Proportionally Representative Clustering

TL;DR

This work proposes a new axiom ``proportionally representative fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together.

Abstract

In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on centroid clustering--one of the fundamental tasks in unsupervised machine learning. We propose a new axiom ``proportionally representative fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair clustering algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom. Our algorithm for the discrete setting also matches the best known approximation factor for PF.

Figures (9)

• Figure 1: An example instance with 6 agents and $k=3$ for which a PF outcome does not exist.
• Figure 2: Some of the requirements of PRF for instance in Example \ref{['example:PFnott']}.
• Figure 3: The k-means solution may not satisfy PRF.
• Figure F.4: Buddy dataset: MSD to closest 1, k/2, and k centroids.
• Figure F.5: Seeds dataset: MSD to closest 1, k/2, and k centroids.

Theorems & Definitions (31)

• Definition 1: Proportional Fairness CFLM19a
• Definition 2: Unanimous Proportionality (UP)
• Example 1: Proportional fairness and core fairness
• Definition 3: $\rho$-approximate Proportional Fairness
• Example 2
• Definition 4: Proportionally Representative Fairness (PRF) for Unconstrained Clustering
• Example 3: Requirements of PRF
• Proposition 1
• proof
• Example 4: $k$-means does not satisfy PRF