Source-linked AI summary
Fair Clustering Through Fairlets
Flavio Chierichetti, Ravi Kumar, Silvio Lattanzi, Sergei Vassilvitskii
TL;DR
The paper asks how to cluster data while maintaining approximately equal protected-class representation in every cluster. It introduces fairlets as a preprocessing reduction to classical k-center and k-median clustering, develops approximation algorithms, and studies the computational cost of fairness. The paper establishes efficient fairlet-decomposition approximations alongside hardness results and empirically quantifies fairness costs on real datasets.
Problem
Fair clustering seeks approximately balanced protected-class representation in every cluster, but colorblind clustering can produce unfair outcomes and fairness may require non-nearest-center assignments.
Method
The paper defines fairlets and reduces fair clustering to fairlet decomposition followed by a classical clustering algorithm, with approximation algorithms based on minimum-cost flow.
Results
The paper provides efficient approximation algorithms for fairlet decompositions, proves that finding good decompositions can be NP-hard, and empirically quantifies the price of fairness.
Takeaways & Limitations
Fair clustering can be structured through fairlets, while fairness introduces both computational bottlenecks and measurable objective cost.
Takeaways & Limitations
The paper assumes binary protected classes, and it identifies extending the framework to multiple protected-class values as a future direction.
Abstract
from arXiv · showhide
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 approximately preserving the clustering objective. We show that any fair clustering problem can be decomposed into first finding good fairlets, and then using existing machinery for traditional clustering algorithms. While finding good fairlets can be NP-hard, we proceed to obtain efficient approximation algorithms based on minimum cost flow. We empirically quantify the value of fair clustering on real-world datasets with sensitive attributes.
1 Introduction
The paper formulates fair k-center and k-median clustering under disparate impact, where clusters must have approximately balanced protected-class representation. It introduces fairlets to reduce fair clustering to classical clustering while developing approximation algorithms and studying computational cost.
- Fairness formulation: Fair clustering requires approximately color-balanced clusters under the disparate-impact notion of fairness.The paper focuses on two protected classes and translates fair representation into color balance within each cluster.
- Fairness formulation: Colorblind clustering can produce unfair clusters, so protected attributes must be explicitly used to obtain a fair solution.Fairness may also increase the objective cost relative to a colorblind solution.
- Technical challenge: Fair k-center and k-median differ from traditional formulations because fairness may require assigning a point to a non-nearest center.A fair clustering therefore includes both center locations and an explicit assignment function.
- Fairlet reduction: Fairlets are minimal fair sets, and any fair clustering problem can be reduced to finding a fairlet decomposition followed by a classical clustering algorithm.The reduction is designed so the resulting clustering satisfies fairness guarantees.
- Algorithms and complexity: The paper develops approximation algorithms for fairlet decompositions across many fairness values and complements them with NP-hardness results.The computational hardness concerns finding a good fairlet decomposition itself.
- Empirical evaluation: Experiments on real data quantify the price of fairness for k-center and k-median clustering.The paper compares the costs of traditional and fair clustering solutions.
2 Preliminaries
The paper defines fair clustering for red and blue points using balance constraints alongside k-center and k-median objectives. It establishes fairlets as small, minimally balanced subsets that preserve the dataset’s overall balance.
- Clustering setup: Clustering partitions metric-space points into k disjoint clusters evaluated by objectives such as k-center and k-median.The k-median objective passage is truncated, but both objectives are identified as clustering criteria.
- Balance and fairness: Each point is colored red or blue, and balance measures how evenly those colors are represented.Perfectly balanced subsets have equal red and blue counts, while monochromatic subsets have balance 0.
- Balance properties: A clustering’s balance cannot exceed the balance of the full dataset, so an imbalanced dataset cannot be partitioned into perfectly balanced clusters.This gives a necessary upper bound on any achievable fairness level.
- Fairlets: When the overall balance is b/r, the dataset admits a decomposition into clusters of size at most b + r with balance b/r.The construction removes balanced red-blue groups iteratively and pairs remaining one-to-one points into size-2 clusters.
- Fairlets: These small balanced clusters are called fairlets, and the resulting partition is a (b, r)-fairlet decomposition.Fairlets provide minimal fair sets for subsequent fair clustering.
