Clustering algorithms are a class of unsupervised machine learning (ML) algorithms that feature ubiquitously in modern data science, and play a key role in many learning-based application pipelines. Recently, research in the ML community has pivoted to analyzing the fairness of learning models, including clustering algorithms. Furthermore, research on fair clustering varies widely depending on the choice of clustering algorithm, fairness definitions employed, and other assumptions made regarding models. Despite this, a comprehensive survey of the field does not exist. In this paper, we seek to bridge this gap by categorizing existing research on fair clustering, and discussing possible avenues for future work. Through this survey, we aim to provide researchers with an organized overview of the field, and motivate new and unexplored lines of research regarding fairness in clustering.

/pdf/an-overview-of-fairness-in-clustering-2h6p05t9vi.pdf

An Overview of Fairness in Clustering

The (k,z)-clustering problem consists of finding a set of k points called centers, such that the sum of distances raised to the power of z of every data point to its closest center is minimized. Among the most commonly encountered special cases are k-median problem (z=1) and k-means problem (z=2). The k-median and k-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multiplicative (1± ε) factor, hence reducing from large to small scale the input to the problem. While there has been an intensive effort to understand what is the best coreset size possible for both problems in various metric spaces, there is still a significant gap between the state-of-the-art upper and lower bounds. In this paper, we make progress on both upper and lower bounds, obtaining tight bounds for several cases, namely: • In finite n point general metrics, any coreset must consist of Ω(k logn / ε2) points. This improves on the Ω(k logn /ε) lower bound of Braverman, Jiang, Krauthgamer, and Wu [ICML’19] and matches the upper bounds proposed for k-median by Feldman and Langberg [STOC’11] and k-means by Cohen-Addad, Saulpic and Schwiegelshohn [STOC’21] up to polylog factors. • For doubling metrics with doubling constant D, any coreset must consist of Ω(k D/ε2) points. This matches the k-median and k-means upper bounds by Cohen-Addad, Saulpic, and Schwiegelshohn [STOC’21] up to polylog factors. • In d-dimensional Euclidean space, any coreset for (k,z) clustering requires Ω(k/ε2) points. This improves on the Ω(k/√ε) lower bound of Baker, Braverman, Huang, Jiang, Krauthgamer, and Wu [ICML’20] for k-median and complements the Ω(kmin(d,2z/20)) lower bound of Huang and Vishnoi [STOC’20]. We complement our lower bound for d-dimensional Euclidean space with the construction of a coreset of size Õ(k/ε2· min(ε−z,k)). This improves over the Õ(k2 ε−4) upper bound for general power of z proposed by Braverman Jiang, Krauthgamer, and Wu [SODA’21] and over the Õ(k/ε4) upper bound for k-median by Huang and Vishnoi [STOC’20]. In fact, ours is the first construction breaking through the ε−2· min(d,ε−2) barrier inherent in all previous coreset constructions. To do this, we employ a novel chaining based analysis that may be of independent interest. Together our upper and lower bounds for k-median in Euclidean spaces are tight up to a factor O(ε−1 polylog k/ε).

/pdf/towards-optimal-lower-bounds-for-k-median-and-k-means-wc6rkzod.pdf

Towards optimal lower bounds for k-median and k-means coresets

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of $k$ ReLUs minimizing the squared error (for $k>1$) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error $\epsilon$. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest $\kappa$-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in $1/\epsilon^2$. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on $\epsilon$.

pdf/tight-hardness-results-for-training-depth-2-relu-networks-rttkclpogu.pdf

Tight Hardness Results for Training Depth-2 ReLU Networks

40th international colloquium on automata, languages and programming

Given a set of $n$ points in $d$ dimensions, the Euclidean $k$-means problem (resp. the Euclidean $k$-median problem) consists of finding $k$ centers such that the sum of squared distances (resp. sum of distances) from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a $(1\pm \varepsilon)$ factor. The current state of the art coreset size is $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for Euclidean $k$-means and $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for Euclidean $k$-median. The best known lower bound for both problems is $\Omega(k \varepsilon^{-2})$. In this paper, we improve the upper bounds $\tilde O(\min(k^{3/2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for $k$-means and $\tilde O(\min(k^{4/3} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for $k$-median. In particular, ours is the first provable bound that breaks through the $k^2$ barrier while retaining an optimal dependency on $\varepsilon$.

