Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Improved sampling via learned diffusions (2307.01198v2)

Published 3 Jul 2023 in cs.LG, math.OC, math.PR, and stat.ML

Abstract: Recently, a series of papers proposed deep learning-based approaches to sample from target distributions using controlled diffusion processes, being trained only on the unnormalized target densities without access to samples. Building on previous work, we identify these approaches as special cases of a generalized Schr\"odinger bridge problem, seeking a stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (59)
  1. Anderson, B. D. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982.
  2. A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. Numerische Mathematik, 84(3):375–393, 2000.
  3. An optimal control perspective on diffusion-based generative modeling. arXiv preprint arXiv:2211.01364, 2022.
  4. Schrödinger Bridge Samplers. arXiv preprint arXiv:1912.13170, 2019.
  5. Wasserstein proximal algorithms for the Schrödinger bridge problem: Density control with nonlinear drift. IEEE Transactions on Automatic Control, 67(3):1163–1178, 2021.
  6. Likelihood training of Schrödinger Bridge using Forward-Backward SDEs theory. arXiv preprint arXiv:2110.11291, 2021a.
  7. Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schrödinger bridge. SIAM Review, 63(2):249–313, 2021b.
  8. Cuturi, M. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, pp. 2292–2300, 2013.
  9. Dai Pra, P. A stochastic control approach to reciprocal diffusion processes. Applied mathematics and Optimization, 23(1):313–329, 1991.
  10. Diffusion Schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695–17709, 2021.
  11. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411–436, 2006.
  12. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009.
  13. Score-based diffusion meets annealed importance sampling. In Advances in Neural Information Processing Systems, 2022.
  14. Shooting Schrödinger’s cat. In Fourth Symposium on Advances in Approximate Bayesian Inference, 2021.
  15. Controlled Markov processes and viscosity solutions, volume 25. Springer Science & Business Media, 2006.
  16. Föllmer, H. Random fields and diffusion processes. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp.  101–203. Springer, 1988.
  17. Variational characterization of free energy: Theory and algorithms. Entropy, 19(11):626, 2017.
  18. Time reversal of diffusions. The Annals of Probability, pp.  1188–1205, 1986.
  19. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840–6851, 2020.
  20. Path integral stochastic optimal control for sampling transition paths. arXiv preprint arXiv:2207.02149, 2022.
  21. A variational perspective on diffusion-based generative models and score matching. Advances in Neural Information Processing Systems, 34, 2021.
  22. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4), 2005.
  23. Torsional diffusion for molecular conformer generation. arXiv preprint arXiv:2206.01729, 2022.
  24. Variational diffusion models. Advances in Neural Information Processing Systems, 34:21696–21707, 2021.
  25. Neural Lagrangian Schrödinger bridge. arXiv preprint arXiv:2204.04853, 2022.
  26. Léonard, C. Some properties of path measures. In Séminaire de Probabilités XLVI, pp.  207–230. Springer, 2014.
  27. Deep generalized Schrödinger bridge. arXiv preprint arXiv:2209.09893, 2022.
  28. Monte Carlo strategies in scientific computing, volume 10. Springer, 2001.
  29. Learning deformation trajectories of Boltzmann densities. arXiv preprint arXiv:2301.07388, 2023.
  30. Flow annealed importance sampling bootstrap. In NeurIPS 2022 AI for Science: Progress and Promises, 2022.
  31. Minka, T. et al. Divergence measures and message passing. Technical report, Citeseer, 2005.
  32. Neal, R. M. Annealed importance sampling. Statistics and computing, 11(2):125–139, 2001.
  33. Neal, R. M. Slice sampling. The Annals of Statistics, 31(3):705–767, 2003.
  34. Nelson, E. Dynamical theories of Brownian motion. Press, Princeton, NJ, 1967.
  35. Improved denoising diffusion probabilistic models. In International Conference on Machine Learning, pp. 8162–8171. PMLR, 2021.
  36. Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space. Partial Differential Equations and Applications, 2(4):1–48, 2021.
  37. Normalizing flows for probabilistic modeling and inference. J. Mach. Learn. Res., 22(57):1–64, 2021.
  38. Pavon, M. Stochastic control and nonequilibrium thermodynamical systems. Applied Mathematics and Optimization, 19(1):187–202, 1989.
  39. Pavon, M. On local entropy, stochastic control and deep neural networks. arXiv preprint arXiv:2204.13049, 2022.
  40. On free energy, stochastic control, and Schrödinger processes. In Modeling, Estimation and Control of Systems with Uncertainty, pp.  334–348. Springer, 1991.
  41. Pham, H. Continuous-time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability. Springer Berlin Heidelberg, 2009.
  42. Richter, L. Solving high-dimensional PDEs, approximation of path space measures and importance sampling of diffusions. PhD thesis, BTU Cottbus-Senftenberg, 2021.
  43. Robust SDE-based variational formulations for solving linear PDEs via deep learning. In International Conference on Machine Learning, pp. 18649–18666. PMLR, 2022.
  44. VarGrad: low-variance gradient estimator for variational inference. Advances in Neural Information Processing Systems, 33:13481–13492, 2020.
  45. Monte Carlo statistical methods, volume 2. Springer, 1999.
  46. Sticking the landing: Simple, lower-variance gradient estimators for variational inference. Advances in Neural Information Processing Systems, 30, 2017.
  47. Diffusions, Markov processes and martingales: Volume 2, Itô calculus, volume 2. Cambridge university press, 2000.
  48. Functional analysis with applications. Springer, 1986.
  49. Improved techniques for training score-based generative models. Advances in neural information processing systems, 33:12438–12448, 2020.
  50. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2020.
  51. Free energy computations: A mathematical perspective. World Scientific, 2010.
  52. Theoretical guarantees for sampling and inference in generative models with latent diffusions. In Conference on Learning Theory, pp.  3084–3114. PMLR, 2019.
  53. Score-based generative modeling in latent space. Advances in Neural Information Processing Systems, 34:11287–11302, 2021.
  54. Vargas, F. Machine-learning approaches for the empirical Schrödinger bridge problem. Technical report, University of Cambridge, Computer Laboratory, 2021.
  55. Denoising diffusion samplers. In International Conference on Learning Representations, 2023a.
  56. Bayesian learning via neural Schrödinger–Föllmer flows. Statistics and Computing, 33(1):1–22, 2023b.
  57. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1(1–2):1–305, 2008.
  58. Stochastic normalizing flows. Advances in Neural Information Processing Systems, 33:5933–5944, 2020.
  59. Path Integral Sampler: a stochastic control approach for sampling. In International Conference on Learning Representations, 2022.
Citations (35)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets