We study the canonical fair clustering problem where each cluster is constrained to have close to population-level representation of each group. Despite significant attention, the salient issue of having incomplete knowledge about the group membership of each point has been superficially addressed. In this paper, we consider a setting where the assigned group memberships are noisy. We introduce a simple noise model that requires a small number of parameters to be given by the decision maker. We then present an algorithm for fair clustering with provable \emph{robustness} guarantees. Our framework enables the decision maker to trade off between the robustness and the clustering quality. Unlike previous work, our algorithms are backed by worst-case theoretical guarantees. Finally, we empirically verify the performance of our algorithm on real world datasets and show its superior performance over existing baselines.
Central China Normal University（华中师范大学）；Kharkiv National University of Radio Electronics（哈尔科夫国立无线电电子大学）
A task of clustering data given in the ordinal scale under conditions of overlapping clusters has been considered. It's proposed to use an approach based on memberhsip and likelihood functions sharing. A number of performed experiments proved effectiveness of the proposed method. The proposed method is characterized by robustness to outliers due to a way of ordering values while constructing membership functions.
IIT Kharagpur（印度理工学院卡哈拉格普尔分校）；University of California, Santa Barbara（加州大学圣巴巴拉分校）；MPI-SWS（马克斯·普朗克软件系统研究所）；Rice University（莱斯大学）
Ensuring fairness in machine learning algorithms is a challenging and essential task. We consider the problem of clustering a set of points while satisfying fairness constraints. While there have been several attempts to capture group fairness in the $k$-clustering problem, fairness at an individual level is relatively less explored. We introduce a new notion of individual fairness in $k$-clustering based on features not necessarily used for clustering. We show that this problem is NP-hard and does not admit a constant factor approximation. Therefore, we design a randomized algorithm that guar…
China University of Mining and Technology（中国矿业大学）；Nankai University（南开大学）
Community detection in network analysis is an attractive research area recently. Here, under the degree-corrected mixed membership (DCMM) model, we propose an efficient approach called mixed regularized spectral clustering (Mixed-RSC for short) based on the regularized Laplacian matrix. Mixed-RSC is designed based on an ideal cone structure of the variant for the eigen-decomposition of the population regularized Laplacian matrix. We show that the algorithm is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node. As a byproduc…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Technical University of Dortmund（多特蒙德工业大学）
We consider the $k$-clustering problem with $\ell_p$-norm cost, which includes $k$-median, $k$-means and $k$-center, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points $P$ of size $n$, a set of $k$ centers induces a fair clustering if every point in $P$ has a center among its $n/k$ closest neighbors. Mahabadi and Vakilian [2020] presented a $(p^{O(p)},7)$-bicriteria approximation for fair clustering with $\ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $\ell_p$-norm…
Xidian University（西安电子科技大学）；University of Alberta（阿尔伯塔大学）；Systems Research Institute, Polish Academy of Sciences（波兰科学院系统研究所）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）
Fuzzy C-Means (FCM) is a widely used clustering method. However, FCM and its many accelerated variants have low efficiency in the mid-to-late stage of the clustering process. In this stage, all samples are involved in the update of their non-affinity centers, and the fuzzy membership grades of the most of samples, whose assignment is unchanged, are still updated by calculating the samples-centers distances. All those lead to the algorithms converging slowly. In this paper, a new affinity filtering technique is developed to recognize a complete set of the non-affinity centers for each sample wi…
In this paper, we study correlation clustering under fairness constraints. Fair variants of $k$-median and $k$-center clustering have been studied recently, and approximation algorithms using a notion called fairlet decomposition have been proposed. We obtain approximation algorithms for fair correlation clustering under several important types of fairness constraints. Our results hinge on obtaining a fairlet decomposition for correlation clustering by introducing a novel combinatorial optimization problem. We define a fairlet decomposition with cost similar to the $k$-median cost and this all…
Indian Institute of Science（印度科学学院）；University of Maryland, College Park（马里兰大学帕克分校）；Microsoft Research（微软研究院）
Center-based clustering (e.g., $k$-means, $k$-medians) and clustering using linear subspaces are two most popular techniques to partition real-world data into smaller clusters. However, when the data consists of sensitive demographic groups, significantly different clustering cost per point for different sensitive groups can lead to fairness-related harms (e.g., different quality-of-service). The goal of socially fair clustering is to minimize the maximum cost of clustering per point over all groups. In this work, we propose a unified framework to solve socially fair center-based clustering an…
Indian Institute of Technology (IIT) Ropar（印度理工学院罗巴尔分校）；Indian Institute of Technology (IIT) Hyderabad（印度理工学院海得拉巴分校）
