NeurIPS 2020

### Review 1

Summary and Contributions: This paper tackles the problem of reducing the support of a measure, while preserving the expectation (w.r.t the above measure) of n given functions.

Strengths: The authors propose a (uniform) random sampling based algorithm as well as a more greedy approach for this problem. The algorithm

Weaknesses: While the algorithm seems to perform well empirically, the theoretical guarantees on the runtime are not very encouraging. How do these theoretical bounds compare with the bounds (if any) that are available for the algorithms in existing literature e.g. [4,5,6]? The bounds for the geometric greedy algorithm are not substantially better-- I was hoping to see that there would be a decrease in the number of iterations needed.

Correctness: I have not checked the supplementary. The approach seems reasonable, the empirical methodology is fine.

Clarity: Comments on the exposition given below.

Relation to Prior Work: I would like a better discussion of how these techniques relate to prior work. Please see below.

Reproducibility: Yes

Additional Feedback: I think the writing could be a little more expository. While I could follow the individual mathematical steps, the overall intuition behind the reduction is not clear to me, it would be nice if this can be brought out. For example, it might be nice, if the authors could add some pictures in the appendix to indicate what the negative cone really means geometrically. In effect it seems to be that the choice of n-points is random, modulo the fact that an n+1-th point can be found in the intersection of the negative cone and the set \bf{x}. Overall, given the runtimes presented in the paper, the practical significance of the result is not clear to me. In the least square regression example given, if we are fine with an approximation, there are results known that need only O(d*log(d) / eps^2), where (1+eps) = the approximation factor. To be fair, this technique gives an exact solution. But is it clear what benefits we hope to get by getting an exact instead of a close approximation? A related question that I had is the following-- in the general statement of what the authors are trying to prove, if we are with an approximation, is it possible to get a substantially more efficient algorithm? The equation between lines 141 and 142 has a typo, w_1 should be w_{n+1}. While the contribution seems quite nice, I would like to read a better explanation of what the benefit is over the approaches in [4, 5, 6]. It would also be nice to elaborate what the authors mean by saying that this approach is “complementary to previous work”.

### Review 2

Summary and Contributions: This work addresses the problem of finding a discrete measure, which is supported on a subset of all data points, such that its expectation is equal to the original measure over the entire set. Authors propose a "greedy geometric sampling" heuristic to obtain the measure efficiently when the dimension is much smaller than the total number of points. This is validated in their experiments where they can combine their approach with existing methods to further reduce the runtime of a canonical least-squares problem.

Strengths: Concepts from cone geometry are used in the algorithm, which is novel. The experiments show that the reduction in computation time can be significant.

Weaknesses: The randomized algorithm can take a large value of resampling iterations, however authors give a "hybrid" algorithm that switches execution to existing deterministic algorithms when the run time exceeds a threshold. In the empirical evaluation, the advantage of resetting the algorithm over just running Algorithm 2 is not clear (both green and orange curves in Fig 1 look very similar).

Correctness: The method of algorithm 2 seems intuitive and I have checked its complexity statement (Proposition 3).

Clarity: The paper is written ok, though it would be better if some of the proofs could brought into the main text (for eg, Theorem 3, 4). Some typos: *line 125, should say "C^-(x*) \cap x \neq \empty" *display after line 141, should say "\hat{w}_{n+1}" after second equality

Relation to Prior Work: The references are from a wide variety of areas, as the authors say at line 31--32. The authors mention how their approach differs from previous work, but the core idea seems intuitive, and I wonder if it has been employed in other problem areas (some mentioned below)

Reproducibility: Yes

Additional Feedback: *The aim in algorithm 2 seems to be to select points that are as "spread out" as possible. Can that be encoded as an explicit optimization objective, and some "sparse greedy approximation" method [A] be employed to optimize it? *Another paper where vectors are selected to be as spread out is [B]. Eq (7) in [B] tries to make a selection so as to increase the angle between all pairs of vectors. [A] Clarkson, Kenneth L. "Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm." ACM Transactions on Algorithms (TALG) 6.4 (2010): 1-30. [B] Aljundi, Rahaf, et al. "Gradient based sample selection for online continual learning." Advances in Neural Information Processing Systems. 2019.

### Review 3

Summary and Contributions: A well known Theorem, a consequence of Caratheodory's convex hull theorem, states that if we have n functions over a probability measure on a large support, we can always choose a another measure over n+1 points in the support so that the expectation of the functions remains the same. The submission proposes two algorithms to find such a reduced support and measure that are incomparable, a deterministic one and a randomized one. THe submission includes experimental evaluation of the algorithms and some analysis.

Strengths: This is a very natural summarization problem and it is interesting to look at it algorithmically.

Weaknesses: The approaches (based on solving linear systems) seem basic and it is not clear if they offer new techniques. I would have liked to see some motivating discussion of potential applications. After feedback: Thank you authors for the feedback! My main concerns was getting more confirmation of novelly (natural problem, simple solution) but it was addressed by the feedback and other reviews. I increased my score. Still, an extended motivating discussion would be helpful .

Correctness: I did not verify but I did not see issues when spot-checking.

Clarity: It is written ok.

Relation to Prior Work: As far as I know.

Reproducibility: Yes

### Review 4

Summary and Contributions: This paper addresses the problem of reducing the support of discrete measures supported on N atoms to (n+1) atoms while preserving the expectations (integrals) over n given real-valued functions. It is built upon the Caratheodory theorem and illustrates its benefits on a fast least-square solver. The contributions are: 1) Theoretical conditions on the geometry of the n+1 atoms (theorem 3, negative cones). 2) A generic algorithm that uses this geometry (by maximizing the volumes of the cones) to reduce the support of any discrete measure. 3) The ability to combine this method with other existing ones.

Strengths: This paper is very relevant to the Neurips community since it tackles a well defined and well explained ML problem with a theoretically grounded method, which claims are proved and evaluated with care. I have also enjoyed the empirical validation on both real and synthetic data. In particular I have appreciated the analysis of the different regimes and the suggestion of the author to prefer this method for the big data regime and to combine it with previous work when the number of samples is low (the hybrid algorithm in the supplementary materials). The empirical validation also illustrates well the difference between the theoretical worst case complexity and the running time for real applications.

Weaknesses: I do not see real weaknesses in this paper.

Correctness: The claims are supported by proofs (mainly given in the supplementary materials) and I have empirically double checked theorem 3, mostly to better get the geometrical intuition behind the method. The empirical methodology seems solid.

Clarity: I am not an expert in this field but I have enjoyed reading this paper, since it is very well written and easy to follow. I have also appreciated the pedagogical effort of presenting the 2D case whenever needed to help the reader grasp the intuition.

Relation to Prior Work: This paper clearly discuss prior works, clearly states its contributions and also shows how it differs and how it can be combined with prior contributions.

Reproducibility: Yes

Additional Feedback: There is a small mistake on line 125, where it should be the negative cone of x^{\star} instead of x. I would recommend the authors to increase the font size and ticklabels size of all their figures. The labels are sometimes different from one figure to another, which does not help and the colors are sometimes changing for a given method from a figure to another. I would recommend keeping one color and one label per method across all figures. On line 230, the authors comment that the run time is approximately O(n). I believe that to illustrate this claim, it would be easier to plot figure 1 in log-log scale. On figure 3, the distributions of run times are totally concentrated on the left half of the plot and I would suggest to restrict the x axis to this range in order to better visualize the distributions. Also I believe the two rows could be merged into one. Showing the distribution for both the log-random and the randomized algorithm in the same plot.