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Parameter Estimation for Gibbs Distributions

Authors David G. Harris, Vladimir Kolmogorov

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Author Details

David G. Harris
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Vladimir Kolmogorov
  • Institute of Science and Technology Austria, Klosterneuburg, Austria

Acknowledgements

We thank Heng Guo for helpful explanations of algorithms for sampling connected subgraphs and matchings, Maksym Serbyn for bringing to our attention the Wang-Landau algorithm and its use in physics.

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David G. Harris and Vladimir Kolmogorov. Parameter Estimation for Gibbs Distributions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 72:1-72:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.72

Abstract

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We will consider sampling problems which come from Gibbs distributions, which are families of probability distributions over a discrete space Ω with probability mass function of the form μ^Ω_β(ω) ∝ e^{β H(ω)} for β in an interval [β_min, β_max] and H(ω) ∈ {0} ∪ [1, n]. The partition function is the normalization factor Z(β) = ∑_{ω ∈ Ω} e^{β H(ω)}, and the log partition ratio is defined as q = (log Z(β_max))/Z(β_min) We develop a number of algorithms to estimate the counts c_x using roughly Õ(q/ε²) samples for general Gibbs distributions and Õ(n²/ε²) samples for integer-valued distributions (ignoring some second-order terms and parameters), We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • Applied computing → Physics
Keywords
  • Gibbs distribution
  • sampling

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References

  1. K. Adiprasito, J. Huh, and E. Katz. Hodge theory for combinatorial geometries. Annals of Mathematics, 188(2):381-452, 2018. Google Scholar
  2. N. Anari, K. Liu, S. O. Gharan, and C. Vinzant. Log-concave polynomials II: High-dimensional walks and an FPRAS for counting bases of a matroid. In Proc. 51st annual ACM Symposium on Theory of Computing (STOC), pages 1-12, 2019. Google Scholar
  3. R. E. Belardinelli and V. D. Pereyra. Wang-Landau algorithm: A theoretical analysis of the saturation of the error. The Journal of Chemical Physics, 127(18):184105, 2007. Google Scholar
  4. I. Bezáková, D. Štefankovič, V. V. Vazirani, and E. Vigoda. Accelerating simulated annealing for the permanent and combinatorial counting problems. SIAM J. Comput., 37:1429-1454, 2008. Google Scholar
  5. Charlie Carlson, Ewan Davies, Alexandra Kolla, and Will Perkins. Computational thresholds for the fixed-magnetization ising model. In Proc. 54th annual ACM Symposium on Theory of Computing (STOC), pages 1459-1472, 2022. Google Scholar
  6. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. In Proc. 53rd annual ACM Symposium on Theory of Computing (STOC), pages 1537-1550, 2021. Google Scholar
  7. Ewan Davies and Will Perkins. Approximately counting independent sets of a given size in bounded-degree graphs. In Proc. 48th International Colloquium on Automata, Languages, and Programming (ICALP), pages 62:1-62:18, 2021. Google Scholar
  8. W. Feng, H. Guo, Y. Yin, and C. Zhang. Fast sampling and counting k-SAT solutions in the local lemma regime. In Proc. 52nd annual ACM Symposium on Theory of Computing (STOC), pages 854-867, 2020. Google Scholar
  9. G. Fort, B. Jourdain, E. Kuhn, T. Leliévre, and G. Stoltz. Convergence of the Wang-Landau algorithm. Mathematics of Computation, 84(295):2297-2327, 2015. Google Scholar
  10. H. Guo and K. He. Tight bounds for popping algorithms. Random Struct. Algorithms, 57(2):371-392, 2020. Google Scholar
  11. H. Guo and M. Jerrum. A polynomial-time approximation algorithm for all-terminal network reliability. SIAM J. Comput., 48(3):964-978, 2019. Google Scholar
  12. M. Huber. Approximation algorithms for the normalizing constant of Gibbs distributions. The Annals of Applied Probability, 25(2):974-985, 2015. Google Scholar
  13. Vishesh Jain, Will Perkins, Ashwin Sah, and Mehtaab Sawhney. Approximate counting and sampling via local central limit theorems. In Proc. 54th annual ACM Symposium on Theory of Computing (STOC), pages 1473-1486, 2022. Google Scholar
  14. M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989. Google Scholar
  15. M. Jerrum and A. Sinclair. The Markov Chain Monte Carlo method: an approach to approximate counting and integration. Approximation algorithms for NP-hard problems, pages 482-520, 1996. Google Scholar
  16. V. Kolmogorov. A faster approximation algorithm for the Gibbs partition function. Proceedings of Machine Learning Research, 75:228-249, 2018. Google Scholar
  17. Pedro Ojeda, Martin E Garcia, Aurora Londoño, and Nan-Yow Chen. Monte Carlo simulations of proteins in cages: influence of confinement on the stability of intermediate states. Biophysical journal, 96(3):1076-1082, 2009. Google Scholar
  18. L. N. Shchur. On properties of the Wang-Landau algorithm. Journal of Physics: Conference Series, 1252, 2019. Google Scholar
  19. D. Štefankovič, S. Vempala, and E. Vigoda. Adaptive simulated annealing: A near-optimal connection between sampling and counting. J. of the ACM, 56(3) Article #18, 2009. Google Scholar
  20. F. Wang and D. P. Landau. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett., 86(10):2050-2053, 2001. Google Scholar
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