Hamiltonian Monte Carlo with explicit, reversible, and volume-preserving adaptive step size control

Okudo, Michiko; Suzuki, Hideyuki

doi:10.14495/jsiaml.9.33

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…In other words, (x, p) = L(x ′ , p ′ ) is not satisfied in general. The HMC with the detailed balance condition can be realized using a leapfrog operator L adaptively defined with an appropriate waiting time [30]. In contrast, our proposed method does not constrain any leapfrog operator.…”

Section: Summary and Discussionmentioning

confidence: 99%

Stochastic dynamics without detailed balance condition connecting simple gradient method and Hamiltonian Monte Carlo

Ichiki,

Ohzeki

2021

Preprint

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Sampling occupies an important position in theories of various scientific fields, and Markov chain Monte Carlo (MCMC) provides the most common technique of sampling. In the progress of MCMC, a huge number of studies have aimed the acceleration of convergence to the target distribution. Hamiltonian Monte Carlo (HMC) is such a variant of MCMC. In the recent development of MCMC, another approach based on the violation of the detailed balance condition has attracted much attention. Historically, these two approaches have been proposed independently, and their relationship has not been clearly understood. In this paper, the two approaches are seamlessly understood in the framework of generalized Monte Carlo method that violates the detailed balance condition. Furthermore we propose an efficient Monte Carlo method based on our framework.

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Section: Summary and Discussionmentioning

confidence: 99%

Stochastic dynamics without detailed balance condition connecting simple gradient method and Hamiltonian Monte Carlo

Ichiki,

Ohzeki

2021

Preprint

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“…If the composite gradient grows faster than linearly in x then the dynamical system exhibits stiffness, meaning gradients change very quickly and hence numerical approximations in which they are assumed to be constant over small time periods will become unstable. Two possible remedies are to use a more complex implicit numerical method or adaptive step-size control, as discussed in Okudo & Suzuki (2016). With the standard choice of kinetic energy ∇K • ∇U(x) = ∇U(x), meaning these instabilities occur when ∇U(x) grows faster than linearly in x .…”

Section: Illustrative Examplementioning

confidence: 99%

Kinetic energy choice in Hamiltonian/hybrid Monte Carlo

2019

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We consider how different choices of kinetic energy in Hamiltonian Monte Carlo affect algorithm performance. To this end, we introduce two quantities which can be easily evaluated, the composite gradient and the implicit noise. Results are established on integrator stability and geometric convergence, and we show that choices of kinetic energy that result in heavy-tailed momentum distributions can exhibit an undesirable negligible moves property, which we define. A general efficiency-robustness trade off is outlined, and implementations which rely on approximate gradients are also discussed. Two numerical studies illustrate our theoretical findings, showing that the standard choice which results in a Gaussian momentum distribution is not always optimal in terms of either robustness or efficiency.

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“…Our implicit midpoint-based method is able to achieve efficiency gains over leapfrog on a more general class of problems. Okudo and Suzuki (2015) modify the leapfrog integrator in HMC by adding an auxiliary variable that allows for online adaptation of the stepsize. The RMHMC method of Girolami and Calderhead (2011) and the SoftAbs extension introduced by Betancourt (2013) both use the local Hessian of the potential energy function to adaptively change the mass matrix of HMC.…”

Section: Introductionmentioning

confidence: 99%

Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions

Pourzanjani¹,

Petzold²

2019

Preprint

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Hamiltonian Monte Carlo (HMC) has been widely adopted in the statistics community because of its ability to sample high-dimensional distributions much more efficiently than other Metropolis-based methods. Despite this, HMC often performs sub-optimally on distributions with high correlations or marginal variances on multiple scales because the resulting stiffness forces the leapfrog integrator in HMC to take an unreasonably small stepsize. We provide intuition as well as a formal analysis showing how these multiscale distributions limit the stepsize of leapfrog and we show how the implicit midpoint method can be used, together with Newton-Krylov iteration, to circumvent this limitation and achieve major efficiency gains. Furthermore, we offer practical guidelines for when to choose between implicit midpoint and leapfrog and what stepsize to use for each method, depending on the distribution being sampled. Unlike previous modifications to HMC, our method is generally applicable to highly non-Gaussian distributions exhibiting multiple scales. We illustrate how our method can provide a dramatic speedup over leapfrog in the context of the No-U-Turn sampler (NUTS) applied to several examples.

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Hamiltonian Monte Carlo with explicit, reversible, and volume-preserving adaptive step size control

Cited by 4 publications

References 8 publications

Stochastic dynamics without detailed balance condition connecting simple gradient method and Hamiltonian Monte Carlo

Stochastic dynamics without detailed balance condition connecting simple gradient method and Hamiltonian Monte Carlo

Kinetic energy choice in Hamiltonian/hybrid Monte Carlo

Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions

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