Log concave distributions

associated tools

β€œa 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.


Bagnoli, Mark, and Ted Bergstrom. 1989. β€œLog-Concave Probability and Its Applications,” 17.
Brosse, Nicolas, Γ‰ric Moulines, and Alain Durmus. 2018. β€œThe 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. β€œSpin-Glass Theory for Pedestrians.” Journal of Statistical Mechanics: Theory and Experiment 2005 (05): P05012.
Domke, Justin. 2017. β€œA 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. β€œHybrid Monte Carlo.” Physics Letters B 195 (2): 216–22.
Durmus, Alain, and Eric Moulines. 2016. β€œHigh-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. β€œInteracting 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. β€œSimulated 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. β€œRiemann 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. β€œImplicit Langevin Algorithms for Sampling From Log-Concave Densities.” arXiv:1903.12322 [Cs, Stat], March.
Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. β€œStochastic Gradient Descent as Approximate Bayesian Inference.” JMLR, April.
Mangoubi, Oren, and Aaron Smith. 2017. β€œRapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions.” arXiv:1708.07114 [Math, Stat], August.
Norton, Richard A., and Colin Fox. 2016. β€œTuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.” arXiv:1610.00781 [Math, Stat], October.
Saumard, Adrien, and Jon A. Wellner. 2014. β€œLog-Concavity and Strong Log-Concavity: A Review.” arXiv:1404.5886 [Math, Stat], April.
Welling, Max, and Yee Whye Teh. 2011. β€œBayesian 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. β€œLangevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.” Statistics & Probability Letters 91 (Supplement C): 14–19.

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