# Variational Fair Clustering

## TL;DR

This paper introduces a scalable variational framework for fair clustering that balances fairness and clustering objectives, outperforming existing spectral methods in efficiency and flexibility.

## Contribution

It presents a novel variational approach with a tight upper bound for fair clustering, enabling scalable, distributed optimization without eigenvalue decomposition.

## Key findings

- Achieves competitive fairness and clustering quality on benchmarks.
- Offers scalable, distributed optimization suitable for large datasets.
- Does not require spectral eigenvalue computations.

## Abstract

We propose a general variational framework of fair clustering, which integrates an original Kullback-Leibler (KL) fairness term with a large class of clustering objectives, including prototype or graph based. Fundamentally different from the existing combinatorial and spectral solutions, our variational multi-term approach enables to control the trade-off levels between the fairness and clustering objectives. We derive a general tight upper bound based on a concave-convex decomposition of our fairness term, its Lipschitz-gradient property and the Pinsker's inequality. Our tight upper bound can be jointly optimized with various clustering objectives, while yielding a scalable solution, with convergence guarantee. Interestingly, at each iteration, it performs an independent update for each assignment variable. Therefore, it can be easily distributed for large-scale datasets. This scalability is important as it enables to explore different trade-off levels between the fairness and clustering objectives. Unlike spectral relaxation, our formulation does not require computing its eigenvalue decomposition. We report comprehensive evaluations and comparisons with state-of-the-art methods over various fair-clustering benchmarks, which show that our variational formulation can yield highly competitive solutions in terms of fairness and clustering objectives.