Title: Private Geometric Median in Nearly-Linear Time
Venue: NeurIPS 2025
Accepted: yes
Keywords: Differential Privacy

Geometric Median

Stochastic Convex Optimization
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 $(\epsilon, \delta)$-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 $D$ of $x_i$ for $i \in [n]$. Their algorithm requires $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\epsilon}$ samples, which they prove is information-theoretically optimal. This result is surprising because its error scales with the effective radius of $D$ (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 [CLMPS16]. To achieve our results, we use subsampling and geometric aggregation tools inspired by FriendlyCore [TCKMS22] 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of privately estimating the geometric median, a topic that has been investigated previously and an information-theoretically optimal sample complexity is known. In this work, the authors propose an algorithm that achieves the same optimal sample complexity while running in a nearly linear time $\\tilde{O}(nd + d/\\alpha^2)$ for $n$ input data points of dimension $d$, significantly improving the previous result of [HSU24]. The workflow closely follows the framework established by [HSU24], incorporating several novel techniques that enhance runtime efficiency.\n\n[HSU24] Mahdi Haghifam, Thomas Steinke, and Jonathan Ullman. Private geometric median. Advances in Neural Information Processing Systems, 37:46254–46293, 2024.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The estimation of the geometric median is a fundamental problem in computational geometry. This work presents a private algorithm that operates in nearly linear time, marking a substantial advancement over previous results.\n2. The desired runtime is achieved by making several improvements to different parts of [HSU24]'s framework, which are novel and technically solid.\n3. Comprehensive experiments are conducted to show the effectiveness of the proposed techniques.\n\nWeakness:\n\n1. The algorithm requires prior knowledge of parameters $r$ and $R$, which provides lower and upper bounds on the effective radius $r^{(0.9)}$.\n2. In Section 5 the authors evaluate only Algorithm 1 and 3. The presented analysis of Algorithm 2 incurs large constants that largely affect its performance in practice. Therefore, the entire algorithm is not ready for real-world deployment.\n\nQUESTIONS:\n1. Is it possible to remove the requirement of knowledge $r$ and $R$ ? As mentioned in the paper, the resulting sample complexity is independent of these parameters. I wonder if this dependence in the time complexity bound can be removed as well.\n2. This work focuses on approximate DP, where a small probability $\\delta$ of privacy violation is allowed. Is it able to achieve the more stringent notion of pure DP?\n3. The runtime of the private algorithm is $\\tilde{O}(nd + d/\\alpha^2)$ while one of the non-private algorithms in [CLM+16] runs in $O(d/\\alpha^2)$. Is it possible to develop a private algorithm with this time complexity?\n\nSome (possible) typos:\n\n1. [Line 110] \"the FriendlyCore algorithm of [HSU24]\": This seems to be a wrong citation.\n2. In the proof of Lemma 5 [Line 520], the setting of $\\mu$ and $\\epsilon$ may only lead to a probability bound of $\\exp(-k\\cdot \\frac{n-1}{n} /3)$, which is slightly looser than the claimed $\\delta / 4T$.\n3. [Line 537] Lemma 6 should be referenced as Lemma 2.\n4. [Line 563] $f_i^\\star\\ge 0.7 k$ should be corrected to $f_i^\\star \\ge 0.75 k$\n5. [Line 618] what is meant by \"large enough $C$\"?\n6. [Line 619] \"we run Algorithm 2 with this value of $\\hat{r}$.\" However, Lemma 7 requires $\\hat{r} \\ge r^{(0.75)}$ while Lemma 6 only guarantees $\\hat{r}\\ge \\frac{1}{4}r^{(0.75)}$. I think this should be corrected by passing $4\\hat{r}$ to Algorithm $2$, and some related quantities need to be modified accordingly.\n7. [Line 623] Lemma 6 should be referenced as Lemma 7.\n\n[CLM+16] Michael B Cohen, Yin Tat Lee, Gary Miller, Jakub Pachocki, and Aaron Sidford. Geometric median in nearly linear time. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 9–21, 2016.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe task is to compute geometric median of a point set in a differentially private way. The geometric median is the minimizer of $\\sum_i ||\\cdot - x_i ||$ for Euclidean norm, and this paper is happy with approximate minimizer achieving $1+\\alpha$  multiple of the minimum.\n\nA recent work (HSU24) has provided an algorithm solving the task with a certain sample complexity and proved that it is the optimal complexity. However, there was a significant gap in the computational efficiency between private and non-private geometric median computation in the prior work.\n\nThis work provides an algorithm similar in spirit to the one of (HSU24) that runs in a very similar time to the non-private algorithms. Per my understanding, the differences in the algorithms are subtle (but necessary for the improved runtime), but the big picture stays the same.