[Paper Review] Clustering with Fairness Constraints: A Flexible and Scalable Approach

Imtiaz Masud Ziko, Éric Granger+2 more|arXiv (Cornell University)|Jun 19, 2019

Health disparities and outcomes20 references10 citations

TL;DR

This paper proposes a flexible, scalable variational framework for fair clustering that integrates arbitrary demographic proportion constraints into diverse clustering objectives—such as K-means, K-median, and Normalized Cut—via a tight upper bound on a KL-based fairness penalty. The method enables joint optimization with convergence guarantees and independent per-assignment updates, supporting distributed computation and efficient trade-off control between fairness and clustering quality.

ABSTRACT

This study investigates a general variational formulation of fair clustering, which can integrate fairness constraints with a large class of clustering objectives. Unlike the existing methods, our formulation can impose any desired (target) demographic proportions within each cluster. Furthermore, it enables to control the trade-off between the fairness and clustering terms. We derive an auxiliary function (tight upper bound) of our KL-based fairness penalty via its concave-convex decomposition and Lipschitz-gradient property. Our upper bound can be optimized jointly with various clustering objectives, including prototype-based, such as K-means and K-median, or graph-based such as Normalized Cut. Interestingly, at each iteration, our general fair-clustering algorithm performs an independent update for each assignment variable, while guaranteeing convergence. Therefore, it can be easily distributed for large-scale data sets. Such scalability is important as it enables to explore different trade-off levels between the fairness and clustering objectives. Unlike fairness-constrained spectral clustering, our formulation does not need storing an affinity matrix and computing its eigenvalue decomposition. We show the effectiveness, flexibility and scalability of our approach through comprehensive evaluations and comparisons to the existing methods over several data sets.

Motivation & Objective

• To address the limitation of existing fair clustering methods that cannot flexibly enforce arbitrary target demographic proportions within clusters.
• To develop a unified framework that integrates fairness constraints with a broad class of clustering objectives, including prototype-based and graph-based methods.
• To enable scalable optimization through a tight upper bound on the fairness penalty, allowing distributed and efficient computation on large-scale datasets.
• To eliminate the need for computationally heavy steps like eigen-decomposition of an affinity matrix, common in spectral clustering-based fairness methods.
• To provide a flexible trade-off mechanism between fairness and clustering quality, enabling exploration of different fairness levels.

Proposed method

• Formulates a general variational objective that combines clustering loss with a KL-based fairness penalty to enforce desired demographic proportions per cluster.
• Derives a tight upper bound on the fairness penalty using concave-convex decomposition and the Lipschitz-gradient property of the KL divergence.
• Optimizes the upper bound jointly with various clustering objectives, such as K-means, K-median, and Normalized Cut, through a unified algorithmic framework.
• Employs an iterative optimization strategy where each cluster assignment variable is updated independently per iteration, ensuring convergence and enabling parallelization.
• Avoids storing and computing eigen-decompositions of large affinity matrices, unlike fairness-constrained spectral clustering methods.
• Supports flexible trade-off control between fairness and clustering performance via a hyperparameter that balances the two terms in the objective.

Experimental results

Research questions

• RQ1 Can a unified framework enforce arbitrary target demographic proportions within each cluster across diverse clustering objectives?
• RQ2 How can fairness constraints be integrated with clustering objectives in a way that maintains computational scalability and convergence?
• RQ3 To what extent does the proposed method outperform existing fair clustering approaches in terms of fairness, clustering quality, and efficiency?
• RQ4 Can the method avoid the computational bottleneck of eigen-decomposition in spectral clustering while maintaining fairness guarantees?
• RQ5 How effective is the method in balancing fairness and clustering performance across different data distributions and fairness levels?

Key findings

• The proposed method achieves superior fairness enforcement by allowing arbitrary target demographic proportions per cluster, unlike prior methods with fixed or binary constraints.
• The derived upper bound on the fairness penalty enables joint optimization with multiple clustering objectives, including K-means, K-median, and Normalized Cut, with guaranteed convergence.
• The algorithm’s independent assignment updates allow for efficient distributed implementation, significantly improving scalability on large datasets.
• The method avoids the need for affinity matrix storage and eigen-decomposition, reducing computational overhead compared to fairness-constrained spectral clustering.
• Comprehensive evaluations demonstrate the method’s effectiveness, flexibility, and scalability across multiple datasets, outperforming existing approaches in fairness and clustering quality trade-offs.
• The framework enables systematic exploration of fairness-clustering trade-offs by tuning a single hyperparameter, offering practical utility in real-world applications.