Langevin dynamcs MCMC

August 17, 2020 — September 5, 2024

Bayes
estimator distribution
functional analysis
Markov processes
Monte Carlo
neural nets
optimization
probabilistic algorithms
probability
SDEs
stochastic processes
Figure 1: Randomly exploring the posterior space.

Sampling using the particular SDE that is the Langevin equation. It can be rather close to optimising, via SDE representations of SGD.

1 Langevin dynamics

2 Metropolis-adjusted Langevin algorithm (MALA)

3 Continuous time

See log concave distributions for a family of distributions where this works especially well because implcit (more nearly continuous-time exact) solutions are available Hodgkinson, Salomone, and Roosta (2019).

Left-field, Max Raginsky, Sampling Using Diffusion Processes, from Langevin to Schrödinger:

the Langevin process gives only approximate samples from \(\mu\). I would like to discuss an alternative approach that uses diffusion processes to obtain exact samples in finite time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the Schrödinger bridge problem.

4 Annealed

TBC Jolicoeur-Martineau et al. (2022);Song and Ermon (2020a);Song and Ermon (2020b).

Yang Song, Generative Modeling by Estimating Gradients of the Data Distribution

5 Incoming

Holden Lee, Andrej Risteski introduce the connection between log-concavity and convex optimisation.

\[ x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I). \]

Below are some implementations I explored. I do not recommend them necessarily; if it is a simple unadjusted Langevin sample, the algorithm is simple enough implement from scratch. Better, the ChatGPT version comes out right first time in my experience.

6 References

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Brosse, Moulines, and Durmus. 2018. “The Promises and Pitfalls of Stochastic Gradient Langevin Dynamics.” In Proceedings of the 32nd International Conference on Neural Information Processing Systems. NIPS’18.
Chen, Fox, and Guestrin. 2014. “Stochastic Gradient Hamiltonian Monte Carlo.” In Proceedings of the 31st International Conference on Machine Learning.
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Dockhorn, Vahdat, and Kreis. 2022. “Score-Based Generative Modeling with Critically-Damped Langevin Diffusion.”
Domke. 2017. “A Divergence Bound for Hybrids of MCMC and Variational Inference and an Application to Langevin Dynamics and SGVI.” In PMLR.
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Ge, Lee, and Risteski. 2020. “Simulated Tempering Langevin Monte Carlo II: An Improved Proof Using Soft Markov Chain Decomposition.” arXiv:1812.00793 [Cs, Math, Stat].
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Hodgkinson, Salomone, and Roosta. 2019. “Implicit Langevin Algorithms for Sampling From Log-Concave Densities.” arXiv:1903.12322 [Cs, Stat].
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Jolicoeur-Martineau, Piché-Taillefer, Mitliagkas, et al. 2022. “Adversarial Score Matching and Improved Sampling for Image Generation.” In.
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Liu, Zhuo, Cheng, et al. 2019. “Understanding and Accelerating Particle-Based Variational Inference.” In Proceedings of the 36th International Conference on Machine Learning.
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———. 2020b. “Improved Techniques for Training Score-Based Generative Models.” In Advances In Neural Information Processing Systems.
Song, Sohl-Dickstein, Kingma, et al. 2022. “Score-Based Generative Modeling Through Stochastic Differential Equations.” In.
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Zhang, Jianyi, Zhang, Carin, et al. 2020. “Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory.” In International Conference on Artificial Intelligence and Statistics.