Private Geometric Median
This work addresses privacy-preserving robust statistics for data analysis, offering improved error guarantees over existing methods, though it is incremental in advancing DP algorithms for geometric median.
The paper tackles the problem of computing the geometric median under differential privacy without requiring strong a priori bounds on data radius, resulting in polynomial-time algorithms with excess error scaling with the effective diameter of the data points, and includes a lower bound showing optimal sample complexity.
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.