DP-PCA: Statistically Optimal and Differentially Private PCA

• URL: http://arxiv.org/abs/2205.13709v1
• Date: Fri, 27 May 2022 02:02:17 GMT
• Title: DP-PCA: Statistically Optimal and Differentially Private PCA
• Authors: Xiyang Liu, Weihao Kong, Prateek Jain, Sewoong Oh
• Abstract summary: DP-PCA is a single-pass algorithm that overcomes both limitations. For sub-Gaussian data, we provide nearly optimal statistical error rates even for $n=tilde O(d)$.
• Score: 44.22319983246645
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the canonical statistical task of computing the principal component from $n$ i.i.d.~data in $d$ dimensions under $(\varepsilon,\delta)$-differential privacy. Although extensively studied in literature, existing solutions fall short on two key aspects: ($i$) even for Gaussian data, existing private algorithms require the number of samples $n$ to scale super-linearly with $d$, i.e., $n=\Omega(d^{3/2})$, to obtain non-trivial results while non-private PCA requires only $n=O(d)$, and ($ii$) existing techniques suffer from a non-vanishing error even when the randomness in each data point is arbitrarily small. We propose DP-PCA, which is a single-pass algorithm that overcomes both limitations. It is based on a private minibatch gradient ascent method that relies on {\em private mean estimation}, which adds minimal noise required to ensure privacy by adapting to the variance of a given minibatch of gradients. For sub-Gaussian data, we provide nearly optimal statistical error rates even for $n=\tilde O(d)$. Furthermore, we provide a lower bound showing that sub-Gaussian style assumption is necessary in obtaining the optimal error rate.

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