Riemann manifold Langevin and Hamiltonian Monte Carlo methods

By Mark A. Girolami, Ben Calderhead. 2011

1,671 citations recorded by the source corpus.

Summary. The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis–Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log‐Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non‐linear differential equations. Substantial improvements in the time‐normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability, Optimization & Training, Probabilistic & Bayesian Methods

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Citations in the corpus

Flow Contrastive Estimation of Energy-Based Models (2019, methodology): “The difﬁculty lies in the fact that sampling from p θ ( x ) requires MCMC such as Hamiltonian monte carlo or Langevin dynamics [14, 78], which may take a long time to converge, especially on high dimensional and multi-modal space such as image space.”
Learning Energy-Based Spatial-Temporal Generative ConvNets for Dynamic Patterns (2019, methodology): “The Langevin dynamics can be extended to Hamiltonian Monte Carlo [56] or more sophisticated versions [57].”
Generalizing Hamiltonian Monte Carlo with Neural Networks (2017, background): “It is further capable of emulating several common extensions of HMC such as within-trajectory tempering [15] and diagonal mass matrices [5].”
Learning Generative ConvNets via Multi-grid Modeling and Sampling (2017, background): “…of L ( θ ) is The expectation in equation (5) is analytically intractable and has to be approximated by MCMC, such as the Langevin dynamics [48, 7], which iterates the following step: where τ indexes the time of the Langevin dynamics, ∆ τ is the step size, and Z τ ∼ N ( 0 , I ) is Gaussian…”
A-NICE-MC: Adversarial Training for MCMC (2017, background): “We consider three datasets - german (25 covariates, 1000 data points), heart (14 covariates, 532 data points) and australian (15 covariates, 690 data points) - and evaluate the lowest ESS across all covariates (following the settings in [35]), where we obtain 5000 samples after 1000 (4)For R hat val”
Kernel Sequential Monte Carlo (2015, methodology): “en established over the past years, and where more reﬁned gradient-based samplers suited to non-linear targets like Metropolis Adjusted Langevin Algorithm (MALA) [13] or Hamiltonian Monte Carlo (HMC) [14, 15] are not applicable. Outline. The paper proceeds as follows. We review MCMC steps based on p”
Control functionals for Monte Carlo integration (2014, methodology): “…by using population MCMC sampling, involving both (a) ‘within-temperature’ proposals produced by the (simplified) manifold Metropolis-adjusted Langevin algorithm of Girolami and Calderhead (2011), and (b) ‘between-temperature’ proposals, as described previously by Calderhead and Girolami (2009).”
Stochastic Gradient Hamiltonian Monte Carlo (2014, methodology): “Riemann manifold HMC (Girolami and Calderhead, 2011) makes use of the Riemann geometry to adapt the mass M , enabling the algorithm to make use of curvature information to perform more efficient sampling.”
The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo (2011, methodology): “Following Girolami and Calderhead (2011) we compute ESS separately for each dimension and report the minimum ESS across all dimensions, since we want our samplers to eﬀectively explore all dimensions of the target distribution.”
Automatic differentiation in machine learning: a survey (2015, methodology): “The Riemannian manifold Langevin and Hamiltonian Monte Carlo methods (Girolami and Calderhead, 2011) use higher-order derivative information to more closely track the information geometry of the sampled distribution for faster convergence and exploration.”
Learning Energy-Based Models by Diffusion Recovery Likelihood (2020, methodology): “Generating synthesized samples from pθ(x) can be done with Markov Chain Monte Carlo (MCMC) such as Langevin dynamics (or Hamiltonian Monte Carlo [9]), which iterates”
Automatic Differentiation Variational Inference (2016, methodology): “We omit sampling results as both HMC and NUTS struggle with the model; neither produce useful samples within an hour.”
Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression (2012, methodology): “Here we apply our methodology to the model and prior speciﬁed in Girolami and Calderhead (2011).”
Deep Energy Estimator Networks (2018, background): “But there are important new developments on the HMC front (Girolami & Calderhead, 2011; Levy et al., 2017), which DEEN can build on.”
Cooperative Training of Descriptor and Generator Networks (2016, methodology): “Step D1 for Langevin revision : Sampling synthesized examples from the current model by Langevin dynamics (Neal, 2011; Girolami & Calderhead, 2011).”
Privacy for Free: Posterior Sampling and Stochastic Gradient Monte Carlo (2015, background): “…extension of SGLD is Stochastic Gradient Fisher Scoring (SGFS), where Ahn et al. (2012) proposes to adaptively interpolate between a preconditioned SGLD (see preconditioning (Girolami & Calderhead, 2011)) and a Markov Chain that samples from a normal approximation of the posterior distribution.”
Alternating Back-Propagation for Generator Network (2016, methodology): “The Langevin dynamics can be extended to Hamiltonian Monte Carlo (Neal, 2011) or more sophisticated versions (Girolami and Calderhead, 2011).”