Title: A Computationally Viable Numerical Gradient-based Technique for Optimal Covering Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: Optimal cover

Optimal n-net

Sensitivity of nonlinear programs

Numerical gradient algorithm
Abstract: The problem of optimally covering a given compact subset of $\mathbb{R}^N$ with a preassigned number $n$ of Euclidean metric balls has a long-standing history and it is well-recognized to be computationally hard. This article establishes a numerically viable algorithm for obtaining optimal covers of compact sets via two key contributions. The first is a foundational result establishing Lipschitz continuity of the marginal function of a certain parametric non-convex maximization problem in the optimal covering problem, and it provides the substrate for numerical gradient algorithms to be employed in this context. The second is an adaptation of a stochastically smoothed numerical gradient-based (zeroth-order) algorithm for a non-convex minimization problem, that, equipped with randomized restarts, spurs global convergence to an optimal cover. Several numerical experiments with complicated nonconvex compact sets demonstrate the excellent performance of our techniques.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the optimal covering problem, where the goal is to obtain the optimal way to cover a compact subset of Euclidean space with a given number of Euclidean balls, where the optimality is formulated based on the maximum distances of the centers of the balls to the subset. The paper proposes a reformulation of the original problem to make it possess desirable regularity properties, and thus propose an algorithm based on numerical gradient to solve the problem. The efficiency of the algorithm is quantified, and numerical experiments are provided. The experiments demonstrate that the proposed algorithm indeed achieves good performances.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. Sufficient background is provided, motivating the research problem and demonstrating its importance.\n\n2. The proposed algorithm is theoretically grounded, and is proven to yield reasonable performances.\n\nWeaknesses:\n\n1. As acknowledged by the authors, the proposed algorithm is still computationally costly.\n\n2. The empirical results largely serve to demonstrate the correctness of the proposed algorithm, without showing its efficiency.\n\nQUESTIONS:\nI am not that familiar with the tackled problem. As briefly mentioned in Strengths and Weaknesses, I find the empirical results relatively weak. To me the presented empirical results largely show that the algorithm was able to find reasonable results, without showing that it finds really good results or has good convergence properties due to the lack of a ground truth solution or relevant baselines. While the authors argued that no competing algorithms are available in the literature in Section 4, the authors also mentioned some previous approaches in Section 1. It would be ideal if the authors can improve the part on numerical results using as baselines some previous approaches, thus providing empirical evidence that the proposed algorithm is indeed a significant improvement upon the previous ones either in terms of computational efficiency or accuracy or both.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of covering a (possibly nonconvex) bounded set using fixed number, say $n$ balls of smallest possible radius. The quantity of interest are centers of those $n$ balls (called as Chebyshev net of size $n$) and their radius. The main contribution is to reformulate the problem into a  min-max type problem where the inner maximization is nonconcave. The resulting outer value function is nonconcave, nonsmooth and Clarke ``irregular\" in variables that represent the location of ball-centers. They show that under mild assumption, the outer Clarke irregular function is Lipschitz continuous. Hence, they can apply zeroth-order methods developed in the recent literature for such nonconvex nonsmooth problems.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper makes minimal assumptions about the structure of the set $\\mathcal{M}$ that needs to be covered. They provide novel insights on how one can prove Lipschitz continuity outer value function $\\mathcal{G}(x_1, \\dots, x_n)$ using such representation, along with classical results from analysis. This may be of independent interest in other applications. The paper is written with great clarity. I didn't see any obvious typos. All the relevant points are discussed in some detail along with the analysis.\n\nOn the weakness side: The formulation requires solving nonconvex maximization to evaluate value function $\\mathcal{G}(x)$. Indeed we have $\\mathcal{G}(x) = \\max t$ s.t. $h(x, y, t) \\le 0$ where $h$ is  nonconcave in $y$. One requires global solution of this problem in order to evaluate $\\mathcal{G}$ accurately. This is required since zero-th order schemes for problem $\\min_x \\mathcal{G}(x)$ with randomized smoothing requires high quality estimate to estimate numerical \"average\" descent directions. Without the global solves of such problem, the method may fail as illustrated in their own experiment in appendix. The good thing is reader is well-cautioned about it in the main paper.\nAnother weakness is that the algorithmic contribution in this paper are minimal. They are used as blackbox. Some important details and considerations for convergence of these algorithms are missing as well. I have some questions in this regard below.\n\nQUESTIONS:\nIt is necessary for the convergence that the domain of $x$ remains bounded. It is proved that it suffices to search on a bounded space $\\mathbb{B}(\\mathcal{M})$ which is based on Chebyshev center and radius for case of $n =1$. However, not much is said about the computation of these. Can you expand on how to find such a point?