Variational Fair Clustering

Imtiaz Masud Ziko, Jing Yuan, Eric Granger, Ismail Ben Ayed

Abstract: We propose a general variational framework of fair clustering,which integrates an original Kullback-Leibler (KL) fairnessterm with a large class of clustering objectives, including proto-type or graph based. Fundamentally different from the existingcombinatorial and spectral solutions, our variational multi-term approach enables to control the trade-off levels betweenthe fairness and clustering objectives. We derive a generaltight upper bound based on a concave-convex decompositionof our fairness term, its Lipschitz-gradient property and thePinsker’s inequality. Our tight upper bound can be jointly op-timized with various clustering objectives, while yielding ascalable solution, with convergence guarantee. Interestingly,at each iteration, it performs an independent update for eachassignment variable. Therefore, it can be easily distributed forlarge-scale datasets. This scalability is important as it enablesto explore different trade-off levels between the fairness andclustering objectives. Unlike spectral relaxation, our formula-tion does not require computing its eigenvalue decomposition.We report comprehensive evaluations and comparisons withstate-of-the-art methods over various fair-clustering bench-marks, which show that our variational formulation can yieldhighly competitive solutions in terms of fairness and clusteringobjectives