[Paper Review] A Stochastic Alternating Balance $k$-Means Algorithm for Fair Clustering

Suyun Liu, L. N. Vicente|arXiv (Cornell University)|May 29, 2021

Privacy-Preserving Technologies in Data32 references4 citations

TL;DR

This paper proposes a stochastic alternating balance $k$-means (SAfairKM) algorithm for fair clustering that alternates between mini-batch $k$-means updates and group swap updates to balance clustering cost and demographic fairness. The method efficiently generates high-quality, well-spread Pareto fronts across synthetic and real datasets, outperforming a penalized fairness approach in computational efficiency and robustness.

ABSTRACT

In the application of data clustering to human-centric decision-making systems, such as loan applications and advertisement recommendations, the clustering outcome might discriminate against people across different demographic groups, leading to unfairness. A natural conflict occurs between the cost of clustering (in terms of distance to cluster centers) and the balance representation of all demographic groups across the clusters, leading to a bi-objective optimization problem that is nonconvex and nonsmooth. To determine the complete trade-off between these two competing goals, we design a novel stochastic alternating balance fair $k$-means (SAfairKM) algorithm, which consists of alternating classical mini-batch $k$-means updates and group swap updates. The number of $k$-means updates and the number of swap updates essentially parameterize the weight put on optimizing each objective function. Our numerical experiments show that the proposed SAfairKM algorithm is robust and computationally efficient in constructing well-spread and high-quality Pareto fronts both on synthetic and real datasets.

Motivation & Objective

• To address the inherent conflict between minimizing clustering cost and ensuring balanced representation of demographic groups in clusters.
• To develop a robust, computationally efficient method for generating well-spread and high-quality Pareto fronts in fair clustering.
• To provide a flexible in-processing framework that allows trade-offs between clustering quality and fairness without relying on penalty parameters.
• To improve upon existing penalized fairness approaches by using alternating updates that preserve fairness while optimizing clustering cost.
• To enable scalable fair clustering on real-world datasets like Adult and Bank with multiple demographic groups.

Proposed method

• The algorithm alternates between classical mini-batch $k$-means updates to minimize clustering cost and group swap updates to improve demographic balance across clusters.
• The number of $k$-means and swap updates acts as a control parameter for the trade-off between clustering cost and fairness.
• A list update mechanism connects multiple parallel runs from different initializations and update pairs, enhancing robustness and escape from local optima.
• The method is designed as a bi-objective optimization framework, explicitly balancing clustering cost and demographic representation.
• It avoids the use of penalty coefficients by directly enforcing fairness through structured swap operations, unlike prior methods that use KL-divergence penalties.
• The algorithm is applied to both synthetic and real-world datasets, including Adult and Bank, with varying numbers of clusters and demographic group ratios.

Experimental results

Research questions

• RQ1 How can a clustering algorithm balance the trade-off between minimizing clustering cost and ensuring fair representation of demographic groups?
• RQ2 Can alternating updates between $k$-means and group swaps generate higher-quality and better-spread Pareto fronts than penalty-based methods?
• RQ3 How does the proposed algorithm perform in terms of computational efficiency and robustness across diverse data distributions and demographic group imbalances?
• RQ4 What is the impact of varying the number of $k$-means and swap updates on the quality and spread of the Pareto front?
• RQ5 Can the method be effectively scaled to real-world datasets with multiple protected attributes and complex group distributions?

Key findings

• SAfairKM generated more spread-out and higher-quality Pareto fronts than the penalized VfairKM method on all four synthetic datasets, regardless of group imbalance.
• The algorithm achieved significantly lower average CPU time per nondominated solution: 0.80s for Syn_equal_ds1 vs. 1.06s for VfairKM, and 59.12s for Bank (K=10) vs. 76.29s for VfairKM.
• On the Adult dataset with K=10, SAfairKM achieved an average CPU time of 18.52s per solution, compared to 40.43s for VfairKM, demonstrating superior computational efficiency.
• The list update mechanism enhanced robustness by connecting multiple runs, reducing sensitivity to initialization and improving convergence to high-quality solutions.
• For the Bank dataset with K=5, SAfairKM required only 11.97s per solution, compared to 50.31s for VfairKM, highlighting its scalability on larger datasets.
• The method maintained strong performance across datasets with unequal group sizes (e.g., |V₁|:|V₂|=1:3), confirming its robustness under demographic imbalance.