of a sensitive dataset before releasing them. The noise level determines the privacy loss, which quantifies how easily an adversary can detect a target individual's presence in the dataset using the published statistic. Most privacy analyses provide non-asymptotic upper bounds on the privacy loss which hold uniformly across all datasets. Sometimes, these bounds can be pessimistic on a given dataset. In such cases, it can be useful to complement these privacy bounds with sharp privacy characterizations that quantify a mechanism's exact privacy loss on a given dataset. With this goal, we study differentially private principal component analysis (PCA), where the goal is to privatize the leading principal components of a dataset with $n$ samples and $p$ features. We analyze the exponential mechanism and provide sharp asymptotic characterizations of its utility and privacy loss in the high-dimensional limit ($p \rightarrow \infty$). We show that in this limit, detecting a target individual's presence using privatized principal components is asymptotically equivalent to distinguishing between two Gaussians with different means, where the mean difference depends on certain spectral properties of the dataset. Our analysis combines the hypothesis-testing formulation of privacy guarantees proposed by Dong, Roth, and Su (2022) with Le Cam's contiguity arguments.
Abstract:In differential privacy, random noise is introduced to privatize summary statistics of a sensitive dataset before releasing them. The noise level determines the privacy loss, which quantifies how easily an adversary can detect a target individual's presence in the dataset using the published statistic. Most privacy analyses provide non-asymptotic upper bounds on the privacy loss which hold uniformly across all datasets. Sometimes, these bounds can be pessimistic on a given dataset. In such cases, it can be useful to complement these privacy bounds with sharp privacy characterizations that quantify a mechanism's exact privacy loss on a given dataset. With this goal, we study differentially private principal component analysis (PCA), where the goal is to privatize the leading principal components of a dataset with $n$ samples and $p$ features. We analyze the exponential mechanism and provide sharp asymptotic characterizations of its utility and privacy loss in the high-dimensional limit ($p \rightarrow \infty$). We show that in this limit, detecting a target individual's presence using privatized principal components is asymptotically equivalent to distinguishing between two Gaussians with different means, where the mean difference depends on certain spectral properties of the dataset. Our analysis combines the hypothesis-testing formulation of privacy guarantees proposed by Dong, Roth, and Su (2022) with Le Cam's contiguity arguments.
Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)
arXiv:2511.07270 [math.ST]
(or arXiv:2511.07270v4 [math.ST] for this version)
https://doi.org/10.48550/arXiv.2511.07270 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 313 words
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