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Private Geometric Median

Mahdi Haghifam, Thomas Steinke, Jonathan Ullman

TL;DR

This work develops private algorithms for computing the geometric median that adapt to the data’s intrinsic scale rather than an a priori bound. It introduces a two-phase private localization (RadiusFinder + Localization) to privately estimate the quantile radius and fix a localized search region, followed by two polynomial-time fine-tuning methods (DP-GD and LocDPCuttingPlane) that achieve excess error scaling with the effective diameter $\Delta_{\gamma n}(\theta^*)$. A separate pure-DP method (SInvS) based on inverse smooth sensitivity provides a different privacy trade-off, while a lower bound shows near-optimal sample complexity $\tilde{\Omega}(\sqrt{d}/\varepsilon)$. Together, these results demonstrate that private GM can be both accurate and efficient, with decisions adapting to the data’s geometry and robustness to outliers, and they highlight remaining directions such as potential linear-time, optimally-tuned private GM algorithms.

Abstract

In this paper, we study differentially private (DP) algorithms for computing the geometric median (GM) of a dataset: Given $n$ points, $x_1,\dots,x_n$ in $\mathbb{R}^d$, the goal is to find a point $θ$ that minimizes the sum of the Euclidean distances to these points, i.e., $\sum_{i=1}^{n} \|θ- x_i\|_2$. Off-the-shelf methods, such as DP-GD, require strong a priori knowledge locating the data within a ball of radius $R$, and the excess risk of the algorithm depends linearly on $R$. In this paper, we ask: can we design an efficient and private algorithm with an excess error guarantee that scales with the (unknown) radius containing the majority of the datapoints? Our main contribution is a pair of polynomial-time DP algorithms for the task of private GM with an excess error guarantee that scales with the effective diameter of the datapoints. Additionally, we propose an inefficient algorithm based on the inverse smooth sensitivity mechanism, which satisfies the more restrictive notion of pure DP. We complement our results with a lower bound and demonstrate the optimality of our polynomial-time algorithms in terms of sample complexity.

Private Geometric Median

TL;DR

This work develops private algorithms for computing the geometric median that adapt to the data’s intrinsic scale rather than an a priori bound. It introduces a two-phase private localization (RadiusFinder + Localization) to privately estimate the quantile radius and fix a localized search region, followed by two polynomial-time fine-tuning methods (DP-GD and LocDPCuttingPlane) that achieve excess error scaling with the effective diameter . A separate pure-DP method (SInvS) based on inverse smooth sensitivity provides a different privacy trade-off, while a lower bound shows near-optimal sample complexity . Together, these results demonstrate that private GM can be both accurate and efficient, with decisions adapting to the data’s geometry and robustness to outliers, and they highlight remaining directions such as potential linear-time, optimally-tuned private GM algorithms.

Abstract

In this paper, we study differentially private (DP) algorithms for computing the geometric median (GM) of a dataset: Given points, in , the goal is to find a point that minimizes the sum of the Euclidean distances to these points, i.e., . Off-the-shelf methods, such as DP-GD, require strong a priori knowledge locating the data within a ball of radius , and the excess risk of the algorithm depends linearly on . In this paper, we ask: can we design an efficient and private algorithm with an excess error guarantee that scales with the (unknown) radius containing the majority of the datapoints? Our main contribution is a pair of polynomial-time DP algorithms for the task of private GM with an excess error guarantee that scales with the effective diameter of the datapoints. Additionally, we propose an inefficient algorithm based on the inverse smooth sensitivity mechanism, which satisfies the more restrictive notion of pure DP. We complement our results with a lower bound and demonstrate the optimality of our polynomial-time algorithms in terms of sample complexity.
Paper Structure (16 sections, 12 theorems, 15 equations, 1 figure, 1 table, 5 algorithms)

This paper contains 16 sections, 12 theorems, 15 equations, 1 figure, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Notation
  5. Private Localization
  6. Step 1: Private Estimation of Quantile Radius
  7. Step 2: Fast Localization
  8. Private Fine-tuning
  9. Fine-tuning Using DPGD
  10. Fine-tuning Using Noisy Cutting Plane Method
  11. Pure-DP Algorithm for Geometric Median
  12. Lower Bound on the Sample Complexity
  13. Numerical Example
  14. Conclusion and Limitations
  15. Preliminaries
  16. ...and 1 more sections

Key Result

Lemma 2.2

Fix $n \in \mathbb{N}$. For every dataset $\mathbf{X}^{(n)}$, for every $1/2\leq \gamma \leq 1$ and for every fixed $\nu$, the query $N(\nu) \triangleq \frac{1}{m}\mathop{\mathrm{\mathrm{max}\newline}}\limits_{\{i_1,\dots,i_m\}\subseteq [n] } \mathopen{}\mathclose{\left\{ N_{i_1}\mathopen{}\mathcl

Figures (1)

  • Figure :

Theorems & Definitions (15)

  • Definition 1.1: Quantile Radius
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Theorem 3.1
  • Lemma 3.2
  • ...and 5 more