[Paper Review] Approximate Group Fairness for Clustering

Bo Li, Lijun Li+3 more|arXiv (Cornell University)|Mar 31, 2022

Privacy-Preserving Technologies in Data32 references4 citations

TL;DR

This paper introduces core fairness as a refined group fairness notion for centroid clustering, where no coalition of at least $ n/k $ agents can collectively reduce their total distance by deviating to a new center. The authors propose approximate core fairness via two relaxations—on the size of blocking coalitions and the degree of distance improvement—and design approximation algorithms that achieve strong fairness with bounded efficiency loss in metric spaces including lines, trees, and general metrics.

ABSTRACT

We incorporate group fairness into the algorithmic centroid clustering problem, where $k$ centers are to be located to serve $n$ agents distributed in a metric space. We refine the notion of proportional fairness proposed in [Chen et al., ICML 2019] as {\em core fairness}, and $k$-clustering is in the core if no coalition containing at least $n/k$ agents can strictly decrease their total distance by deviating to a new center together. Our solution concept is motivated by the situation where agents are able to coordinate and utilities are transferable. A string of existence, hardness and approximability results is provided. Particularly, we propose two dimensions to relax core requirements: one is on the degree of distance improvement, and the other is on the size of deviating coalition. For both relaxations and their combination, we study the extent to which relaxed core fairness can be satisfied in metric spaces including line, tree and general metric space, and design approximation algorithms accordingly.

Motivation & Objective

• To address group fairness in centroid clustering by formalizing a stronger fairness concept than proportionality.
• To study the existence and approximability of core clusterings in metric spaces, including lines, trees, and general metrics.
• To relax the core fairness condition via two dimensions: minimum distance improvement and minimum coalition size, enabling practical approximation algorithms.
• To design algorithms that balance fairness (core stability) and efficiency (low social cost) in real-world clustering scenarios.
• To empirically validate that the proposed algorithms outperform standard clustering methods (e.g., k-means++, Lloyd’s algorithm) in fairness metrics while maintaining competitive efficiency.

Proposed method

• Define core fairness as a stability condition: no coalition of size at least $ n/k $ can strictly reduce its total distance by deviating to a new center.
• Introduce two relaxations: $ (eta, eta) $-core (coalition size threshold) and $ (eta, eta) $-core (distance improvement threshold), generalizing the core concept.
• Design approximation algorithms for $ (eta, eta) $-core in line, tree, and general metric spaces, with performance guarantees on fairness and social cost.
• Propose a two-stage algorithm $ ALG_g^+ $ that first identifies fair candidate centers and then assigns agents to minimize total distance.
• Use a greedy selection strategy to iteratively improve fairness while bounding the social cost relative to optimal k-means/k-medians.
• Conduct extensive experiments on synthetic (Gaussian, S-sets) and real-world (Mopsi locations) datasets to evaluate fairness ($ \alpha, \beta $) and efficiency (social cost ratio).

Experimental results

Research questions

• RQ1 Can core fairness be guaranteed in metric spaces such as lines, trees, and general metrics, and what are the existence and approximability limits?
• RQ2 How can core fairness be meaningfully relaxed to enable practical approximation algorithms while preserving fairness guarantees?
• RQ3 What is the trade-off between fairness (measured by $ \alpha $ and $ \beta $) and efficiency (social cost) in clustering algorithms?
• RQ4 How do the proposed algorithms compare to standard methods like k-means++ and Lloyd’s algorithm in terms of fairness and efficiency on real and synthetic data?
• RQ5 In what scenarios does the proposed algorithm achieve exact core fairness ($ \alpha = 1, \beta = 1 $), and when are relaxations necessary?

Key findings

• The proposed $ ALG_g^+ $ algorithm achieves significantly lower $ \alpha $ and $ \beta $ values than k-means++ on both Gaussian and S-sets datasets, indicating stronger approximate core fairness.
• On the Mopsi locations dataset with $ k=10 $, both Algorithm 4 and Lloyd’s algorithm produced clusterings with $ \alpha = 1, \beta = 1 $, indicating exact core fairness, though social cost ratio was 1.38.
• For $ k $-medians objective, Algorithm 4 outperformed Lloyd’s algorithm in both fairness ($ \alpha $ and $ \beta $) and social cost on the Gaussian dataset.
• On S1 dataset with $ k $-means objective, the proposed algorithm maintained better fairness than k-means++ across all $ k $, with a social cost ratio of 1.42.
• The algorithm achieves a social cost ratio bounded by a small constant compared to k-means++ on the Gaussian dataset, and is even better in some cases.
• In general, the proposed algorithms ensure better core fairness than standard clustering methods while maintaining competitive efficiency, especially in high-overlap and real-world settings.