## Langevin MCMC

βa Markov Chain reminiscent of noisy gradient descentβ. Holden Lee, Andrej Risteski introduce this the connection between log-concavity and convex optimisation.

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

Rob Salomone explains this well; see Hodgkinson, Salomone, and Roosta (2019).

Andrej Risteskiβs Beyond log-concave sampling series is a also a good introduction to log-concave sampling.

## References

Bagnoli, Mark, and Ted Bergstrom. 1989. βLog-Concave Probability and Its Applications,β 17.

Brosse, Nicolas, Alain Durmus, and Eric Moulines. n.d. βThe Promises and Pitfalls of Stochastic Gradient Langevin Dynamics,β 11.

Castellani, Tommaso, and Andrea Cavagna. 2005. βSpin-Glass Theory for Pedestrians.β

*Journal of Statistical Mechanics: Theory and Experiment*2005 (05): P05012.Domke, Justin. 2017. βA Divergence Bound for Hybrids of MCMC and Variational Inference and an Application to Langevin Dynamics and SGVI.β In

*PMLR*, 1029β38.Duane, Simon, A. D. Kennedy, Brian J. Pendleton, and Duncan Roweth. 1987. βHybrid Monte Carlo.β

*Physics Letters B*195 (2): 216β22.Durmus, Alain, and Eric Moulines. 2016. βHigh-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.β

*arXiv:1605.01559 [Math, Stat]*, May.Garbuno-Inigo, Alfredo, Franca Hoffmann, Wuchen Li, and Andrew M. Stuart. 2020. βInteracting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler.β

*SIAM Journal on Applied Dynamical Systems*19 (1): 412β41.Ge, Rong, Holden Lee, and Andrej Risteski. 2020. βSimulated Tempering Langevin Monte Carlo II: An Improved Proof Using Soft Markov Chain Decomposition.β

*arXiv:1812.00793 [Cs, Math, Stat]*, September.Girolami, Mark, and Ben Calderhead. 2011. βRiemann Manifold Langevin and Hamiltonian Monte Carlo Methods.β

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*73 (2): 123β214.Hodgkinson, Liam, Robert Salomone, and Fred Roosta. 2019. βImplicit Langevin Algorithms for Sampling From Log-Concave Densities.β

*arXiv:1903.12322 [Cs, Stat]*, March.Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. βStochastic Gradient Descent as Approximate Bayesian Inference.β

*JMLR*, April.Mangoubi, Oren, and Aaron Smith. 2017. βRapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions.β

*arXiv:1708.07114 [Math, Stat]*, August.Norton, Richard A., and Colin Fox. 2016. βTuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.β

*arXiv:1610.00781 [Math, Stat]*, October.Saumard, Adrien, and Jon A. Wellner. 2014. βLog-Concavity and Strong Log-Concavity: A Review.β

*arXiv:1404.5886 [Math, Stat]*, April.Welling, Max, and Yee Whye Teh. n.d. βBayesian Learning via Stochastic Gradient Langevin Dynamics,β 8.

Xifara, T., C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. 2014. βLangevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.β

*Statistics & Probability Letters*91 (Supplement C): 14β19.
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