Improved Coresets for Euclidean k-Means

Fair clustering is a constrained variant of clustering where the goal is to partition a set of colored points, such that the fraction of points of any color in every cluster is more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS, 2017] in a seminal work and became widely popular in the clustering literature. In this paper, we propose a new construction of coresets for fair clustering based on random sampling. The new construction allows us to obtain the first coreset for fair clustering in general metric spaces. For Euclidean spaces, we obtain the first coreset whose size does not depend exponentially on the dimension. Our coreset results solve open questions proposed by Schmidt et al. [WAOA, 2019] and Huang et al. [NeurIPS, 2019]. The new coreset construction helps to design several new approximation and streaming algorithms. In particular, we obtain the first true constant-approximation algorithm for metric fair clustering, whose running time is fixed-parameter tractable (FPT). In the Euclidean case, we derive the first $(1+\epsilon)$-approximation algorithm for fair clustering whose time complexity is near-linear and does not depend exponentially on the dimension of the space. Besides, our coreset construction scheme is fairly general and gives rise to coresets for a wide range of constrained clustering problems. This leads to improved constant-approximations for these problems in general metrics and near-linear time $(1+\epsilon)$-approximations in the Euclidean metric.

pdf/on-coresets-for-fair-clustering-in-metric-and-euclidean-1ylw9olnri.pdf

On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications

Fair clustering is a variant of constrained clustering where the goal is to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS 2017] and became widely popular. This paper proposes a new construction of coresets for fair k-means and k-median clustering for Euclidean and general metrics based on random sampling. For the Euclidean space ℝ^d, we provide the first coresets whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt, Schwiegelshohn, and Sohler [WAOA 2019] and Huang, Jiang, and Vishnoi [NeurIPS 2019]. For general metric, our construction provides the first coreset for fair k-means and k-median.
New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. In particular, we obtain 
- the first fixed-parameter tractable (FPT) PTAS for fair k-means and k-median clustering in ℝ^d. The near-linear time of our PTAS improves over the previous scheme of Bohm, Fazzone, Leonardi, and Schwiegelshohn [ArXiv 2020] with running time n^{poly(k/e)};
- FPT "true" constant-approximation for metric fair clustering. All previous algorithms for fair k-means and k-median in general metric are bicriteria and violate the fairness constraints; 
- FPT 3-approximation for lower-bounded k-median improving the best-known 3.736 factor of Bera, Chakrabarty, and Negahbani [ArXiv 2019];
- the first FPT constant-approximations for metric chromatic clustering and 𝓁-Diversity clustering; 
- near linear-time (in n) PTAS for capacitated and lower-bounded clustering improving over PTAS of Bhattacharya, Jaiswal, and Kumar [TOCS 2018] with super-quadratic running time;
- a streaming (1+e)-approximation for fair k-means and k-median of space complexity polynomial in k, d, e and log{n} (the previous algorithms have exponential space complexity on either d or k).

https://drops.dagstuhl.de/opus/volltexte/2021/14092/pdf/LIPIcs-ICALP-2021-23.pdf/

On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications.

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs. and Seminar 21293: Parameterized Complexity in Graph Drawing [19].

Parameterized Algorithms for Upward Planarity

In k-Clustering we are given a multiset of n vectors X subset Z^d and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost sum_{i=1}^k min_{c_i in R^d} sum_{x in C_i} |x-c_i|_p^p <= D, where |*|_p is the Minkowski (L_p) norm of order p. For p=1, k-Clustering is the well-known k-Median. For p=2, the case of the Euclidean distance, k-Clustering is k-Means. We study k-Clustering from the perspective of parameterized complexity. The problem is known to be NP-hard for k=2 and it is also NP-hard for d=2. It is a long-standing open question, whether the problem is fixed-parameter tractable (FPT) for the combined parameter d+k. In this paper, we focus on the parameterization by D. We complement the known negative results by showing that for p=0 and p=infty, k-Clustering is W1-hard when parameterized by D. Interestingly, the complexity landscape of the problem appears to be more intricate than expected. We discover a tractability island of k-Clustering: for every p in (0,1], k-Clustering is solvable in time 2^O(D log D) (nd)^O(1).

https://drops.dagstuhl.de/opus/volltexte/2019/11576/pdf/LIPIcs-FSTTCS-2019-14.pdf/

Parameterized k-Clustering: Tractability Island

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On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications

On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications.

Parameterized Algorithms for Upward Planarity

Parameterized k-Clustering: Tractability Island

Parameterized k-Clustering: Tractability Island