Document Open Access Logo

Hardness of Sampling for the Anti-Ferromagnetic Ising Model on Random Graphs

Authors Neng Huang , Will Perkins , Aaron Potechin

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.61.pdf
  • Filesize: 0.82 MB
  • 23 pages

Document Identifiers

Author Details

Neng Huang
  • University of Michigan, Ann Arbor, MI, USA
Will Perkins
  • Georgia Institute of Technology, Atlanta, GA, USA
Aaron Potechin
  • University of Chicago, IL, USA

Funding

  • Huang, Neng: Work was primarily done when NH was affiliated with the University of Chicago, supported partly by the Institute for Data, Econometrics, Algorithms, and Learning (IDEAL) with NSF grant ECCS-2216912.
  • Perkins, Will: Supported in part by NSF grant CCF-2309708.

Cite As Get BibTex

Neng Huang, Will Perkins, and Aaron Potechin. Hardness of Sampling for the Anti-Ferromagnetic Ising Model on Random Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.61

Abstract

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree d for large constant d, proving that when the normalized inverse temperature satisfies β > 1 (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in W₂ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions.
Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree d for large d. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Random graph
  • spin glass
  • sampling algorithm

Metrics

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

Related Versions

References

  1. Ahmed El Alaoui and David Gamarnik. Hardness of sampling solutions from the Symmetric Binary Perceptron, July 2024. arXiv:2407.16627 [cs, math]. URL: https://doi.org/10.48550/arXiv.2407.16627.
  2. Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Local algorithms for Maximum Cut and Minimum Bisection on locally treelike regular graphs of large degree, November 2021. arXiv:2111.06813 [math-ph]. URL: https://doi.org/10.48550/arXiv.2111.06813.
  3. Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Optimization of mean-field spin glasses. The Annals of Probability, 49(6):2922-2960, November 2021. Publisher: Institute of Mathematical Statistics. URL: https://doi.org/10.1214/21-AOP1519.
  4. Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization, March 2022. arXiv:2203.05093 [cond-mat]. URL: https://doi.org/10.48550/arXiv.2203.05093.
  5. 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 arXiv:2106.04105, 2021. Google Scholar
  6. 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, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649622.
  7. Roland Bauerschmidt and Thierry Bodineau. A very simple proof of the LSI for high temperature spin systems. Journal of Functional Analysis, 276(8):2582-2588, 2019. Google Scholar
  8. Roland Bauerschmidt, Thierry Bodineau, and Benoit Dagallier. Kawasaki dynamics beyond the uniqueness threshold. arXiv preprint arXiv:2310.04609, 2023. URL: https://doi.org/10.48550/arXiv.2310.04609.
  9. Ivona Bezáková, Daniel Štefankovič, Vijay V Vazirani, and Eric Vigoda. Accelerating simulated annealing for the permanent and combinatorial counting problems. SIAM Journal on Computing, 37(5):1429-1454, 2008. URL: https://doi.org/10.1137/050644033.
  10. Charlie Carlson, Ewan Davies, Alexandra Kolla, and Will Perkins. Computational thresholds for the fixed-magnetization Ising model. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1459-1472, 2022. URL: https://doi.org/10.1145/3519935.3520003.
  11. Michael Celentano. Sudakov–Fernique post-AMP, and a new proof of the local convexity of the TAP free energy. The Annals of Probability, 52(3):923-954, 2024. URL: https://doi.org/10.1214/23-AOP1675.
  12. Sourav Chatterjee. Disorder chaos and multiple valleys in spin glasses. arXiv preprint, 2009. URL: https://arxiv.org/abs/0907.3381.
  13. Antares Chen, Neng Huang, and Kunal Marwaha. Local algorithms and the failure of log-depth quantum advantage on sparse random CSPs, October 2023. arXiv:2310.01563 [quant-ph]. URL: https://doi.org/10.48550/arXiv.2310.01563.
  14. Louis H. Y. Chen. Poisson Approximation for Dependent Trials. The Annals of Probability, 3(3):534-545, June 1975. Publisher: Institute of Mathematical Statistics. URL: https://doi.org/10.1214/aop/1176996359.
  15. Wei-Kuo Chen. Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound. The Annals of Probability, 45(6A):3929-3966, November 2017. Publisher: Institute of Mathematical Statistics. URL: https://doi.org/10.1214/16-AOP1154.
