Private Geometric Median in Nearly-Linear Time

• URL: http://arxiv.org/abs/2505.20189v1
• Date: Mon, 26 May 2025 16:32:49 GMT
• Title: Private Geometric Median in Nearly-Linear Time
• Authors: Syamantak Kumar, Daogao Liu, Kevin Tian, Chutong Yang,
• Abstract summary: Estimating the geometric median of a dataset is a fundamental problem in computational geometry.<n> [HSU24] gave an $alpha$-multiplicative approximation to the geometric median objective.<n>We give an improved algorithm that obtains the same approximation quality.
• Score: 11.537965585323304
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $(\varepsilon, \delta)$-differentially private algorithm obtaining an $\alpha$-multiplicative approximation to the geometric median objective, $\frac 1 n \sum_{i \in [n]} \|\cdot - \mathbf{x}_i\|$, given a dataset $\mathcal{D} := \{\mathbf{x}_i\}_{i \in [n]} \subset \mathbb{R}^d$. Their algorithm requires $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\varepsilon}$ samples, which they prove is information-theoretically optimal. This result is surprising because its error scales with the \emph{effective radius} of $\mathcal{D}$ (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 $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\epsilon}$ samples, but in time $\widetilde{O}(nd + \frac d {\alpha^2})$. 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.

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