Private Geometric Median in Nearly-Linear Time

Estimating the geometric median of a dataset is a robust counterpart to mean estimation, and is a fundamental problem in computational geometry. Recently, [HSU24] gave an (ε, δ)-differentially private algorithm obtaining an α-multiplicative approximation to the geometric median objective, 1 n ∑i ∈ [n] \|· - xᵢ\|, given a dataset D := ᵢ ∈ [n] ⊂ Rᵈ. Their algorithm requires n √ d · 1 αε samples, which they prove is information-theoretically optimal. This result is surprising because its error scales with the effective radius of D (i.e., of a ball capturing most points), rather than the worst-case radius. We give an improved algorithm that obtains the same approximation quality, also using n √ d · 1 αε samples, but in time O(nd + d α²). Our runtime is nearly-linear, plus the cost of the cheapest non-private first-order method due to [CLM+16]. To achieve our results, we use subsampling and geometric aggregation tools inspired by FriendlyCore [TCK+22] to speed up the "warm start" component of the [HSU24] algorithm, combined with a careful custom analysis of DP-SGD's sensitivity for the geometric median objective.