- Fair objectives: Fair clustering requires k clusters with balance at least t while minimizing the chosen center or median objective.The formulation explicitly incorporates both the fairness threshold and the clustering cost.
3 Fairlet decomposition and fair clustering
The paper reduces fair clustering to preprocessing the data into fairlets and then clustering their centers with a conventional algorithm. The resulting assignment remains fair, while its cost decomposes into fairlet and center-clustering costs.
- Reduction: Fair clustering is reduced to first forming small balance-preserving fairlets and then applying vanilla clustering to those smaller clusters.This preprocessing ensures that any resulting solution satisfies the fairness requirement.
- Reduction: The choice of fairlet partition directly affects the approximation guarantees of the final clustering algorithm.Different fairlet decompositions can therefore lead to different quality guarantees.
- Fairlet costs: A fairlet decomposition designates one point from each fairlet as its center and assigns every original point to its fairlet.The decomposition cost is defined from distances between points and their designated fairlet centers.
- Induced clustering: The algorithm clusters fairlet centers into k groups, then assigns each original point according to its fairlet’s assigned center.Because fairlets preserve balance, the induced clustering has balance t.
- Cost guarantee: For both k-median and k-center, the final cost equals the fairlet-decomposition cost plus the cost of clustering the fairlet-center multiset.Lemma 6 states this additive decomposition explicitly for both objectives.
- Cost guarantee: The reduction uses the triangle inequality to relate the original-point objective to the fairlet and center-clustering costs.The proof is shown for k-median, with the k-center argument described as similar.
4 Algorithms
The algorithms reduce fair clustering to fairlet decomposition followed by classical clustering, using matching or minimum-cost flow to obtain approximation guarantees and expose computational limits.
- (1,1)-fairlets: A minimum-cost perfect matching finds an optimal (1,1)-fairlet decomposition for k-center in polynomial time.Bichromatic pairs form matching edges, and minimizing the maximum matching-edge weight minimizes fairlet k-center cost.
- (1,1)-fairlets: 3-approximation: fairlets followed by clustering solves the (1,k)-fair center problem.The guarantee combines fairlet-cost control with the 2-approximation for classical k-center.
- (1,t′)-fairlets: Minimum-cost flow yields a 2-approximation for integral (1,t′)-fairlet decomposition and a 4-approximation for (1/t′,k)-fair center.The flow construction has integral optimal solutions and converts them into fairlet decompositions.
- Hardness: For each fixed t′ ≥3, optimal (1,t′)-fairlet decomposition and minimum-cost (1/t′,k)-fair median clustering are NP-hard.The hardness reduction uses partitioning a graph into induced star subgraphs of order t′.
5 Experiments
Experiments on three UCI datasets compare classical and fair k-center and k-median algorithms, showing a fairness-cost trade-off and convergence toward the fairlet lower bound.
- Datasets: Experiments use three UCI datasets with sensitive attributes: gender in Diabetes and Census, and marital status in Bank.Diabetes and Bank contain 1,000 records each, while Census contains 600 records.
- Classical clustering: Classical algorithms produce very low balance, reaching 0 in four of six cases for larger k and producing monochromatic clusters.In the Bank k-median instance, three monochromatic clusters appear starting at k = 12.
- Fair clustering: Fair clustering maintains balance as k increases, but its solutions cost more than their unfair counterparts.The experiments therefore display an explicit fairness-cost trade-off.
- Fair clustering: In all scenarios, fair-clustering cost converges to the fairlet-decomposition cost, which is a lower bound on the optimal fair-clustering cost.The fairlet cost is shown alongside each experimental result as the limiting reference.
6 Conclusions
The paper initiates fair clustering, reducing it to fairlet decomposition and providing efficient approximations while identifying computational bottlenecks and open extensions beyond binary protected classes.
- Contributions: Fair clustering is reduced to classical clustering through fairlets, with efficient approximation algorithms for finding fairlet decompositions.The paper also proves lower bounds showing that fairness can introduce a computational bottleneck.
- Future directions: Tightening decomposition approximation ratios and strengthening hardness results are identified as future directions.The conclusion also proposes extending the framework to protected classes with multiple values.