# Log concave distributions

associated tools

August 28, 2017 — March 11, 2021

Bayes

estimator distribution

Markov processes

Monte Carlo

probabilistic algorithms

probability

“a Markov Chain reminiscent of noisy gradient descent”. Holden Lee, Andrej Risteski introduce this the connection between log-concavity and convex optimisation.

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

## 1 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.

## 2 References

Bagnoli, and Bergstrom. 1989. “Log-Concave Probability and Its Applications.”

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.
Castellani, and Cavagna. 2005. “Spin-Glass Theory for Pedestrians.”

*Journal of Statistical Mechanics: Theory and Experiment*.
Domke. 2017. “A Divergence Bound for Hybrids of MCMC and Variational Inference and an Application to Langevin Dynamics and SGVI.” In

*PMLR*.
Duane, Kennedy, Pendleton, et al. 1987. “Hybrid Monte Carlo.”

*Physics Letters B*.
Durmus, and Moulines. 2016. “High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.”

*arXiv:1605.01559 [Math, Stat]*.
Garbuno-Inigo, Hoffmann, Li, et al. 2020. “Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler.”

*SIAM Journal on Applied Dynamical Systems*.
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]*.
Girolami, and Calderhead. 2011. “Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.”

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
Hodgkinson, Salomone, and Roosta. 2019. “Implicit Langevin Algorithms for Sampling From Log-Concave Densities.”

*arXiv:1903.12322 [Cs, Stat]*.
Mandt, Hoffman, and Blei. 2017. “Stochastic Gradient Descent as Approximate Bayesian Inference.”

*JMLR*.
Mangoubi, and Smith. 2017. “Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions.”

*arXiv:1708.07114 [Math, Stat]*.
Norton, and Fox. 2016. “Tuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.”

*arXiv:1610.00781 [Math, Stat]*.
Saumard, and Wellner. 2014. “Log-Concavity and Strong Log-Concavity: A Review.”

*arXiv:1404.5886 [Math, Stat]*.
Welling, and Teh. 2011. “Bayesian Learning via Stochastic Gradient Langevin Dynamics.” In

*Proceedings of the 28th International Conference on International Conference on Machine Learning*. ICML’11.
Xifara, Sherlock, Livingstone, et al. 2014. “Langevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.”

*Statistics & Probability Letters*.