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Scalable Algorithms for Individual Preference Stable Clustering

Ron Mosenzon, Ali Vakilian

TL;DR

By refining the local search approach, it is shown that a $O(\log n)$-IP stability guarantee for this algorithm is confirmed, where $n$ denotes the number of points in the input.

Abstract

In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is $α$-IP stable when each data point's average distance to its cluster is no more than $α$ times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a $O(\log n)$-IP stability guarantee for this algorithm, where $n$ denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, $\tilde{O}(nk)$.

Scalable Algorithms for Individual Preference Stable Clustering

TL;DR

By refining the local search approach, it is shown that a -IP stability guarantee for this algorithm is confirmed, where denotes the number of points in the input.

Abstract

In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is -IP stable when each data point's average distance to its cluster is no more than times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a -IP stability guarantee for this algorithm, where denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, .
Paper Structure (75 sections, 39 theorems, 157 equations, 9 algorithms)

This paper contains 75 sections, 39 theorems, 157 equations, 9 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition and Background
  3. Our Results
  4. Simple Local Search.
  5. Fast Algorithm.
  6. Bounding Approximation Factor.
  7. Other IP Stability Functions.
  8. Our Techniques
  9. Faster Algorithm.
  10. Approximate IP Stability.
  11. Median-IP Stability.
  12. Max-IP Stability.
  13. Related Work
  14. Clustering Stability.
  15. Individual Fairness in Clustering.
  16. ...and 60 more sections

Key Result

Theorem 1

For every $n \geq 3$, the natural local search algorithm (Algorithm :alg:natural-local-search) always terminates with parameter $\alpha \geq 2\log n$ on any metric space with $n$ points, regardless of the value of $k$.

Theorems & Definitions (141)

  • Definition 1: IP Stability ahmadi2022individual
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Theorem 4
  • Theorem 5
  • Definition 3: average distance
  • Lemma 1
  • proof
  • ...and 131 more