Related papers: Private Geometric Median

Private Geometric Median

URL: http://arxiv.org/abs/2406.07407v1
Date: Tue, 11 Jun 2024 16:13:09 GMT
Title: Private Geometric Median
Authors: Mahdi Haghifam, Thomas Steinke, Jonathan Ullman,
Abstract summary: We study differentially private (DP) algorithms for computing the geometric median (GM) of a dataset. Our main contribution is a pair of DP algorithms for the task of private GM with an excess error guarantee that scales with the effective diameter of the datapoints.
Score: 10.359525525715421
License: http://creativecommons.org/licenses/by/4.0/
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 $\theta$ that minimizes the sum of the Euclidean distances to these points, i.e., $\sum_{i=1}^{n} \|\theta - x_i\|_2$. 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.

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