Fair Clustering via Alignment

TL;DR

This paper tackles fair clustering by proposing FCA, an in-processing method that aligns data from protected groups into an aligned space and then performs clustering there. The key idea is a novel decomposition of the perfectly fair clustering objective into a transport (alignment) term and a clustering term, enabling a two-phase alternating algorithm with theoretical guarantees of near-optimal utility for any fairness level. The authors further introduce FCA-C to flexibly control the fairness level, establish its approximation properties, and demonstrate through extensive experiments that FCA achieves superior fairness-utility trade-offs, numerical stability, and scalability across tabular and visual datasets. Overall, FCA provides a practical, high-utility approach to fair clustering with interpretable alignment-based matching and robust performance advantages over existing methods.

Abstract

Algorithmic fairness in clustering aims to balance the proportions of instances assigned to each cluster with respect to a given sensitive attribute. While recently developed fair clustering algorithms optimize clustering objectives under specific fairness constraints, their inherent complexity or approximation often results in suboptimal clustering utility or numerical instability in practice. To resolve these limitations, we propose a new fair clustering algorithm based on a novel decomposition of the fair $K$-means clustering objective function. The proposed algorithm, called Fair Clustering via Alignment (FCA), operates by alternately (i) finding a joint probability distribution to align the data from different protected groups, and (ii) optimizing cluster centers in the aligned space. A key advantage of FCA is that it theoretically guarantees approximately optimal clustering utility for any given fairness level without complex constraints, thereby enabling high-utility fair clustering in practice. Experiments show that FCA outperforms existing methods by (i) attaining a superior trade-off between fairness level and clustering utility, and (ii) achieving near-perfect fairness without numerical instability.

Figures (9)

• Figure 1: Comparison between the fairlet-based method and our approach with $n_{0} = n_{1} = 4$ and $K = 2.$ The representative of each fairlet is set as the mean vector of the data within that fairlet, and the standard $K$-means algorithm is then applied to this set of representatives. The clustering results are visualized using contours. While both the fairlet-based method and ours result in perfectly fair clustering, i.e., balance (Bal) $= 1 = \min (n_{0}/n_{1}, n_{1}/n_{0})$, our approach yields a lower cost ($9.82 < 10.22$), due to more efficient matchings.
• Figure 2: Example illustration of $\mathcal{W}$ and $\Gamma$ (or equivalently $\mathbb{Q}$) when $(n_{0}, n_{1}) = (2, 5)$ and $\varepsilon = \gamma_{1, 4} + \gamma_{2, 2} + \gamma_{2, 5}.$
• Figure 3: Bal vs. Cost trade-offs on Adult dataset. Black square ($\blacksquare$) is from the standard clustering, orange circle ($\bullet$) is from VFC, green star ($\star$) is from FCA-C, orange dashed line (- -) is the maximum of Bal that VFC can achieve, and blue line (--) is the maximum achievable balance $\texttt{Bal}^{\star}.$
• Figure 4: Bal vs. Cost trade-offs on (left) Adult, (center) Bank and (right) Census datasets. Black square ($\blacksquare$) is from the standard clustering, orange circle ($\bullet$) is from VFC, green star ($\star$) is from FCA-C, orange dashed line (- -) is the maximum of Bal that VFC can achieve, and blue line (--) is the maximum achievable balance $\texttt{Bal}^{\star}.$
• Figure 5: Bal vs. Cost trade-offs on (left) Adult, (center) Bank and (right) Census datasets. Black square ($\blacksquare$) is from the standard clustering, orange circle ($\bullet$) is from VFC, green star ($\star$) is from FCA-C, orange dashed line (- -) is the maximum of Bal that VFC can achieve, and blue line (--) is the maximum achievable balance $\texttt{Bal}^{\star}.$ The data are not $L_{2}$-normalized.

Theorems & Definitions (15)

• Theorem 3.1
• Remark 3.2: Comparison to the fairlet-based methods
• Remark 3.4: $\mathcal{A}^{*}$ becomes deterministic when $n_{0} = n_{1}$
• Theorem 4.1: Equivalence between $\tilde{C}$ and constrained $C$
• Proposition 4.2: Relationship between balance and $\varepsilon$
• Theorem 4.3: Approximation guarantee of FCA-C
• Remark 4.4: Comparison of the approximation rate with existing approaches
• proof : Proof of \ref{['thm:equalcase']}
• proof : Proof of \ref{['thm:unequalcase']}
• proof : Proof of \ref{['thm:fca-c']}