Individual Fairness for $k$-Clustering

Mahabadi, Sepideh; Vakilian, Ali

doi:10.48550/arxiv.2002.06742

Cited by 5 publications

(13 citation statements)

References 5 publications

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“…Besides explainability, many other clustering variants have received recent attention, such as fair clustering [7,10,21,24,31,43], online clustering [12,15,20,25,35], and the use of same-cluster queries [2,5,22,33].…”

Section: Related Workmentioning

confidence: 99%

Explainable $k$-Means and $k$-Medians Clustering

Dasgupta,

Frost,

Moshkovitz

et al. 2020

Preprint

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Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a complicated way. To improve interpretability, we consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We study this problem from a theoretical viewpoint, measuring cluster quality by the k-means and k-medians objectives: Must there exist a tree-induced clustering whose cost is comparable to that of the best unconstrained clustering, and if so, how can it be found? In terms of negative results, we show, first, that popular top-down decision tree algorithms may lead to clusterings with arbitrarily large cost, and second, that any tree-induced clustering must in general incur an Ω(log k) approximation factor compared to the optimal clustering. On the positive side, we design an efficient algorithm that produces explainable clusters using a tree with k leaves. For two means/medians, we show that a single threshold cut suffices to achieve a constant factor approximation, and we give nearly-matching lower bounds. For general k ≥ 2, our algorithm is an O(k) approximation to the optimal k-medians and an O(k 2 ) approximation to the optimal k-means. Prior to our work, no algorithms were known with provable guarantees independent of dimension and input size.

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Section: Related Workmentioning

confidence: 99%

Explainable $k$-Means and $k$-Medians Clustering

Dasgupta,

Frost,

Moshkovitz

et al. 2020

Preprint

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“…None of these works address the question of individual fairness and are orthogonal to the direction we take in this paper. Recently, [25,37] consider a notion of individual fairness which requires every point j to have a center within a distance of r j where r j is the minimum radius ball centered at j that contains at least n/k points. Our notion of individual fairness differs significantly from this notion and is not directly comparable.…”

Section: Related Workmentioning

confidence: 99%

“…Then, we define F 2 (i, j) = d(i, j)/r i , ∀j ∈ B i and F 2 (i, j) = 1, otherwise. The motivation behind F 2 is inspired by the individual fairness notion in [25,37]. More specifically, in F 2 , each point is required to be treated similarly to its closest |V |/k neighbours.…”

Section: Experimental Evaluationmentioning

confidence: 99%

Distributional Individual Fairness in Clustering

Anderson,

Bera,

Das

et al. 2020

Preprint

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In this paper, we initiate the study of fair clustering that ensures distributional similarity among similar individuals. In response to improving fairness in machine learning, recent papers have investigated fairness in clustering algorithms and have focused on the paradigm of statistical parity/group fairness. These efforts attempt to minimize bias against some protected groups in the population. However, to the best of our knowledge, the alternative viewpoint of individual fairness, introduced by Dwork et al. (ITCS 2012) in the context of classification, has not been considered for clustering so far. Similar to Dwork et al., we adopt the individual fairness notion which mandates that similar individuals should be treated similarly for clustering problems. We use the notion of f -divergence as a measure of statistical similarity that significantly generalizes the ones used by Dwork et al. We introduce a framework for assigning individuals, embedded in a metric space, to probability distributions over a bounded number of cluster centers. The objective is to ensure (a) low cost of clustering in expectation and (b) individuals that are close to each other in a given fairness space are mapped to statistically similar distributions. We provide an algorithm for clustering with p-norm objective (k-center, k-means are special cases) and individual fairness constraints with provable approximation guarantee. We extend this framework to include both group fairness and individual fairness inside the protected groups. Finally, we observe conditions under which individual fairness implies group fairness. We present extensive experimental evidence that justifies the effectiveness of our approach.Preprint. Under review.

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“…We now provide the idea behind a dynamic programming approach for choosing p products that minimize the weighted regret of a population S = {τ i } n i=1 . The approach relies on the simple observation that there exists an optimal solution such that if the consumers in set S(p ) are assigned to the p -th product c p , c p ∈ argmin c∈R + R S(p ) (c) (for a single product c) 10 .…”

Section: B a More General Regret Notionmentioning

confidence: 99%

“…However, we are not aware of any technical connections between our work and this line of research. Within the fairness in machine learning literature, our work is closest to fair facility location problems [9,10], which attempt to choose a small number of "centers" to serve a large and diverse population and "fair allocation" problems that arise in the context of predictive policing [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Algorithms and Learning for Fair Portfolio Design

Diana¹,

Dick²,

Elzayn³

et al. 2020

Preprint

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We consider a variation on the classical finance problem of optimal portfolio design. In our setting, a large population of consumers is drawn from some distribution over risk tolerances, and each consumer must be assigned to a portfolio of lower risk than her tolerance. The consumers may also belong to underlying groups (for instance, of demographic properties or wealth), and the goal is to design a small number of portfolios that are fair across groups in a particular and natural technical sense.Our main results are algorithms for optimal and near-optimal portfolio design for both social welfare and fairness objectives, both with and without assumptions on the underlying group structure. We describe an efficient algorithm based on an internal two-player zero-sum game that learns near-optimal fair portfolios ex ante and show experimentally that it can be used to obtain a small set of fair portfolios ex post as well. For the special but natural case in which group structure coincides with risk tolerances (which models the reality that wealthy consumers generally tolerate greater risk), we give an efficient and optimal fair algorithm. We also provide generalization guarantees for the underlying risk distribution that has no dependence on the number of portfolios and illustrate the theory with simulation results.

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Individual Fairness for $k$-Clustering

Cited by 5 publications

References 5 publications

Explainable $k$-Means and $k$-Medians Clustering

Explainable $k$-Means and $k$-Medians Clustering

Distributional Individual Fairness in Clustering

Algorithms and Learning for Fair Portfolio Design

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