Document Open Access Logo

Efficient Parallel Ising Samplers via Localization Schemes

Authors Xiaoyu Chen , Hongyang Liu , Yitong Yin , Xinyuan Zhang

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.46.pdf
  • Filesize: 0.94 MB
  • 22 pages
HTML5 Logo HTML (experimental)
LIPIcs.APPROX-RANDOM.2025.46.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Parallel algorithms
Keywords
  • Localization scheme
  • parallel sampling
  • Ising model

Metrics

Abstract

We introduce efficient parallel algorithms for sampling from the Gibbs distribution and estimating the partition function of Ising models. These algorithms achieve parallel efficiency, with polylogarithmic depth and polynomial total work, and are applicable to Ising models in the following regimes: (1) Ferromagnetic Ising models with external fields; (2) Ising models with interaction matrix J of operator norm ‖J‖₂ < 1.
Our parallel Gibbs sampling approaches are based on localization schemes, which have proven highly effective in establishing rapid mixing of Gibbs sampling. In this work, we employ two such localization schemes to obtain efficient parallel Ising samplers: the field dynamics induced by negative-field localization, and restricted Gaussian dynamics induced by stochastic localization. This shows that localization schemes are powerful tools, not only for achieving rapid mixing but also for the efficient parallelization of Gibbs sampling.

Cite As Get BibTex

Xiaoyu Chen, Hongyang Liu, Yitong Yin, and Xinyuan Zhang. Efficient Parallel Ising Samplers via Localization Schemes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 46:1-46:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.46

Author Details

Xiaoyu Chen
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Hongyang Liu
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Yitong Yin
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Xinyuan Zhang
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China

Funding

This work has been partially supported by the New Cornerstone Science Foundation.
  • Zhang, Xinyuan: NSFC Young Student Basic Research Program (PhD candidate, No. 623B2051).