\n\nThe algorithm runs in three phases, first, an effective radius r (i.e., the majority of points are contained in a ball of this radius) of the input is computed, then, a crude estimate of the geometric median (in a distance at most r) is computed. Finally, a custom DP-SGD analysis is performed leveraging the structure of geometric median problem.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is easy to read and I like the way how is it written, that it provides the intuition and sketches results before diving into the technicalities. I would only suggest to introduce the $r^{(0.9)}$ notation more explicitly.\n\nI find this to be a valuable contribution since (private) geometric median problem is quite important and the improvements in runtime are significant. The DP-SGD analysis technique used in the paper taking advantage of the fact that the gradients are unit vectors can potentially also be used elsewher.\n\nOn the other hand, the scope of the paper is very narrow and the improvements are only in the run time and that is why I do not lean to the highest score.\n\nQUESTIONS:\n-\n\nLIMITATIONS:\n-", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a DP optimization algorithm to solve the following task: given (x1,...,xn) in (R^d)^n, such that the data points have a norm of R, we aim to solve x* = argmin_x sum_i ||x - xi||. This problem is known as private geometric median. The previous work [Haghifam-Steinke-Ullman'24] proposes an algorithm to provide (1+alpha) multiplicative error with the sample complexity of sqrt(d)/(alpha*epsilon). However, the runtime of their proposed algorithms is either quadratic in n or quadratic in d. The main result of this paper is to push the time complexity to nd + d/alpha^2. Notice that d/alpha^2 is the optimal time complexity of the first-order method even non-privately.\nThe idea behind the proposed algorithm is as follows: it has three stages. \n\n1- In the first stage, an algorithm similar to the algorithm proposed in [Haghifam-Steinke-Ullman'24] (these algorithms are inspired by GoodRadius of [Stemmer-Nissim-Vadhan'16]). The goal is to find a radius privately around the geometric median that contains 70% of the data points. The main idea is instead of computing all the n^2 pairwise distances is sub-sampling that makes the algorithm run in linear time.\n\n2- The second stage is inspired by the FriendlyCore of Tsfadia et al. and it consists of finding a center with the promise that there exists a radius that contains 70% of the data points. The new idea is that FriendlyCore can be sped up to run in O~(nd) time (independently of R/r) using subsampled weights.\n\n3- Finally, the third stage, which is very interesting, is a bespoke analysis of the DP-SGD for the geometric median loss function. The idea here is to \"warm-start\" DP-SGD from the centerpoint given by the output of Stage 2 and the radius given by Stage 1. The main idea is to show that when gradient descent steps are contractive for an appropriate step size, we can use the reduction in [Koren-Feldman-Talwar'20] to convert the contractivity to a private algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nThe result of this paper is interesting: it combines lots of interesting techniques to provide a very intuitive algorithm that improves the prior work's runtime.\n\nThere are a few weaknesses:\n\n- If we view this problem from the standpoint of DP optimization, it would be interesting to find a broader class of loss functions for which we can obtain an algorithm with multiplicative guarantee.\n\n- The optimal dependence of the sample complexity on R is unclear. Note that in one dimension there are algorithms that require log*(R) samples.\n\n- The problem of geometric median is very related to the problem of privately finding a point in the convex hull. A helpful comparison for the community is to compare the problem of private geometric median with the problem of privately finding a point in the convex hull, for instance, see the following paper:\nKaplan, H., Mansour, Y., Moran, S., Stemmer, U., & Tur, N. (2025, June). On differentially private linear algebra. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (pp. 2362-2373).\n\nLine 110 - the citation is wrong for the FriendlyCore algorithm.\n\nQUESTIONS:\nplease see above the weakness part\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a novel and practically relevant algorithm for computing a differentially private geometric median in high dimensions, achieving nearly-linear runtime. The authors build upon the FriendlyCore warm-start technique and integrate a refined DP-SGD procedure with a smoothed subgradient to overcome the non-smoothness of the geometric median objective.\nTheir proposed algorithm achieves:\n- Near-optimal sample complexity\n- Strong error guarantees comparable to prior work\n- A significant improvement in runtime from prior quadratic-time algorithms to near-linear.