DP-PCA: Statistically Optimal and Differentially Private PCA

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 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.