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Local Search k-means++ with Foresight

Authors Theo Conrads, Lukas Drexler , Joshua Könen , Daniel R. Schmidt , Melanie Schmidt

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LIPIcs.SEA.2024.7.pdf
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Author Details

Theo Conrads
  • Department of Computer Science, University of Cologne, Germany
Lukas Drexler
  • Faculty of Mathematics and Natural Sciences, Department of Computer Science, Heinrich Heine University Düsseldorf, Germany
Joshua Könen
  • Institute of Computer Science, University of Bonn, Germany
Daniel R. Schmidt
  • Faculty of Mathematics and Natural Sciences, Department of Computer Science, Heinrich Heine University Düsseldorf, Germany
Melanie Schmidt
  • Faculty of Mathematics and Natural Sciences, Department of Computer Science, Heinrich Heine University Düsseldorf, Germany

Funding

  • Drexler, Lukas: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 456558332 - funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 456558332.
  • Könen, Joshua: funded by Lamarr Institute for Machine Learning and Artificial Intelligence - lamarr-institute.org
  • Schmidt, Melanie: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 456558332 - funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 456558332.

Cite AsGet BibTex

Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt. Local Search k-means++ with Foresight. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SEA.2024.7

Abstract

Since its introduction in 1957, Lloyd’s algorithm for k-means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed k-means++ which enhances Lloyd’s algorithm by a seeding method which guarantees a 𝒪(log k)-approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of k-means++ by local search techniques to obtain a 𝒪(1)-approximation. On the practical side, the greedy variant of k-means++ is often used although its worst-case behaviour is provably worse than for the standard k-means++ variant. We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy k-means++ instead of k-means++, and (b) Improving LS++ by better entangling it with Lloyd’s algorithm. Option (a) worsens the theoretical guarantees of k-means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Facility location and clustering
  • Information systems → Clustering
Keywords
  • k-means clustering
  • kmeans++
  • greedy
  • local search

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References

  1. Available at https://www.kdd.org/kdd-cup/view/kdd-cup-2004/data. Google Scholar
  2. Sara Ahmadian, Ashkan Norouzi-Fard, Ola Svensson, and Justin Ward. Better guarantees for k-means and Eucl. k-median by primal-dual alg. SIAM J. Comput., 49(4), 2020. Google Scholar
  3. Daniel Aloise, Pierre Hansen, and Leo Liberti. An improved column generation algorithm for minimum sum-of-squares clustering. Mathematical Programming, 131:195-220, 2012. Google Scholar
  4. David Arthur and Sergei Vassilvitskii. K-means++: The advantages of careful seeding. In Proceedings of the 18th SODA, pages 1027-1035, USA, 2007. Google Scholar
  5. Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of Euclidean k-means. In Lars Arge and János Pach, editors, Proc. of the 31st SoCG, volume 34 of LIPIcs, pages 754-767, 2015. Google Scholar
  6. Anup Bhattacharya, Jan Eube, Heiko Röglin, and Melanie Schmidt. Noisy, greedy and not so greedy k-means++. In Proc. of the 28th ESA, 2020. Google Scholar
  7. M. Emre Celebi, Hassan A. Kingravi, and Patricio A. Vela. A comparative study of efficient initialization methods for the k-means clustering algorithm. Expert Syst. Appl., 40(1):200-210, 2013. Google Scholar
  8. Davin Choo, Christoph Grunau, Julian Portmann, and Václav Rozhon. k-means++: few more steps yield constant approximation. In International Conference on Machine Learning, pages 1909-1917. PMLR, 2020. Google Scholar
  9. Theo Conrads. Lokale Such- und Samplingmethoden für das k-Means- und k-Median-Problem. Master’s thesis, Universität zu Köln, 2021. Google Scholar
  10. Sanjoy Dasgupta. The hardness of k-means clustering, 2008. Technical report. Google Scholar
  11. Lukas Drexler, Joshua Könen, Daniel R. Schmidt, Melanie Schmidt, and Giulia Baldini. algo-hhu/FLSpp. Software, swhId: https://archive.softwareheritage.org/swh:1:dir:39136777e542456572a51515813b6cb377ed0940;origin=https://github.com/algo-hhu/FLSpp;visit=swh:1:snp:18f736f0d6ee924ac77fcfdbeed3e15015a6b867;anchor=swh:1:rev:72d58dc26e599c190373274ed6c22009d07a911b (visited on 2024-06-27). URL: https://github.com/algo-hhu/FLSpp.
  12. Charles Elkan. Using the triangle inequality to accelerate k-means. In Tom Fawcett and Nina Mishra, editors, Proc. of the 20th ICML, pages 147-153, 2003. Google Scholar
  13. Gereon Frahling and Christian Sohler. A fast k-means implementation using coresets. In Proc. of the 22nd SoCG, pages 135-143, 2006. Google Scholar
  14. Pasi Fränti and Sami Sieranoja. K-means properties on six clustering benchmark datasets. Appl. Intell., 48(12):4743-4759, 2018. Google Scholar
  15. Pasi Fränti and Sami Sieranoja. How much can k-means be improved by using better initialization and repeats? Pattern Recognition, 93:95-112, 2019. Google Scholar
  16. Bernd Fritzke. The k-means-u* algorithm: non-local jumps and greedy retries improve k-means++ clustering. CoRR, abs/1706.09059, 2017. Google Scholar
  17. Christoph Grunau, Ahmet Alper Özüdoğru, Václav Rozhoň, and Jakub Tětek. A nearly tight analysis of greedy k-means++. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1012-1070. SIAM, 2023. Google Scholar
  18. Greg Hamerly. Making k-means even faster. In SDM, pages 130-140. SIAM, 2010. Google Scholar
  19. Grete Heinz, Louis J. Peterson, Roger W. Johnson, and Carter J. Kerk. Exploring relationships in body dimensions. Journal of Statistics Education, 11(2), 2003. Google Scholar
  20. Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Comput. Geom., 28(2-3):89-112, 2004. Google Scholar
  21. Silvio Lattanzi and Christian Sohler. A better k-means++ algorithm via local search. In Proc. of the 36th ICML, volume 97 of Proceedings of Machine Learning Research, pages 3662-3671. PMLR, 09-15 June 2019. Google Scholar
  22. Euiwoong Lee, Melanie Schmidt, and John Wright. Improved and simplified inapproximability for k-means. Inf. Process. Lett., 120:40-43, 2017. Google Scholar
  23. Meena Mahajan, Prajakta Nimbhorkar, and Kasturi R. Varadarajan. The planar k-means problem is np-hard. Theor. Comput. Sci., 442:13-21, 2012. Google Scholar
  24. Manfred Padberg and Giovanni Rinaldi. A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Review, 33(1):60-100, 1991. Google Scholar
  25. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825-2830, 2011. Google Scholar
  26. J.M Peña, J.A Lozano, and P Larrañaga. An empirical comparison of four initialization methods for the k-means algorithm. Pattern Recognition Letters, 20(10):1027-1040, 1999. Google Scholar
  27. Dennis Wei. A constant-factor bi-criteria approximation guarantee for k-means++. In Advances in Neural Information Processing Systems, volume 29, 2016. Google Scholar
  28. I.-C. Yeh. Modeling of strength of high-performance concrete using artificial neural networks. Cement and Concrete Research, 28(12):1797-1808, 1998. Google Scholar
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