- Physical Review Journals
All JournalsPhysics Magazine

PRX Intelligence

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Training Energy-Based Models with Non-MCMC Samplers and Efficient Temperature Estimation

Kentaro Kubo1,* and Hayato Goto1,2

  • *Contact author: kentaro.kubo.b91@mail.toshiba
PDF

PRX Intelligence 1, 013021 – Published 17 September, 2026

DOI: https://doi.org/10.1103/2vb5-2lt8

Abstract

Efficient sampling from Boltzmann distributions over discrete variables is a fundamental operation in a wide range of applications. While fast non-Markov chain Monte Carlo (MCMC) samplers have recently emerged as promising alternatives to conventional MCMC methods, their practical use for probabilistic learning remains hindered by the difficulty of estimating the effective temperature of the generated samples. In this work, we begin by introducing Langevin simulated bifurcation (LSB), a Boltzmann sampler that enables fast and parallel sampling with accuracy comparable to sequential MCMC methods. To address the challenge of unknown effective temperature, we propose conditional expectation matching (CEM), an efficient estimation method applicable to energy-based models (EBMs) with exploitable conditional independence structures. Building on these components, we further develop a learning framework, termed sampler adaptive learning (SAL), which adaptively adjusts the model temperature to match that of the distribution induced by fast non-MCMC sampling. We demonstrate the effectiveness of LSB, CEM, and SAL on semirestricted Boltzmann machines (SRBMs), a class of EBMs that are difficult to train using conventional approaches. LSB achieves orders-of-magnitude acceleration over Gibbs sampling while maintaining comparable or higher accuracy, and CEM enables accurate temperature estimation of the resulting distribution with negligible computational overhead. As a consequence, SAL enables efficient training of SRBMs and outperforms conventional Boltzmann machine learning methods on synthetic spin-glass datasets. In addition, the trained models achieve strong performance across multiple tasks, including image generation, reconstruction, and classification. These results establish LSB as a fast and accurate Boltzmann sampler and provide key insights that enable practical applications of fast non-MCMC sampling methods via efficient temperature estimation with CEM.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (52)

  1. Y. LeCun, S. Chopra, R. Hadsell, M. Ranzato, and F. Huang, A tutorial on energy-based learning, 2006, http://yann.lecun.com/exdb/publis/orig/lecun-06.pdf.
  2. Y. Song and D. P. Kingma, How to train your energy-based models, arXiv:2101.03288.
  3. D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, A learning algorithm for Boltzmann machines, Cogn. Sci. 9, 147 (1985).
  4. G. E. Hinton, Training products of experts by mini mizing contrastive divergence, Neural Comput. 14, 1771 (2002).
  5. G. E. Hinton, A practical guide to training restricted Boltzmann machines, in Neural Networks: Tricks of the Trade, Lecture Notes in Computer Science (Springer, Berlin, 2012), Vol. 7700, pp. 599–619.
  6. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  7. D. P. Landau and K. Binder, A Guide to Monte Carlo Methods in Statistical Physics (Cambridge University Press, Cambridge, England, 2000).
  8. R. M. Neal, Bayesian Learning for Neural Networks (Springer Science and Business Media, New York, 2012), Vol. 118.
  9. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983).
  10. S. Chowdhury, A. Grimaldi, N. A. Aadit, S. Niazi, M. Mohseni, and S. Kanai, A full-stack view of probabilistic computing with p-bits: Devices, architectures, and algorithms, IEEE J. Explor. Solid-State Comput. Devices Circuits 9, 1 (2023).
  11. C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, An introduction to MCMC for machine learning, Mach. Learn. 50, 5 (2003).
  12. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
  13. W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57, 97 (1970).
  14. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  15. H. Goto, K. Tatsumura, and A. R. Dixon, Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems, Sci. Adv. 5, eaav2372 (2019).
  16. H. Goto, K. Endo, M. Suzuki, Y. Sakai, T. Kanao, Y. Hamakawa, R. Hidaka, M. Yamasaki, and K. Tatsumura, High-performance combinatorial optimization based on classical mechanics, Sci. Adv. 7, eabe7953 (2021).
  17. T. Kanao and H. Goto, Simulated bifurcation assisted by thermal fluctuation, Commun. Phys. 5, 153 (2022).
  18. T. Kanao and H. Goto, Simulated bifurcation for higher-order cost functions, Appl. Phys. Express 16, 014501 (2023).
  19. H. Goto, R. Hidaka, and K. Tatsumura, Edge-of-chaos enhanced quantum-inspired algorithm for combinatorial optimization, Phys. Rev. Appl. 25, 044011 (2026).
  20. M. W. Johnson et al., Quantum annealing with manufactured spins, Nature (London) 473, 194 (2011).
  21. R. Harris et al., Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor, Phys. Rev. B 82, 024511 (2010).
  22. P. I. Bunyk, E. M. Hoskinson, M. W. Johnson, E. Tolkacheva, F. Altomare, A. J. Berkley, R. Harris, J. P. Hilton, T. Lanting, A. J. Przybysz, and J. Whittaker, Architectural considerations in the design of a superconducting quantum annealing processor, IEEE Trans. Appl. Supercond. 24, 1 (2014).
  23. K. Boothby, P. Bunyk, J. Raymond, and A. Roy, Next-generation topology of D-wave quantum processors, arXiv:2003.00133.
  24. M. H. Amin, Searching for quantum speedup in quasistatic quantum annealers, Phys. Rev. A 92, 052323 (2015).
  25. S. H. Adachi and M. P. Henderson, Application of quantum annealing to training of deep neural networks, arXiv:1510.06356.
  26. V. Dixit, R. Selvarajan, T. Aldwairi, Y. Koshka, M. A. Novotny, T. S. Humble, M. A. Alam, and S. Kais, Training a quantum annealing based restricted Boltzmann machine on cybersecurity data, IEEE Trans. Emerg. Top. Comput. Intell. 6, 417 (2021).
  27. T. Sato, M. Ohzeki, and K. Tanaka, Assessment of image generation by quantum annealer, Sci. Rep. 11, 13523 (2021).
  28. T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, A coherent Ising machine for 2000-node optimization problems, Science 354, 603 (2016).
  29. P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, A fully programmable 100-spin coherent Ising machine with all-to-all connections, Science 354, 614 (2016).
  30. T. Hiroki, Y. Yamada, K. Inaba, T. Ikuta, Y. Yonezu, T. Inagaki, T. Honjo, T. Kazama, K. Enbutsu, T. Umeki, and R. Kasahara, Observing a phase transition in a coherent Ising machine, Phys. Rev. Appl. 19, L031001 (2023).
  31. E. S. Tiunov, A. E. Ulanov, and A. I. Lvovsky, Annealing by simulating the coherent Ising machine, Opt. Express 27, 10288 (2019).
  32. A. E. Ulanov, E. S. Tiunov, and A. I. Lvovsky, Quantum-inspired annealers as Boltzmann generators for machine learning and statistical physics, arXiv:1912.08480.
  33. H. Goto, Z. Lin, and Y. Nakamura, Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators, Sci. Rep. 8, 7154 (2018).
  34. F. Böhm, D. Alonso-Urquijo, G. Verschaffelt, and G. Vander Sande, Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning, Nat. Commun. 13, 5847 (2022).
  35. J. Tosca, M. C. Strinati, C. Conti, and C. Ciuti, Emergent equilibrium in all-optical single quantum-trajectory Ising machines, Phys. Rev. Lett. 134, 230404 (2025).
  36. E. Raimondo, A. Grimaldi, V. Tyberkevych, R. Tomasello, A. Giordano, M. Carpentieri, A. Slavin, M. Chiappini, and G. Finocchio, Adaptive Ising Machine based on phase-locking of an auto-oscillator to a biharmonic external driving with noise, Phys. Rev. Lett. 136, 177201 (2026).
  37. J. Raymond, S. Yarkoni, and E. Andriyash, Global warming: Temperature estimation in annealers, Front. ICT 3, 23 (2016).
  38. M. Benedetti, J. Realpe-Gómez, R. Biswas, and A. Perdomo-Ortiz, Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning, Phys. Rev. A 94, 022308 (2016).
  39. M. Benedetti, J. Realpe-Gómez, R. Biswas, and A. Perdomo-Ortiz, Quantum-assisted learning of hardware-embedded probabilistic graphical models, Phys. Rev. X 7, 041052 (2017).
  40. S. Osindero and G. E. Hinton, Modeling image patches with a directed hierarchy of Markov random fields, Adv. Neural Inf. Process. Syst. 20, 1121 (2008).
  41. R. Salakhutdinov, Learning and evaluating Boltzmann machines, Technical Report UTML TR 2008-002 (Department of Computer Science, University of Toronto, 2008).
  42. H. Schulz, A. Müller, and S. Behnke, Exploiting local structure in stacked Boltzmann machines, in European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN) (d-side, Evere, Belgium, 2010).
  43. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
  44. M. Mézard and A. Montanari, Information, Physics, and Computation (Oxford University Press, Oxford, 2009).
  45. G. E. Hinton and T. J. Sejnowski, Learning and Relearning in Boltzmann Machines, in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, MA, 1986), Vol. 1.
  46. D. J. C. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, 2003).
  47. M. Lichman, M. Lichman, UCI Machine Learning Repository, 2013, http://archive.ics.uci.edu/ml/.
  48. P. J. Rossky, J. D. Doll, and H. L. Friedman, Brownian dynamics as smart Monte Carlo simulation, J. Chem. Phys. 69, 4628 (1978).
  49. G. O. Roberts and O. Stramer, Langevin diffusions and Metropolis–Hastings algorithms, Methodol. Comput. Appl. Probab. 4, 337 (2002).
  50. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics (Cambridge University Press, Cambridge, 2004).
  51. P. Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  52. See Supplemental Material at http://link.aps.org/supplemental/10.1103/2vb5-2lt8 for details of the data preparation, hyperparameter settings, additional training procedures, and additional numerical results.

Outline

Information

PRX Intelligence 1, 013021– Published 17 September, 2026
  • Received 6 January 2026
  • Accepted 31 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation