Document Open Access Logo

An Optimal Randomized Algorithm for Finding the Saddlepoint

Authors Justin Dallant , Frederik Haagensen , Riko Jacob , László Kozma , Sebastian Wild

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.44.pdf
  • Filesize: 0.75 MB
  • 12 pages

Document Identifiers

Author Details

Justin Dallant
  • Department of Computer Science, Université libre de Bruxelles, Belgium
Frederik Haagensen
  • Department of Computer Science, IT University of Copenhagen, Denmark
Riko Jacob
  • Department of Computer Science, IT University of Copenhagen, Denmark
László Kozma
  • Institut für Informatik, Freie Universität Berlin, Germany
Sebastian Wild
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Dallant, Justin: Supported by the French Community of Belgium via the funding of a FRIA grant.
  • Haagensen, Frederik: Supported by Independent Research Fund Denmark, grant 0136-00144B, "DISTRUST" project.
  • Kozma, László: Supported by DFG grant KO 6140/1-2.
  • Wild, Sebastian: Supported by EPSRC grant EP/X039447/1.

Acknowledgements

This work was initiated at Dagstuhl Seminar 23211 "Scalable Data Structures".

Cite AsGet BibTex

Justin Dallant, Frederik Haagensen, Riko Jacob, László Kozma, and Sebastian Wild. An Optimal Randomized Algorithm for Finding the Saddlepoint. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 44:1-44:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.44

Abstract

A saddlepoint of an n × n matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the value of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a strict saddlepoint (an entry that is the strict maximum of its row and the strict minimum of its column) an O(n log* n)-time algorithm was recently obtained by Dallant, Haagensen, Jacob, Kozma, and Wild, improving the O(n log n) bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal O(n)-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial O(n²) runtime cannot be improved even with the use of randomness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • saddlepoint
  • matrix
  • comparison
  • search
  • randomized algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Mihir Bellare and John Rompel. Randomness-efficient oblivious sampling. In Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 276-287. IEEE, 1994. Google Scholar
  2. Daniel Bienstock, Fan Chung, Michael L. Fredman, Alejandro A. Schäffer, Peter W. Shor, and Subhash Suri. A note on finding a strict saddlepoint. Am. Math. Monthly, 98(5):418-419, April 1991. Google Scholar
  3. Guy E. Blelloch. Prefix sums and their applications. Technical Report CMU-CS-90-190, School of Computer Science, Carnegie Mellon University, November 1990. Google Scholar
  4. Christopher C. Byrne and Leonid N. Vaserstein. An improved algorithm for finding saddlepoints of two-person zero-sum games. Int. J. Game Theory, 20(2):149-159, June 1991. Google Scholar
  5. Antonin Chambolle and Thomas Pock. A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of mathematical imaging and vision, 40:120-145, 2011. Google Scholar
  6. Antonin Chambolle and Thomas Pock. On the ergodic convergence rates of a first-order primal-dual algorithm. Mathematical Programming, 159(1-2):253-287, 2016. Google Scholar
  7. Justin Dallant, Frederik Haagensen, Riko Jacob, László Kozma, and Sebastian Wild. Finding the saddlepoint faster than sorting. In 2024 Symposium on Simplicity in Algorithms (SOSA), pages 168-178. SIAM, 2024. Google Scholar
  8. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. Google Scholar
  9. Yijie Han. Optimal parallel selection. ACM Transactions on Algorithms (TALG), 3(4):38-es, 2007. Google Scholar
  10. Donald E. Knuth. The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms. Addison Wesley Longman Publishing Co., Inc., USA, 1997. Google Scholar
  11. Donald E. Knuth. The Art of Computer Programming, Volume 2 (3rd Ed.): Seminumerical Algorithms. Addison Wesley Longman Publishing Co., Inc., USA, 1997. Google Scholar
  12. Tianyi Lin, Chi Jin, and Michael I. Jordan. Near-optimal algorithms for minimax optimization. In Conference on Learning Theory, pages 2738-2779. PMLR, 2020. Google Scholar
  13. Donna Crystal Llewellyn, Craig Tovey, and Michael Trick. Finding saddlepoints of two-person, zero sum games. The American Mathematical Monthly, 95(10):912-918, 1988. Google Scholar
  14. Arnab Maiti, Ross Boczar, Kevin Jamieson, and Lillian J. Ratliff. Near-optimal pure exploration in matrix games: A generalization of stochastic bandits & dueling bandits, 2023. URL: https://arxiv.org/abs/2310.16252.
  15. Arnab Maiti, Ross Boczar, Kevin Jamieson, and Lillian J. Ratliff. Query-efficient algorithms to find the unique nash equilibrium in a two-player zero-sum matrix game, 2023. URL: https://arxiv.org/abs/2310.16236.
  16. Michael Maschler, Shmuel Zamir, and Eilon Solan. Game theory. Cambridge University Press, 2020. Google Scholar
  17. Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms. Cambridge University Press, 1995. Google Scholar
  18. Angelia Nedić and Asuman Ozdaglar. Subgradient methods for saddle-point problems. Journal of optimization theory and applications, 142:205-228, 2009. Google Scholar
  19. Meisam Razaviyayn, Tianjian Huang, Songtao Lu, Maher Nouiehed, Maziar Sanjabi, and Mingyi Hong. Nonconvex min-max optimization: Applications, challenges, and recent theoretical advances. IEEE Signal Processing Magazine, 37(5):55-66, 2020. Google Scholar
  20. Salil P. Vadhan. Pseudorandomness. Foundations and Trendsregistered in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact