Source-linked AI summary
Proportionally Fair Clustering
Xingyu Chen, Brandon Fain, Liang Lyu, Kamesh Munagala
TL;DR
The paper asks how to define fairness for metric centroid clustering without prespecifying protected groups. It introduces proportionality, develops algorithms to compute, optimize, and audit proportional solutions, and finds that exact solutions may not exist while proportionality can trade off against k-means objectives.
Problem
Centroid clustering lacks a fairness notion that protects sufficiently large groups without assuming predefined protected subsets.
Method
The paper defines proportionality through the absence of Pareto-improving complaints by sufficiently large coalitions and develops algorithms for approximate computation, constrained optimization, sampling, and auditing.
Results
ρ-proportional solutions need not exist for ρ < 2, while they always exist for ρ ≥ 1 +
Takeaways & Limitations
Proportionality meaningfully constrains clustering choices and exposes a data-dependent tradeoff with the k-means objective.
Takeaways & Limitations
Proportionality can be incompatible with k-center, k-means, and k-median objectives, and exact proportional solutions may not exist.
Abstract
from arXiv · showhide
We extend the fair machine learning literature by considering the problem of proportional centroid clustering in a metric context. For clustering $n$ points with $k$ centers, we define fairness as proportionality to mean that any $n/k$ points are entitled to form their own cluster if there is another center that is closer in distance for all $n/k$ points. We seek clustering solutions to which there are no such justified complaints from any subsets of agents, without assuming any a priori notion of protected subsets. We present and analyze algorithms to efficiently compute, optimize, and audit proportional solutions. We conclude with an empirical examination of the tradeoff between proportional solutions and the $k$-means objective.
1 Introduction
The paper introduces proportionality as a fairness concept for centroid clustering, requiring sufficiently large groups to receive proportional center entitlements without predefined protected attributes. It develops algorithms and auditing methods, while showing that proportionality can conflict sharply with standard clustering objectives.
- Fairness concept: The guarantee applies to all sufficiently large subsets, so it does not require prior knowledge of protected attributes.This makes proportionality oblivious to the definition of sensitive attributes or protected subgroups.
- Motivation: A three-cluster example shows that k-means and k-median can allocate two centers to a large, dispersed cluster while two smaller clusters share one.The resulting solution can become arbitrarily non-proportional as the smaller clusters’ radii shrink.
- Motivation: Proportionality can protect groups such as dense urban communities or patient clusters even when their protected status is unavailable to the algorithm.The motivating examples contrast proportional center allocation with minimizing total distance, which can obscure differences between smaller groups.
- Fairness concept: Proportionality gives any sufficiently large group entitlement to choose a proportional number of centers if those centers would improve every member’s distance.A blocking coalition consists of at least r⌈n/k⌉ points that can jointly use at most r centers to strictly improve their distances.
- Algorithms and auditing: Approximate proportional solutions can be computed, optimized with k-median as a constraint, and audited from a random sample of size ˜O(k^3).For constant k, sampling supports near-linear-time auditing and approximate computation comparable to running the classic k-means heuristic.
2 Existence and Computation of Proportional Solutions
Exact proportional solutions may not exist, so the paper develops approximation and local-search methods for finding proportional clusterings. The greedy method is efficient and approximately proportional, while local capture searches for more proportional solutions but lacks convergence guarantees.
- Existence: For all ρ < 2, a ρ-proportional solution is not guaranteed to exist.This impossibility holds in the general setting with separate individuals and feasible centers.
- Existence: When N = M, for all ρ < 1.5, a ρ-proportional solution is not guaranteed to exist.The construction uses three identical clusters and shows that one cluster must violate proportionality below this factor.
- Approximate computation: Greedy Capture grows balls around feasible centers and opens a center when its ball captures at least ⌈n/k⌉ unmatched points.Open centers continue capturing points as their balls expand; the algorithm runs in ˜O(mn) time.
- Approximate computation: The greedy algorithm always produces an approximately proportional solution, with a constant-factor guarantee that is tight in the worst case.A matching instance makes a coalition’s improvement approach the algorithm’s bound as ϵ tends to zero.
- Local improvement: Local Capture swaps out the least-demanded current center whenever a candidate center is demanded by a coalition violating ρ-fairness.If it terminates, it returns a ρ-proportional solution, but convergence is not guaranteed and each iteration takes ˜O(mn^2) time.
3 Proportionality as a Constraint
The paper treats proportionality as a constraint in a k-median linear program, then rounds fractional solutions while preserving approximate proportionality and bounding objective cost.
