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We study the canonical statistical task of computing the principal componentfrom $n$ i.i.d.~data in $d$ dimensions under$(\varepsilon,\delta)$-differential privacy. Although extensively studied inliterature, existing solutions fall short on two key aspects: ($i$) even forGaussian data, existing private algorithms require the number of samples $n$ toscale super-linearly with $d$, i.e., $n=\Omega(d^{3/2})$, to obtain non-trivialresults while non-private PCA requires only $n=O(d)$, and ($ii$) existingtechniques suffer from a non-vanishing error even when the randomness in eachdata point is arbitrarily small. We propose DP-PCA, which is a single-passalgorithm that overcomes both limitations. It is based on a private minibatchgradient ascent method that relies on {\em private mean estimation}, which addsminimal noise required to ensure privacy by adapting to the variance of a givenminibatch of gradients. For sub-Gaussian data, we provide nearly optimalstatistical error rates even for $n=\tilde O(d)$. Furthermore, we provide alower bound showing that sub-Gaussian style assumption is necessary inobtaining the optimal error rate.

DP-PCA: Statistically Optimal and Differentially Private PCA