We revisit the problem of fair clustering, first introduced by Chierichetti et al., that requires each protected attribute to have approximately equal representation in every cluster; i.e., a balance property. Existing solutions to fair clustering are either not scalable or do not achieve an optimal trade-off between clustering objective and fairness. In this paper, we propose a new notion of fairness, which we call $tau$-fair fairness, that strictly generalizes the balance property and enables a fine-grained efficiency vs. fairness trade-off. Furthermore, we show that simple greedy round-robi…
Oregon State University（俄勒冈州立大学）；School of Electrical Engineering and Computer Science（电气工程与计算机科学学院）
This work considers clustering nodes of a largely incomplete graph. Under the problem setting, only a small amount of queries about the edges can be made, but the entire graph is not observable. This problem finds applications in large-scale data clustering using limited annotations, community detection under restricted survey resources, and graph topology inference under hidden/removed node interactions. Prior works tackled this problem from various perspectives, e.g., convex programming-based low-rank matrix completion and active query-based clique finding. Nonetheless, many existing methods…
University of California, Davis（加州大学戴维斯分校）；Brandeis University（布兰迪斯大学）
Clustering algorithms are widely used in many societal resource allocation applications, such as loan approvals and candidate recruitment, among others, and hence, biased or unfair model outputs can adversely impact individuals that rely on these applications. To this end, many fair clustering approaches have been recently proposed to counteract this issue. Due to the potential for significant harm, it is essential to ensure that fair clustering algorithms provide consistently fair outputs even under adversarial influence. However, fair clustering algorithms have not been studied from an adver…
Fair clustering is the process of grouping similar entities together, while satisfying a mathematically well-defined fairness metric as a constraint. Due to the practical challenges in precise model specification, the prescribed fairness constraints are often incomplete and act as proxies to the intended fairness requirement, leading to biased outcomes when the system is deployed. We examine how to identify the intended fairness constraint for a problem based on limited demonstrations from an expert. Each demonstration is a clustering over a subset of the data. We present an algorithm to ident…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；Queen’s University Belfast（贝尔法斯特女王大学）
A clustering may be considered as fair on pre-specified sensitive attributes if the proportions of sensitive attribute groups in each cluster reflect that in the dataset. In this paper, we consider the task of fair clustering for scenarios involving multiple multi-valued or numeric sensitive attributes. We propose a fair clustering method, \textit{FairKM} (Fair K-Means), that is inspired by the popular K-Means clustering formulation. We outline a computational notion of fairness which is used along with a cluster coherence objective, to yield the FairKM clustering method. We empirically evalua…
MPI-SWS（马克斯·普朗克软件系统研究所）；University of California, Davis（加利福尼亚大学戴维斯分校）
Clustering algorithms are ubiquitous in modern data science pipelines, and are utilized in numerous fields ranging from biology to facility location. Due to their widespread use, especially in societal resource allocation problems, recent research has aimed at making clustering algorithms fair, with great success. Furthermore, it has also been shown that clustering algorithms, much like other machine learning algorithms, are susceptible to adversarial attacks where a malicious entity seeks to subvert the performance of the learning algorithm. However, despite these known vulnerabilities, there…
Deep clustering has the potential to learn a strong representation and hence better clustering performance compared to traditional clustering methods such as $k$-means and spectral clustering. However, this strong representation learning ability may make the clustering unfair by discovering surrogates for protected information which we empirically show in our experiments. In this work, we study a general notion of group-level fairness for both binary and multi-state protected status variables (PSVs). We begin by formulating the group-level fairness problem as an integer linear programming form…
Department of Statistics, Seoul National University, Republic of Korea（统计系，首尔国立大学，大韩民国）；School of Law, Seoul National University, Republic of Korea（法学院，首尔国立大学，大韩民国）
Within the relatively busy area of fair machine learning that has been dominated by classification fairness research, fairness in clustering has started to see some recent attention. In this position paper, we assess the existing work in fair clustering and observe that there are several directions that are yet to be explored, and postulate that the state-of-the-art in fair clustering has been quite parochial in outlook. We posit that widening the normative principles to target for, characterizing shortfalls where the target cannot be achieved fully, and making use of knowledge of downstream p…
Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学）；Climax Data Pattern（Climax数据模式公司）
