A Private Approximation of the 2nd-Moment Matrix of Any Subsamplable Input

We study the problem of differentially private second moment estimation and present a new algorithm that achieve strong privacy-utility trade-offs even for worst-case inputs under subsamplability assumptions on the data. We call an input (m, α, β)-subsamplable if a random subsample of size m (or larger) preserves w.p ≥ 1 − β the spectral structure of the original second moment matrix up to a multiplicative factor of 1 ± α. Building upon subsamplability, we give a recursive algorithmic framework similar to Kamath et al. (2019) that abides zero-Concentrated Differential Privacy (zCDP) while preserving w.h.p the accuracy of the second moment estimation upto an arbitrary factor of (1 ± γ). We then show how to apply our algorithm to approximate the second moment matrix of a distribution D, even when a noticeable fraction of the input are outliers.