On the cost of essentially fair clusterings

Abstract

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -especially if the data is already biased.At NIPS 2017, Chierichetti et al. [14] proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair k-center problem and a O(t)-approximation for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair k-center [34].We extend and improve the known results. Firstly, we give a 5-approximation for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives kcenter, k-supplier, k-median, k-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed.

Fair Correlation Clustering

“…In this work, we initiate a study of a fair variant of the correlation clustering problem where each vertex has a given feature, and the goal is to make sure that the distribution of the features is the same as the global distribution in each cluster. This is the same notion of fairness studied by Chierichetti et al [14], Bercea et al [7] and Bera et al [6] on k-center and k-median. In another variation, the goal is to make sure the number of nodes of a specific feature c i in a cluster of size n is between n qi , n pi where p i ≤ q i ∈ Z ≥1 , and p i , q i are specified per each feature c i .…”

“…The first direction includes results on statistical parity [22], disparate impact [17], and individual fairness [16]. Second direction includes a bulk of work including fair rankings [10], fair clusterings [14,25,7,6,2], fair voting [9], and fair optimization with matroid constraints [15].…”

Rényi Fair Inference

“…For K-means clustering problem, we show that sufficiently large regularization coefficient yields perfect fairness under disparate impact doctrine. Unlike the twophase methods proposed in [13,5,43,7,47], our method does not require any pre-processing step, is scalable, and allows for regulating the trade-off between the clustering quality and fairness.…”

“…For example, we say a predictor Ŷθ satisfies equalized odds condition if the predictor Ŷθ is conditionally independent of the sensitive attribute S given the true label Y . Similar to formulation (7), the equalized odds fairness notion can be achieved by the following min-max problem…”

“…In addition to classification, we apply our regularization framework to the fair K-means clustering problem under disparate impact doctrine. Many recent works on fair clustering propose a two-phase algorithm for solving the K-centers and K-median fair clustering problems [13,5,43,7]. In the first phase, data points are partitioned into small subsets, referred to as fairlets, that satisfy fairness requirements.…”