\n\nIn line 60, you mentioned a second version in equation (2) where the $n$ centers must belong $\\mathcal{M}$ itself. However, not much is said about this problem later on. How would Algorithm 1 change is this problem needs to be solved. Does that also mean projection is required onto set $\\mathcal{M}^n$ instead of line 13 of Algorithm 1. Moreover, if $\\mathcal{M}$ is nonconvex, does the convergence analysis borrowed from Liu et al. 2024 still hold if one needs to project onto a nonconvex set?\n\nTheorem 3.6 proves (or borrows) the convergence result for $\\min_x \\mathcal{G}$ problem. However, it does not explicitly state the impact of $b_1, b_2$. Although, it is clear that bigger $b_1,b_2$ will give more smoothing and possibly better estimates of descent direction. Also, I am wondering what are the conditions required on step-size $\\gamma$. Usually that is critical to get any convergence/complexity guarantees. Without it, the development looks incomplete to me.\n\nI see that you are treating the variable $x =(x_1, \\dots, x_n)$ as a full vector in dimension $Nn$. However, due to the obvious structure of the problem, it probably makes sense to consider block-wise updates and sample $b_1/n$ in unit spheres in $\\mathbb{R}^N$, a total of $n$-times. Would that benefit in scaling this approach?\n\nLIMITATIONS:\nI think the authors clearly mentioned some of the limitations, as those in experiment on Example D.1 where they use IPOPT instead of ``global optimizer\" BARON to solve the inner maximization problem.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers an optimally covering problem, which is generally NP-hard and computationally challenging. To challenge this problem, this paper proposes a new approach which consists of (according to the authors claim in the paper) (i) A new reformulation of the problem which realizes a smooth objective function, (ii) a new gradient-based algorithm is proposed to minimize the objective function in (i), and its computational complexity is analyzed, and (iii) numerical experiments verifies the actual efficiency of the proposed framework.\n\nSTRENGTHS AND WEAKNESSES:\nFirst of all, I have to confess that I am not an expert in this field, and I may be misunderstanding something.  I also have to say that I did not necessarily get into the mathematical detail of the theorems.\n\nStrength: As far as I went through from the abstract to the conclusions, I felt that this paper is very well-organized (well-written). Although I myself is not an expert on this topic, the introduction is clearly written such that the situation in the research field becomes visible. If we completely believe the authors' claim on their contributions, the algorithm is new, which is based on a new reformulation (and thus the development of the new algorithm is not straightforward), and the algorithm is good both theoretically and numerically.\n\nWeakness: Since I could not get into the technical detail, I cannot think of any defects. One thing I noticed is that no information on the actual computational time in the numerical experiments is given. The reason is clarified in the beginning of the numerical experiments section, where it is claimed that no direct comparison of computation time is feasible (since there is no competing existing algorithm). Still, I think some information on actual computation time might be informative for those who hope to employ their algorithm.\n\nQUESTIONS:\nI would like to ask the authors how they think about my suggestion written in Weakness.\n\nFrom the reasons I wrote above (including the fact that I am not an expert), I begin with a neutral rate. I would like to reconsider my rating depending on the additional information in the discussion stages.\n\nLIMITATIONS:\nThe authors do not explicitly state any limitations. \nI did not notice any other limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript describes an algorithm to construct a Chebyshev net\nwith $n$ balls for a point set in $\\mathbb{R}^N$. Formally, the\noptimization problem for the optimal cover amounts to a min max min\nproblem. The authors suggest a convex relaxation of the inner\nminimization in the spirit of quadratic inequality constraints.\n\nThe central theorem 3.2 establishes regularity properties that justify\nthe problem relaxation with Lagrangian (9).\nThe mansucript presents simple examples of covering an ellipse and a\n2-dim. non-convex shape.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe corage problem plays an important role in statistical learning\ntheory (usually expnential covers) as well as in computational\ngeometry. The manuscript has a distinct computational geometry flavor\nbut it might be relevant for approximations in machine learning.\n\nTheoretically the manuscript describes the algorithm quite well. It is\nmathematically solid as far as I can judge.\n\nThe algorithm is demonstrated by constructing two simple coverings. Scaling \n\nWeaknesses:\n\nI find the topic not central to machine learning or neural\ncomputation. It provides theoretical concepts and an algorithm which\nhas a weak relation to approximation theory, e.g., \\epsilon covers.\n\nThe numerical demonstrations are only illustrations but do not refer\nto reaL computational problems in ML. The examples in two dimensions\nwith 8 or 6 covers might not even be interesting for computational\ngeometry.\n\nQUESTIONS:\nIt would be good to receive some\nexamples of coverings in high dimensions ($N \\approx 100$) with 1000\nto 10000 balls. How does the algorithm scale in this parameter range.\nOne might consgtruct concrete covers by finite size scaling in such a\nparameter range and this would be a real application with relevance\nfor ML.\n\nThe authors should say something about the scaling in high dimensions,\ni.e., point clouds in N dimensions like in coding theory.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-18"}