\nEmpirical evaluation demonstrates a 30 times speedup on synthetic and real data benchmarks, without significant loss of accuracy, and with robust error performance across multiple trials.\n\nSTRENGTHS AND WEAKNESSES:\nThe geometric median problem is a fundamental problem in many domains including clustering, robust statistics and social choice. This paper makes a technically sound and valuable contribution offering a novel and efficient . The combination of a warm-start approximation with a customly analyzed DP-SGD routine is elegant and well-justified, addressing the challenge of non-smooth optimization under differential privacy.\n\nThe authors do a rigorous analysis using mathematical proofs but also give an efficient implementation and empirical evaluation of their method. The writing is clear and well-organized, with helpful figures (notably Figures 1 and 2) to illustrate empirical gains.\n\nQUESTIONS:\nThe algorithm depends on the parameters r, R? The upper bound is more justified as there may be a natural bound on the points or one could normalize them. However, r is less justified. How can one compute it and what happens if it is not correct? Could you obtain an error with an additive term relative to r?\n\n- Lemma 9 seems strange. How does n >= 20 relate to (eps,delta)-dp? Seems like the condition is artificial and It would be nice to remove it.\n\n- I am never a fan of misusing the O tilde notation. Theorem 1 writes O tilde to hide all polylogarithmic factors (even for unrelated variables) but it should only be used to mean hiding polylogarithmic factors of the quantity inside the O tilde. It would be nice to fix the theorems appropriately rather than adding a footnote.\n\nLIMITATIONS:\nYes, the authors describe the potential suboptimality of the runtime as a limitation tied to using first order methods for optimization. Alternative methods could potentially have better dependency.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-08"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new efficient algorithm for computing in a\ndifferentially private way the geometric median of a set of\ndata. Recent work has proposed two-phases (warm-up and boosting)\nalgorithms for a multiplicative approximation of the geometric median\nwith optimal sample complexity but running in polynomial time. The\nalgorithm proposed in this paper achieve similar accuracy and sample\ncomplexity but runs in almost linear time (for some range of the\nparameters).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The paper address an important problem: improving the runtime of\n algorithms for differentially private geometric median with good\n average error rates.\n\n- The paper improves the runtime of the warm up phase by using\n  subsampling in a \n    \n- In the boosting phase the paper gives an analysis of DPSGD\n  leveraging the properties of the geometric median to obtain an\n  improved runtime.\n\n\nWeaknesses\n\n- The proposed algorithm uses as a subroutine an adaptation of the\n  FriendlyCore procedure from [TCK+22] but it is unclear why this\n  departure from the original algorithm is needed.\n\nRecent work proposed differentially private algorithms for the\ngeometric median with average error guarantees but those algorithm\nhave a polynomial runtime, which can be problematic in large data\nsettings. It is thus important to have more efficient algorithms and\nthis paper does exactly that.The proposed algorithm uses a two-phases\napproach, based on warm-up and boosting, as the ones in previous work\nbut it optimizes the running time of each step.\n\nTo optimize the runtime of the warm-up, which has itself two phases,\none for estimating the radius and one for estimating the centerpoints,\nthe paper uses subsampling. In the first phase the algorithm subsample\npoints to estimate the radius. This change is relatively\nstraightforward.  In the second phase, the proposed algorithm departs\nfrom previous work and uses as a subcomponent the FriendlyCore\nprocedure from [TCK+22].  In orde to improve the efficiency of this\nprocedure the algorithm uses subsample in the computation of the\nweights associated with each data point. This change is non-trivial\nand needs an interesting change in the privacy proof. However, it is\nunclear why this departure from the original algorithm is needed since\nthe obtained improvement is quite limited.\nTo optimize the runtime of the boosting phase the paper proposes an\ninteresting ad-hoc analysis of DP-SGD specific to the geometric\nmedian.  Specifically, the paper uses the properties of the geometric\nmedian to control the sensitivity of DP-SGD and obtain an optimization\nin nearly-linear time.\nOverall, the proposed algorithm shows considerable improvement thanks\nto non-trivial changes to the algorithms from previous work.\n\nA couple of minor points:\n- in equation (2) the notation for the indicator function is not yet\n  defined\n- line 110 has a wrong citation\n\nQUESTIONS:\nCan you clarify why you use the FriendlyCore procedure for estimating the centerpoints? What do you gain?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-19"}