“a Markov Chain reminiscent of noisy gradient descent”. Holden Lee, Andrej Risteski introduce this the connection between log-concavity and convex optimisation.
\[ x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I). \]
Langevin MCMC
See SGD MCMC for now.
Rob Salomone explains this well; see Hodgkinson, Salomone, and Roosta (2019).
Andrej Risteski’s Beyond log-concave sampling series is a also a good introduction to log-concave sampling.
References
Bagnoli, Mark, and Ted Bergstrom. 1989. “Log-Concave Probability and Its Applications,” 17.
Brosse, Nicolas, Éric Moulines, and Alain Durmus. 2018. “The Promises and Pitfalls of Stochastic Gradient Langevin Dynamics.” In Proceedings of the 32nd International Conference on Neural Information Processing Systems, 8278–88. NIPS’18. Red Hook, NY, USA: Curran Associates Inc.
Castellani, Tommaso, and Andrea Cavagna. 2005. “Spin-Glass Theory for Pedestrians.” Journal of Statistical Mechanics: Theory and Experiment 2005 (05): P05012.
Domke, Justin. 2017. “A Divergence Bound for Hybrids of MCMC and Variational Inference and an Application to Langevin Dynamics and SGVI.” In PMLR, 1029–38.
Duane, Simon, A. D. Kennedy, Brian J. Pendleton, and Duncan Roweth. 1987. “Hybrid Monte Carlo.” Physics Letters B 195 (2): 216–22.
Durmus, Alain, and Eric Moulines. 2016. “High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.” arXiv:1605.01559 [Math, Stat], May.
Garbuno-Inigo, Alfredo, Franca Hoffmann, Wuchen Li, and Andrew M. Stuart. 2020. “Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler.” SIAM Journal on Applied Dynamical Systems 19 (1): 412–41.
Ge, Rong, Holden Lee, and Andrej Risteski. 2020. “Simulated Tempering Langevin Monte Carlo II: An Improved Proof Using Soft Markov Chain Decomposition.” arXiv:1812.00793 [Cs, Math, Stat], September.
Girolami, Mark, and Ben Calderhead. 2011. “Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2): 123–214.
Hodgkinson, Liam, Robert Salomone, and Fred Roosta. 2019. “Implicit Langevin Algorithms for Sampling From Log-Concave Densities.” arXiv:1903.12322 [Cs, Stat], March.
Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. “Stochastic Gradient Descent as Approximate Bayesian Inference.” JMLR, April.
Mangoubi, Oren, and Aaron Smith. 2017. “Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions.” arXiv:1708.07114 [Math, Stat], August.
Norton, Richard A., and Colin Fox. 2016. “Tuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.” arXiv:1610.00781 [Math, Stat], October.
Saumard, Adrien, and Jon A. Wellner. 2014. “Log-Concavity and Strong Log-Concavity: A Review.” arXiv:1404.5886 [Math, Stat], April.
Welling, Max, and Yee Whye Teh. 2011. “Bayesian Learning via Stochastic Gradient Langevin Dynamics.” In Proceedings of the 28th International Conference on International Conference on Machine Learning, 681–88. ICML’11. Madison, WI, USA: Omnipress.
Xifara, T., C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. 2014. “Langevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.” Statistics & Probability Letters 91 (Supplement C): 14–19.
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