  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.