References

  1. Nima Anari, Callum Burgess, Kevin Tian, and Thuy-Duong Vuong. Quadratic speedups in parallel sampling from determinantal distributions. In Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2023), pages 367-377. ACM, 2023. URL: https://doi.org/10.1145/3558481.3591104.
  2. Nima Anari, Sinho Chewi, and Thuy-Duong Vuong. Fast parallel sampling under isoperimetry. In The Thirty Seventh Annual Conference on Learning Theory (ICML 2024), volume 247 of Proceedings of Machine Learning Research, pages 161-185. PMLR, 2024. URL: https://proceedings.mlr.press/v247/anari24a.html.
  3. Nima Anari, Ruiquan Gao, and Aviad Rubinstein. Parallel sampling via counting. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 537-548. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649744.
  4. Nima Anari, Nathan Hu, Amin Saberi, and Aaron Schild. Sampling arborescences in parallel. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of LIPIcs, pages 83:1-83:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.83.
  5. Nima Anari, Yizhi Huang, Tianyu Liu, Thuy-Duong Vuong, Brian Xu, and Katherine Yu. Parallel discrete sampling via continuous walks. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023), pages 103-116. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585207.
  6. Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, and Thuy-Duong Vuong. Entropic independence I: Modified log-Sobolev inequalities for fractionally log-concave distributions and high-temperature Ising models. arXiv preprint, 2021. URL: https://arxiv.org/abs/2106.04105.
  7. Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, and Thuy-Duong Vuong. Entropic independence: optimal mixing of down-up random walks. In Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC 2022), pages 1418-1430. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520048.
  8. Nima Anari, Frederic Koehler, and Thuy-Duong Vuong. Trickle-down in localization schemes and applications. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 1094-1105. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649622.
  9. Xiaoyu Chen, Zongchen Chen, Yitong Yin, and Xinyuan Zhang. Rapid mixing at the uniqueness threshold. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC 2025), pages 879-890. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718260.
  10. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Rapid mixing of Glauber dynamics via spectral independence for all degrees. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2021), pages 137-148. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00022.
  11. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Optimal mixing for two-state anti-ferromagnetic spin systems. In 63rd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 588-599. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00062.
  12. Xiaoyu Chen and Xinyuan Zhang. A near-linear time sampler for the Ising model with external field. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 4478-4503. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH170.
  13. Yuansi Chen and Ronen Eldan. Localization schemes: A framework for proving mixing bounds for markov chains (extended abstract). In 63rd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 110-122. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00018.
  14. Richard Cole. Parallel merge sort. SIAM J. Comput., 17(4):770-785, 1988. URL: https://doi.org/10.1137/0217049.
  15. Martin Dyer, Alan Frieze, and Ravi Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. Journal of the ACM (JACM), 38(1):1-17, 1991. URL: https://doi.org/10.1145/102782.102783.
  16. Robert G. Edwards and Alan D. Sokal. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3), 38(6):2009-2012, 1988. Google Scholar
  17. Ahmed El Alaoui and Andrea Montanari. An information-theoretic view of stochastic localization. IEEE Trans. Inform. Theory, 68(11):7423-7426, 2022. URL: https://doi.org/10.1109/TIT.2022.3180298.
  18. Ronen Eldan, Frederic Koehler, and Ofer Zeitouni. A spectral condition for spectral gap: fast mixing in high-temperature Ising models. Probability Theory and Related Fields, pages 1-17, 2021. URL: https://doi.org/10.1007/s00440-021-01085-x.
  19. Weiming Feng, Heng Guo, and Jiaheng Wang. Swendsen-wang dynamics for the ferromagnetic ising model with external fields. Inf. Comput., 294:105066, 2023. URL: https://doi.org/10.1016/J.IC.2023.105066.
  20. Weiming Feng, Thomas P. Hayes, and Yitong Yin. Distributed Metropolis sampler with optimal parallelism. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 2121-2140. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.127.
  21. Weiming Feng, Yuxin Sun, and Yitong Yin. What can be sampled locally? Distributed Comput., 33(3-4):227-253, 2020. URL: https://doi.org/10.1007/S00446-018-0332-8.
  22. Weiming Feng, Nisheeth K. Vishnoi, and Yitong Yin. Dynamic sampling from graphical models. SIAM J. Comput., 50(2):350-381, 2021. URL: https://doi.org/10.1137/20M1315099.
  23. Manuela Fischer and Mohsen Ghaffari. A simple parallel and distributed sampling technique: Local glauber dynamics. In 32nd International Symposium on Distributed Computing (DC 2018), volume 121 of LIPIcs, pages 26:1-26:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.26.
  24. Andreas Galanis, Alkis Kalavasis, and Anthimos Vardis Kandiros. On sampling from Ising models with spectral constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024), volume 317 of LIPIcs, pages 70:1-70:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.70.
  25. Thomas P. Hayes and Eric Vigoda. Coupling with the stationary distribution and improved sampling for colorings and independent sets. Ann. Appl. Probab., 16(3):1297-1318, 2006. URL: https://doi.org/10.1214/105051606000000330.
  26. Ernst Ising. Beitrag zur theorie des ferro-und paramagnetismus. PhD thesis, Grefe & Tiedemann Hamburg, 1924. Google Scholar
  27. Joseph JéJé. An introduction to parallel algorithms. Reading, MA: Addison-Wesley, 10:133889, 1992. Google Scholar
  28. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989. URL: https://doi.org/10.1137/0218077.
  29. Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput., 22(5):1087-1116, 1993. URL: https://doi.org/10.1137/0222066.
  30. Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM (JACM), 51(4):671-697, 2004. URL: https://doi.org/10.1145/1008731.1008738.
  31. Dmitriy Kunisky. Optimality of Glauber dynamics for general-purpose Ising model sampling and free energy approximation. In Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms (SODA 2024), pages 5013-5028. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.180.
  32. Holden Lee. Parallelising glauber dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024), volume 317 of LIPIcs, pages 49:1-49:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.49.
  33. Yin Tat Lee, Ruoqi Shen, and Kevin Tian. Structured logconcave sampling with a restricted Gaussian oracle. In The Annual Conference on Learning Theory (COLT 2021), pages 2993-3050, 2021. URL: http://proceedings.mlr.press/v134/lee21a.html.
  34. David A. Levin, Malwina J. Luczak, and Yuval Peres. Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Related Fields, 146(1-2):223-265, 2010. URL: https://doi.org/10.1007/s00440-008-0189-z.
  35. David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2017. Google Scholar
  36. Hongyang Liu and Yitong Yin. Simple parallel algorithms for single-site dynamics. In STOC, pages 1431-1444. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519999.
  37. Hongyang Liu and Yitong Yin. Parallelize single-site dynamics up to dobrushin criterion. J. ACM, 72(1), January 2025. URL: https://doi.org/10.1145/3708558.
  38. Hongyang Liu, Yitong Yin, and Yiyao Zhang. Work-efficient parallel counting via sampling. arXiv preprint arXiv:2408.09719, 2024. URL: https://doi.org/10.48550/arXiv.2408.09719.
  39. Ketan Mulmuley, Umesh V Vazirani, and Vijay V Vazirani. Matching is as easy as matrix inversion. In STOC, pages 345-354. ACM, 1987. URL: https://doi.org/10.1145/28395.383347.
  40. Yossi Shiloach and Uzi Vishkin. An O(logn) parallel connectivity algorithm. J. Algorithms, 3(1):57-67, 1982. URL: https://doi.org/10.1016/0196-6774(82)90008-6.
  41. Shang-Hua Teng. Independent sets versus perfect matchings. Theoret. Comput. Sci., 145(1-2):381-390, 1995. URL: https://doi.org/10.1016/0304-3975(94)00289-U.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact