Core-sets for Fair and Diverse Data Summarization

We study core-set construction algorithms for the task of Diversity Maximization under fairness/partition constraint. Given a set of points $P$ in a metric space partitioned into $m$ groups, and given $k_1,\ldots,k_m$, the goal of this problem is to pick $k_i$ points from each group $i$ such that the overall diversity of the $k=\sum_i k_i$ picked points is maximized. We consider two natural diversity measures: sum-of-pairwise distances and sum-of-nearest-neighbor distances, and show improved core-set construction algorithms with respect to these measures. More precisely, we show the first constant factor core-set w.r.t. sum-of-pairwise distances whose size is independent of the size of the dataset and the aspect ratio. Second, we show the first core-set w.r.t. the sum-of-nearest-neighbor distances. Finally, we run several experiments showing the effectiveness of our core-set approach. In particular, we apply constrained diversity maximization to summarize a set of timed messages that takes into account the messages' recency. Specifically, the summary should include more recent messages compared to older ones. This is a real task in one of the largest communication platforms, affecting the experience of hundreds of millions daily active users. By utilizing our core-set method for this task, we achieve a 100x speed-up while losing the diversity by only a few percent. Moreover, our approach allows us to improve the space usage of the algorithm in the streaming setting.

Paper

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Peer review

Reviewer aN6A6/10 · confidence 3/52023-07-05

Summary

The study presents improved core-set construction algorithms for maximizing diversity under partition constraints. Two diversity measures were considered: sum-of-pairwise distances and sum-of-nearest-neighbor distances. The algorithms are validated through real-world application to time-sensitive message data, resulting in a 100x speedup and minor diversity loss while improving space usage in streaming settings.

Strengths

Algorithms 2, 3, and 4, despite their relative simplicity, provide significant insights, primarily stemming from the proofs of approximations related to their definitions. These algorithms' straightforward yet effective design and the resultant analytical insights shed new light on the problem. These aspects, therefore, make these algorithms especially appealing to a broad research community. Conference attendees will find these insights valuable, sparking potential discussions around their contributions and their implications in the field.

Weaknesses

While the paper offers some contributions, its structure requires revision for improved clarity, particularly moving the detailed 'Overview of the Algorithms' from the first section to the respective sessions dealing with the algorithms and balancing the disproportionate length of the first section within the overall context. The renaming of 'Diversity Maximization under Partition Constraints' to 'Fair Diversity Maximization' is somewhat perplexing. The 'fairness constraints' applied in this study do not differ significantly from the 'partition constraints' defined in the existing literature. Moreover, in other works (e.g., [22]), the term' fairness' is applied based on specific definitions like group fairness or statistical parity. The decision to redefine 'partition constraints' as' fairness constraints' without a clear distinction is somewhat confusing, making it more difficult to discern the precise contributions of this paper. Still, regarding the use of "fairness constraints" throughout the paper, the paper presents two main experiments. The first involves grouping a dataset of messages by creation time to compute a diverse summary that favors recently over older messages. The second uses the method to control each genre's contribution to a movie recommendation system. However, neither of these experiments demonstrates the necessity of applying fairness constraints (as defined in the literature) to solve the proposed problem. The GMM Algorithm, a fundamental building block of the method, must be correctly defined. Since it initially selects some arbitrary point as the first element of S, [11]'s definition is accurate. In fact, because S begins as an empty set, the execution of line 3 in the authors' definition is inconsistent. The text contains some English usage errors that affect clarity. Specifically, the phrase "based on greedy" found in line 107 could benefit from a more explicit definition. The part containing "along the way" in line 101 is vague and could be replaced with a more descriptive phrase. Furthermore, key terminologies, such as the "aspect ratio" of the dataset mentioned in lines 9, 86, and 94, are misused or not sufficiently defined. Some details of the experiments in the main text, such as the choice of 21,474 samples mentioned in line 294, appear arbitrary, and the rationale for their selection is buried in the appendix. Finally, the paper would greatly benefit from maintaining consistent naming and notation across all sections, enhancing its comprehensibility and overall cohesiveness. For example, the "div" definition in line 231 should use the same notation as in line 238; many references to core-set vs. coreset.

Questions

Section 1.3 defines the problem as selecting subsets S_i of k_i points from P_i to maximize div(S). However, the subsets S_i don't strictly contain k_i points in your final results. Instead, they include the core-set and the closest k_i points to each seed, effectively increasing the core-set size to k_i^2, as observed in the 'Core-set for SUM-PAIRWISE DIST' case. Could you please clarify how these results align with your original problem definition? Furthermore, could you please provide additional detail in your problem definition to help readers understand how the algorithm's results can serve as a potential solution to the problem?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

3 good

Limitations

The authors adequately addressed the limitations. There are no potential negative societal impacts from their work.

Reviewer 9Vmw6/10 · confidence 1/52023-07-07

Summary

A set of metric data points is comprised of m disjoint groups. The manuscript seeks for each group a subset of some given size such that the union of subsets the minimal pairwise distance (or sum of pairwise / nearest neighbour distances). For constant factor approximation of minimising the sum of pairwise distances Section 2 of the manuscript reports a union of core-sets per group which for each group is quadratic in size. This result is achieved by applying Gonzalez's algorithm for each group and then simple post-processing steps over the obtained cluster centroids. Further results are achieved for sum of nearest neighbour distances as a diversity measure.

Strengths

S1) Challenging and natural problem: fairness constraint akin to stratified sampling to achieve good group representation and diversity constraint to achieve good overall representation S2) Non-trivial analysis: size of core-sets for min-pair-sum independent of the data size and distribution unlike in prior work. S3) Practical: analysed methods are simple and sufficiently efficient for practical purposes S4) Presentation is for the most part clear and prior work seems to be thoroughly discussed and compared against.

Weaknesses

W1) Presentation could be clearer in some key points Fairness constraint is not fully clear. The empirical results seem to indicate that the sizes per group dictate the sizes of the core-sets, but other parts of the paper indicate that the size of the core-sets could be larger than the dictated group sizes. Are the core-set sizes meant to be the intermediate sizes before the post-processing step?

Questions

Q1) How exactly does the fairness constraint relate to the core-set / "solution" sizes and how can the core-sets be larger than the group sizes specified in the fairness constraints? (see W1)

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

1: Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.

Soundness

3 good

Presentation

1 poor

Contribution

3 good

Limitations

Relevant limitations seem to be discussed.

Reviewer S4pc6/10 · confidence 4/52023-07-12

Summary

This paper gives coresets for two different fair diversity measures (FDM). Fairness is defined as a partition matroid where the number of items from each part is fixed. The authors give a coreset for FDM under the sum-pairwise distance. They also give a coreset for sum-NN-distance. The proposed algorithms are compared against baselines for a real dataset. The main technique appears to be a based on the greedy k-center clustering (which the authors term as GMM).

Strengths

Coresets for these measures have not been studied, hence these are new results. Based on the experiments, the results are fairly practical. We get both a theoretical guarantee and a practical boost in performance. In the experiments, the authors compare with existing baselines to show that their coresets are much smaller while giving comparable performance. While the technical contribution relies mostly on the k-center algorithm, I like the fact that the algorithms are simple.

Weaknesses

The theoretical guarantee for the NN-distance diversity notion seems fairly weak, O(m log k). We already have an O(1) approximation guarantee algorithm anyway, so it is unclear why this is useful.

Questions

1. Why is the O(m log k) coreset guarantee useful?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The theoretical guarantees are not very impressive.

Reviewer CoyK7/10 · confidence 4/52023-07-31

Summary

This paper investigates data summarization problems. Given a set of points in metric space, the goal is to select k points that are diverse. Diversity could be measured as the sum of squared distances, the sum of nearest neighbor distances, etc. Constant approximation algorithms for these problem are known. This paper investigates core-sets for the problem. A shortcoming of prior work is that say given two data sets that have been summarized, how can one combine their summarizations? Simply taking two solutions and combining them can give poor performance. This paper offers coresets that are composable; that is, can be combined. The core sets are of size poly(k), independent of n. The approximation ratio is constant for the sum of pairwise distances and is O(m log k) when there are m classes for the NN metric. The ideas in the paper are simple natural approaches to the problem. A clear application of the core-sets is to get faster algorithms for big data sets. Indeed, this approach naturally leads to much better run times practically. Overall, the paper is reasonable. On one hand, they theoretical ideas are simple. On the other, it seems to me that these results should be known and they clearly make the problem scalable to larger data sets. Comment: The paper would be even stronger by pointing out more applications of the coresets. They should have further applications in the design of distributed or streaming algorithms.

Strengths

The paper gives a new approach to a natural problem. Coresets construction should be known They can achieve speed up practically using their ideas

Weaknesses

The algorithmic and analysis ideas are simple

Questions

Have the authors thought of applying these ideas to MPC for parallel computing algorithms? I think they should immediately lead to results in such settings.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

3 good

Contribution

3 good

Limitations

yes

Authorsrebuttal2023-08-17

Follow up on the rebuttal

Dear reviewer aN6A, We wanted to ask if we have addressed your concerns satisfactorily or if you have further concerns. Could you kindly let us know so that we have a chance to respond before the deadline? Thank you very much.

Reviewer 9Vmw2023-08-17

Thanks, it follows my rough understanding based on the reply and a quick revisit of the manuscript. The abstract states: - "Given a set of points $P$ in a metric space partitioned into $m$ groups, and given $k_1, . . . , k_m$, the goal of this problem is to pick $k_i$ points from each group $i$ such that the overall diversity of the $k = \sum_i k_i$ picked points is maximized." A similar problem is described in Definition 1.1. The goal in this manuscript is to build a core-set $T$ that may no longer contain the exact solution of the problem in Definition 1.1., but is guaranteed to contain an $\alpha$-approximate solution (cf. Definition 1.4) in terms of the diversity measure. The core-set $T$ in this work is the union of group-wise core-sets $T_i$ for each group $i$, i.e., $T = \bigcup_i T_i$. As an $\alpha$-approximate solution still needs to have $k_i$ points from each group, this follows that $T$ must have at least $k_i$ elements per color, but does not preclude having additional elements as the core-set is an intermediate result that can contain more than $k$ elements (e.g., achieved core-set sizes reported in Table 8 of the appendix). As in other core-set tasks the goal is to (approximately) solve some optimisation problem more efficiently by first narrowing it down to the most relevant data points. Table 3 reports the time gains and approximation quality (diversity loss/gain) achieved by running FDM on the core-set $T$ rather than the full set of points $P$, where the time gain implies that the core-set $T$ is much smaller than $P$ and the approximation quality implies that the core-sets contains the elements needed to build an approximate solution.

Authorsrebuttal2023-08-17

Follow up with Reviewer 9Vmw

Dear reviewer 9Vmw, Precisely, your understanding in your last message is fully correct. We very much appreciate your follow up with our rebuttal. We are happy to add these clarifications around the key concepts to make the presentation more clear in the final version.

Reviewer aN6A2023-08-17

After authors' response

Firstly, I'd like to thank the authors for their detailed responses to my review. I decided to increase my score based on the clarifications provided in your comments. Additionally, after reviewing the feedback from other reviewers and your corresponding responses, I have gained a more comprehensive understanding that positively influences my assessment.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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