  16. Wei-Kuo Chen, David Gamarnik, Dmitry Panchenko, and Mustazee Rahman. Suboptimality of local algorithms for a class of max-cut problems. The Annals of Probability, 47(3):1587-1618, May 2019. Publisher: Institute of Mathematical Statistics. URL: https://doi.org/10.1214/18-AOP1291.
  17. Wei-Kuo Chen and Dmitry Panchenko. Disorder chaos in some diluted spin glass models. The Annals of Applied Probability, 28(3):1356-1378, June 2018. URL: https://doi.org/10.1214/17-AAP1331.
  18. Yuansi Chen and Ronen Eldan. Localization schemes: A framework for proving mixing bounds for Markov chains. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 110-122. IEEE, 2022. Google Scholar
  19. Amin Coja-Oghlan, Philipp Loick, Balázs F. Mezei, and Gregory B. Sorkin. The Ising Antiferromagnet and Max Cut on Random Regular Graphs. SIAM Journal on Discrete Mathematics, 36(2):1306-1342, June 2022. Publisher: Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/20M137999X.
  20. Aurelien Decelle, Florent Krzakala, Cristopher Moore, and Lenka Zdeborová. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 84(6):066106, 2011. Google Scholar
  21. Amir Dembo, Andrea Montanari, and Subhabrata Sen. Extremal cuts of sparse random graphs. The Annals of Probability, 45(2):1190-1217, March 2017. Publisher: Institute of Mathematical Statistics. URL: https://doi.org/10.1214/15-AOP1084.
  22. 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, 182(3):1035-1051, 2022. Google Scholar
  23. Francesco Guerra. Broken replica symmetry bounds in the mean field spin glass model. Communications in mathematical physics, 233:1-12, 2003. Google Scholar
  24. Chris Jones, Kunal Marwaha, Juspreet Singh Sandhu, and Jonathan Shi. Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 77:1-77:26, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.77.
  25. Florent Krzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318-10323, 2007. URL: https://doi.org/10.1073/PNAS.0703685104.
  26. Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and slow mixing of the Kawasaki dynamics on bounded-degree graphs. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2405.06209.
  27. M Mézard, G Parisi, and R Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, pages 815-814, 2002. Google Scholar
  28. Marc Mézard and Giorgio Parisi. Mean-field theory of randomly frustrated systems with finite connectivity. Europhysics Letters, 3(10):1067, 1987. Google Scholar
  29. Marc Mézard and Giorgio Parisi. The Bethe lattice spin glass revisited. The European Physical Journal B-Condensed Matter and Complex Systems, 20:217-233, 2001. Google Scholar
  30. Andrea Montanari. Optimization of the Sherrington-Kirkpatrick hamiltonian. SIAM Journal on Computing, 0(0):FOCS19-1-FOCS19-38, 2021. URL: https://doi.org/10.1137/20M132016X.
  31. Elchanan Mossel, Joe Neeman, and Allan Sly. Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields, 162(3):431-461, August 2015. URL: https://doi.org/10.1007/s00440-014-0576-6.
  32. Dmitry Panchenko. The Sherrington-Kirkpatrick Model. Springer Monographs in Mathematics. Springer New York, NY, 2013. Google Scholar
  33. Giorgio Parisi. Infinite number of order parameters for spin-glasses. Physical Review Letters, 43(23):1754, 1979. Google Scholar
  34. Giorgio Parisi. A sequence of approximated solutions to the SK model for spin glasses. Journal of Physics A: Mathematical and General, 13(4):L115, 1980. Google Scholar
  35. Giorgio Parisi. Order parameter for spin-glasses. Physical Review Letters, 50(24):1946, 1983. Google Scholar
  36. David Sherrington and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters, 35(26):1792, 1975. Google Scholar
  37. H. Sompolinsky and Annette Zippelius. Dynamic theory of the spin-glass phase. Phys. Rev. Lett., 47:359-362, August 1981. URL: https://doi.org/10.1103/PhysRevLett.47.359.
  38. Eliran Subag. Following the ground states of full-RSB spherical spin glasses. Communications on Pure and Applied Mathematics, 74(5):1021-1044, 2021. Google Scholar
  39. Michel Talagrand. The Parisi Formula. Annals of Mathematics, 163(1):221-263, 2006. Publisher: Annals of Mathematics. URL: https://www.jstor.org/stable/20159953.
  40. Michel Talagrand. Parisi measures. Journal of Functional Analysis, 231(2):269-286, February 2006. URL: https://doi.org/10.1016/j.jfa.2005.03.001.
  41. Michel Talagrand. Mean Field Models for Spin Glasses: Volume I: Basic Examples. Springer, Berlin, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-15202-3.
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 About DROPS Imprint Privacy Contact