- Motivation: Proportionality is imposed alongside k-median optimization because proportional solutions can otherwise have poor global objectives.The paper gives an example where an algorithm opens both centers in a 60% cluster despite a proportional solution separating 40% and 60% clusters.
- Approximation guarantee: If a ρ-proportional clustering has k-median objective c, polynomial time yields an O(ρ)-proportional clustering with objective at most 8c.The construction solves the augmented linear program and applies a modified rounding algorithm.
- Linear-program formulation: The proportional k-median linear program adds a constraint encoding approximate proportionality to the standard relaxation.Its variables indicate center openings and point-to-center assignments; the objective is k-median cost, while the new constraint uses radius R_j around each candidate center.
- Rounding: In the special case N = M, the rounding analysis tightens the constant factor from 27 to 13.The improvement follows from guaranteed demand at the center of each ball B(i, γR_i).
4 Sampling for Linear-Time Implementations and Auditing
Uniform random samples approximately preserve proportionality across candidate solutions and deviations, enabling near-linear implementations and efficient approximate auditing.
- Sampling guarantee: Uniform random sampling preserves proportionality with high probability for all candidate center sets and deviating centers.The result uses a sample-size bound and a union bound over possible solution and deviation pairs.
- Sampling guarantee: A sample of size approximately O~(k^3) suffices to approximately preserve proportionality for constant k.The paper states that the O~ notation hides lower-order terms.
- Efficient computation: Sampling reduces Algorithm 1 from quadratic input-size time to nearly linear time while preserving approximate proportionality with high probability.This makes the implementation comparable to the running time of the standard k-means heuristic.
- Efficient computation: Local Capture can search for a clustering proportional to (1 + ϵ)-deviations using a random sample of k^3/ϵ^2 points.Without sampling, each iteration takes O~(mn^2) time.
- Auditing: The Audit Problem finds the smallest ρ making a given clustering ρ-proportional, but exact auditing can take quadratic time.The exact method computes the largest relevant distance ratio for each candidate deviating center.
- Auditing: Random sampling also yields an efficient relaxed audit for proportionality to (1 + ϵ)-deviations with probability at least 1 − δ.The relaxed audit estimates the minimum qualifying ρ from a sample rather than the full dataset.
5 Implementations and Empirical Results
Experiments on Iris, Diabetes, and KDD compare proportionality with the k-means objective, revealing a data-dependent tradeoff. A center-union heuristic partially reconciles the objectives on KDD with at most three extra centers for k ≤10.
- Experimental setup: Experiments use Iris, Diabetes, and KDD, comparing Greedy Capture and Local Capture with k-means++ across k values.The KDD evaluation uses a 100,000-point sample for k-means++ and additional samples for Local Capture and proportionality measurement.
- Dataset results: Iris achieves exact proportionality for Local Capture and k-means++, with comparable k-means objectives.Its three natural clusters illustrate that proportionality and the k-means objective need not conflict.
- Dataset results: Diabetes shows a separation in k-means objectives: Local Capture is always better than 1.01-proportional, while k-means++ is not always exactly proportional.The objectives remain generally on the same order of magnitude.
- Dataset results: KDD exhibits a three- to four-order-of-magnitude gap between proportionality and the k-means objective for k-means++ and Local Capture.The authors suspect significant outliers contribute to this conflict; Greedy Capture is comparable to Local Capture and omitted from the figure for clarity.
- Low-objective tradeoff: On KDD, center removal with α = 1.2 and β = 1.5 preserves near-Local-Capture proportionality and near-k-means++ objective using at most 3 extra centers for k ≤10.The heuristic keeps proportionality within 1.2 of Local Capture and the k-means objective within 1.5 of k-means++.
6 Conclusion and Open Directions
The paper introduces proportionality for centroid clustering, develops approximate algorithms and optimization tools, and finds a data-dependent tradeoff with the k-means objective. Exact proportionality may fail to exist, while several computational and theoretical questions remain open.
- Conclusions: Proportionality is introduced as a fair centroid-clustering concept, with approximate algorithms, constrained optimization, and sampling for further applications.The paper also studies proportionality empirically on real data.
- Open directions: ρ-proportional solutions need not exist for ρ < 2, while solutions always exist for ρ ≥ 1 + 2.Closing this approximability gap is identified as an outstanding question.
- Open directions: Open questions include more efficient optimization, other fair clustering concepts, relative advantages and disadvantages, and extensions to supervised learning.The current optimization approach requires solving a linear program on the entire data set.