A Stochastic Alternating Balance k-Means Algorithm for Fair Clustering

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.

L. N. Vicente—Support for this author was partially provided by the Centre for Mathematics of the University of Coimbra under grant FCT/MCTES UIDB/MAT/00324/2020.

Appendices

A Description of an Existing Approach for Comparison

The authors in [32] considered the fairness error computed by the Kullback-Leibler (KL)-divergence, and added it as a penalized term to the classical clustering objective. When using the k-means clustering cost, the resulting problem takes the form:

$$\begin{aligned} \min ~ f_1(s) + \mu \displaystyle \sum _{k=1}^N \mathcal {D}_{KL}(U\Vert \mathbb {P}_k) \quad \text {s.t. } \sum _{k=1}^K s_{p, k} = 1, \forall p \in [N], \end{aligned}$$

(6)

where $\mathcal {D}_{KL}$ is the KL divergence between the desired demographic proportion $U = [u_j, j \in [J]]$ (usually specified by the demographic composition of the whole dataset) and the marginal probability $\mathbb {P}_k = [\mathbb {P}(j|k) = s_k^\top v_j/{e_N}^\top s_k, j \in [J]]$. The penalty coefficient $\mu$ associated with the fairness error is the tool to control the trade-offs between the clustering cost and the clustering balance. To solve problem (6) for a fixed $\mu \ge 0$, the authors in [32] have developed an optimization scheme based on a concave-convex decomposition of the fairness term.

B More Numerical Results

Fig. 2.

Demographic composition of four synthetic datasets.

Fig. 3.

Syn_equal_ds1 data: SAfairKM: 400 iterations, 10 starting labels, and 3 pairs of $(n_a, n_b)$; VfairKM: $\mu _{\max } = 202$.

Fig. 4.

Syn_unequal_ds1 data: SAfairKM: 400 iterations, 10 starting labels, and 3 pairs of $(n_a, n_b)$; VfairKM: $\mu _{\max } = 223$.

Fig. 5.

Syn_equal_ds2 data: SAfairKM: 400 iterations, 10 starting labels, and 3 pairs of $(n_a, n_b)$; VfairKM: $\mu _{\max } = 60$.

Fig. 6.

Syn_unequal_ds2 data: SAfairKM: 400 iterations, 10 starting labels, and 3 pairs of $(n_a, n_b)$; VfairKM: $\mu _{\max } = 0$.

Fig. 7.

Pareto fronts for $K = 5$: SAfairKM: 2500 iterations for Adult and 1500 iterations for Bank, 30 starting labels, and 4 pairs of $(n_a, n_b)$; VfairKM: $\mu _{\max } = 6190$ for Adult and $\mu _{\max } = 4790$ for Bank.