This paper introduces Bounded Fuzzy Possibilistic Method (BFPM) by addressing several issues that previous clustering/classification methods have not considered. In fuzzy clustering, object's membership values should sum to 1. Hence, any object may obtain full membership in at most one cluster. Possibilistic clustering methods remove this restriction. However, BFPM differs from previous fuzzy and possibilistic clustering approaches by allowing the membership function to take larger values with respect to all clusters. Furthermore, in BFPM, a data object can have full membership in multiple clu…
Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportionally represented in each cluster. To this end, we develop variants of both normalized and unnormalized constrained SC and show that they help find fairer clusterings on both synthetic and real data. We also provide a rigorous theoretical analysis of our algorithms on a natural vari…
Humboldt University of Berlin（柏林洪堡大学）；Hasso Plattner Institute, University of Potsdam（波茨坦大学哈索·普拉特纳研究所）；University of Vienna（维也纳大学）
The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that …
Google（谷歌）；University of Maryland（马里兰大学）；Carnegie Mellon University（卡内基梅隆大学）
As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find…
Correlation clustering is a ubiquitous paradigm in unsupervised machine learning where addressing unfairness is a major challenge. Motivated by this, we study Fair Correlation Clustering where the data points may belong to different protected groups and the goal is to ensure fair representation of all groups across clusters. Our paper significantly generalizes and improves on the quality guarantees of previous work of Ahmadi et al. and Ahmadian et al. as follows. - We allow the user to specify an arbitrary upper bound on the representation of each group in a cluster. - Our algorithm allows ind…
MPI-SWS（马克斯·普朗克软件系统研究所）；University of California, Davis（加利福尼亚大学戴维斯分校）
Clustering algorithms are widely utilized for many modern data science applications. This motivates the need to make outputs of clustering algorithms fair. Traditionally, new fair algorithmic variants to clustering algorithms are developed for specific notions of fairness. However, depending on the application context, different definitions of fairness might need to be employed. As a result, new algorithms and analysis need to be proposed for each combination of clustering algorithm and fairness definition. Additionally, each new algorithm would need to be reimplemented for deployment in a rea…
University of Maryland（马里兰大学）；Wellesley College（韦尔斯利学院）
Clustering is a fundamental unsupervised learning problem where a dataset is partitioned into clusters that consist of nearby points in a metric space. A recent variant, fair clustering, associates a color with each point representing its group membership and requires that each color has (approximately) equal representation in each cluster to satisfy group fairness. In this model, the cost of the clustering objective increases due to enforcing fairness in the algorithm. The relative increase in the cost, the ''price of fairness,'' can indeed be unbounded. Therefore, in this paper we propose to…
Harvard University（哈佛大学）；University of Maryland, College Park（马里兰大学帕克分校）
Clustering is a fundamental problem in unsupervised machine learning, and fair variants of it have recently received significant attention due to its societal implications. In this work we introduce a novel definition of individual fairness for clustering problems. Specifically, in our model, each point $j$ has a set of other points $\mathcal{S}_j$ that it perceives as similar to itself, and it feels that it is fairly treated if the quality of service it receives in the solution is $α$-close (in a multiplicative sense, for a given $α\geq 1$) to that of the points in $\mathcal{S}_j$. We begin o…
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 metri…
Brandeis University（布兰迪斯大学）；Michtom School of Computer Science（米奇托姆计算机科学学院）
In this paper, we focus on the fairness issues regarding unsupervised outlier detection. Traditional algorithms, without a specific design for algorithmic fairness, could implicitly encode and propagate statistical bias in data and raise societal concerns. To correct such unfairness and deliver a fair set of potential outlier candidates, we propose Deep Clustering based Fair Outlier Detection (DCFOD) that learns a good representation for utility maximization while enforcing the learnable representation to be subgroup-invariant on the sensitive attribute. Considering the coupled and reciprocal …
We study the question of fair clustering under the {\em disparate impact} doctrine, where each protected class must have approximately equal representation in every cluster. We formulate the fair clustering problem under both the $k$-center and the $k$-median objectives, and show that even with two protected classes the problem is challenging, as the optimum solution can violate common conventions---for instance a point may no longer be assigned to its nearest cluster center! En route we introduce the concept of fairlets, which are minimal sets that satisfy fair representation while approximat…
IIIT Delhi（德里英迪拉·甘地信息技术学院）；UC Santa Cruz（加州大学圣克鲁兹分校）
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 clusteri…
Hong Kong University of Science and Technology（香港科技大学）
In this paper, we study the problem of fair clustering on the $k-$center objective. In fair clustering, the input is $N$ points, each belonging to at least one of $l$ protected groups, e.g. male, female, Asian, Hispanic. The objective is to cluster the $N$ points into $k$ clusters to minimize a classical clustering objective function. However, there is an additional constraint that each cluster needs to be fair, under some notion of fairness. This ensures that no group is either "over-represented" or "under-represented" in any cluster. Our work builds on the work of Chierichetti et al. (NIPS 2…
Northeastern University（东北大学）；University of Pennsylvania（宾夕法尼亚大学）
Motivated by settings in which predictive models may be required to be non-discriminatory with respect to certain attributes (such as race), but even collecting the sensitive attribute may be forbidden or restricted, we initiate the study of fair learning under the constraint of differential privacy. We design two learning algorithms that simultaneously promise differential privacy and equalized odds, a 'fairness' condition that corresponds to equalizing false positive and negative rates across protected groups. Our first algorithm is a private implementation of the equalized odds post-process…
UC Santa Cruz（加州大学圣克鲁兹分校）；Dartmouth College（达特茅斯学院）
We study the problem of finding low-cost Fair Clusterings in data where each data point may belong to many protected groups. Our work significantly generalizes the seminal work of Chierichetti et.al. (NIPS 2017) as follows. - We allow the user to specify the parameters that define fair representation. More precisely, these parameters define the maximum over- and minimum under-representation of any group in any cluster. - Our clustering algorithm works on any $\ell_p$-norm objective (e.g. $k$-means, $k$-median, and $k$-center). Indeed, our algorithm transforms any vanilla clustering solution in…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We give a local search based algorithm for $k$-median and $k$-means (and more generally for any $k$-clustering with $\ell_p$ norm cost function) from the perspective of individual fairness. More precisely, for a point $x$ in a point set $P$ of size $n$, let $r(x)$ be the minimum radius such that the ball of radius $r(x)$ centered at $x$ has at least $n/k$ points from $P$. Intuitively, if a set of $k$ random points are chosen from $P$ as centers, every point $x\in P$ expects to have a center within radius $r(x)$. An individually fair clustering provides such a guarantee for every point $x\in P$…
Ben-Gurion University（本-古里安大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over $W$ of the $\ell_p$-norms of the weighted distances of points in $P$ from the centers. This generalizes various clustering problems, including Socially Fair $k$-Median and $k$-Means, and is closely connected to other problems such as Densest $k$-Subgraph and Min $k$-Union. We utilize convex programming techniques to approximate the $(p,q)$-Fair Clustering pr…
University of Maryland（马里兰大学）；Wellesley College（韦尔斯利学院）；Carnegie Mellon University（卡内基梅隆大学）；Northwestern University（西北大学）
Clustering is a foundational problem in machine learning with numerous applications. As machine learning increases in ubiquity as a backend for automated systems, concerns about fairness arise. Much of the current literature on fairness deals with discrimination against protected classes in supervised learning (group fairness). We define a different notion of fair clustering wherein the probability that two points (or a community of points) become separated is bounded by an increasing function of their pairwise distance (or community diameter). We capture the situation where data points repres…
We study data clustering problems with $\ell_p$-norm objectives (e.g. $k$-Median and $k$-Means) in the context of individual fairness. The dataset consists of $n$ points, and we want to find $k$ centers such that (a) the objective is minimized, while (b) respecting the individual fairness constraint that every point $v$ has a center within a distance at most $r(v)$, where $r(v)$ is $v$'s distance to its $(n/k)$th nearest point. Jung, Kannan, and Lutz [FORC 2020] introduced this concept and designed a clustering algorithm with provable (approximate) fairness and objective guarantees for the $\e…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；Google（谷歌）
A common distinction in fair machine learning, in particular in fair classification, is between group fairness and individual fairness. In the context of clustering, group fairness has been studied extensively in recent years; however, individual fairness for clustering has hardly been explored. In this paper, we propose a natural notion of individual fairness for clustering. Our notion asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. We study several questions related to our proposed notion of individual fairness. On t…
Clustering is a fundamental task in unsupervised learning, one that targets to group a dataset into clusters of similar objects. There has been recent interest in embedding normative considerations around fairness within clustering formulations. In this paper, we propose 'local connectivity' as a crucial factor in assessing membership desert in centroid clustering. We use local connectivity to refer to the support offered by the local neighborhood of an object towards supporting its membership to the cluster in question. We motivate the need to consider local connectivity of objects in cluster…
We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups, whose special cases include the socially fair $k$-median ($p=1$) and socially fair $k$-means ($p=2$) problems. We present (1) a polynomial-time $(5+2\sqrt{6})^p$-approximation with at most $k+m$ centers (2) a $(5+2\sqrt{6}+ε)^p$-approximation with $k$ centers in time $n^{2^{O(p)}\cdot m^2}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{m}\cdot\text{poly}(n)$. The first result is obtained via a refinement of the iterative rounding method using a sequence of linear…
