Adaptive Tuning Of Hamiltonian Monte Carlo Within Sequential Monte Carlo

Buchholz, Alexander; Chopin, Nicolás; Jacob, Pierre

doi:10.48550/arxiv.1808.07730

Cited by 4 publications

(11 citation statements)

References 31 publications

(60 reference statements)

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“…For example, it is possible to select adaptively the annealing parameters β k to ensure the ESS only decreases by a pre-determined percentage (Beskos et al, 2016;Jasra et al, 2011;Schäfer and Chopin, 2013;Zhou et al, 2016). It is also possible to use the approximation of π k obtained at step 13 of Algorithm 1 to adaptively determine the parameters of the MCMC kernel K k (Del Moral et al, 2012b;Buchholz et al, 2021).…”

Section: Extensionsmentioning

confidence: 99%

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Annealed Flow Transport Monte Carlo

Arbel,

Matthews,

Doucet

2021

Preprint

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Annealed Importance Sampling (AIS) and its Sequential Monte Carlo (SMC) extensions are stateof-the-art methods for estimating normalizing constants of probability distributions. We propose here a novel Monte Carlo algorithm, Annealed Flow Transport (AFT), that builds upon AIS and SMC and combines them with normalizing flows (NF) for improved performance. This method transports a set of particles using not only importance sampling (IS), Markov chain Monte Carlo (MCMC) and resampling steps -as in SMC, but also relies on NF which are learned sequentially to push particles towards the successive annealed targets. We provide limit theorems for the resulting Monte Carlo estimates of the normalizing constant and expectations with respect to the target distribution. Additionally, we show that a continuous-time scaling limit of the population version of AFT is given by a Feynman-Kac measure which simplifies to the law of a controlled diffusion for expressive NF. We demonstrate experimentally the benefits and limitations of our methodology on a variety of applications.

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Section: Extensionsmentioning

confidence: 99%

“…This challenging high-dimensional problem is a commonly used benchmark in the SMC literature Buchholz et al, 2021). The mean and covariance function match those estimated by (Møller et al, 1998) and are detailed in the Appendix.…”

Section: Log Gaussian Cox Processmentioning

confidence: 99%

Annealed Flow Transport Monte Carlo

Arbel,

Matthews,

Doucet

2021

Preprint

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“…For this reason, it is preferable that the annealing schedule be chosen adaptively. An adaptive annealing schedule can be chosen by specifying a target effective sample size (ESS) [11] and choosing the next λ such that the ESS is approximately equal to this target [12,13]. For the choice of intermediate distributions given in Equation (2), the ESS used for the weight update is…”

Section: Bayesian Updating With Annealingmentioning

confidence: 99%

“…Resampling can help to reduce variance in the estimator but can cause mode collapse if few particles are used. See [5,12] for more details on resampling. After each annealing sequence is completed (i.e., after the completion of the inner while-loop in the algorithm), one can estimate the ML of the data observed so far by p(y ≤n ) := 1 M ∑ i w i .…”

Section: Bayesian Updating With Annealingmentioning

confidence: 99%

Stochastic Gradient Annealed Importance Sampling for Efficient Online Marginal Likelihood Estimation

Cameron

Eggers

Kroon

2019

Entropy

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We consider estimating the marginal likelihood in settings with independent and identically distributed (i.i.d.) data. We propose estimating the predictive distributions in a sequential factorization of the marginal likelihood in such settings by using stochastic gradient Markov Chain Monte Carlo techniques. This approach is far more efficient than traditional marginal likelihood estimation techniques such as nested sampling and annealed importance sampling due to its use of mini-batches to approximate the likelihood. Stability of the estimates is provided by an adaptive annealing schedule. The resulting stochastic gradient annealed importance sampling (SGAIS) technique, which is the key contribution of our paper, enables us to estimate the marginal likelihood of a number of models considerably faster than traditional approaches, with no noticeable loss of accuracy. An important benefit of our approach is that the marginal likelihood is calculated in an online fashion as data becomes available, allowing the estimates to be used for applications such as online weighted model combination. = ∏ n p(y n |θ), as is common in many parametric models. This restriction is to ensure that a central limit theorem applies to the stochastic likelihood approximation. This can be weakened to any factorization that exhibits a central limit theorem, such as conditionally Markov data and autoregressive models. The posterior distribution over a set of models is proportional to their MLs, and so approximations to ML are sometimes used for model comparison and weighted model averaging [1] (chapter 12). The above integral is typically analytically intractable for any but the simplest models, so one must resort to numerical approximation methods.

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“…For example, Fearnhead and Taylor (2013) and Salomone et al (2018) propose adaptation methods for generic MCMC kernels. Buchholz et al (2018) propose methods for performing the notoriously challenging tuning of HMC kernel parameters in SMC. It is also possible to adaptively choose whether to perform the resampling and move steps based on some measure of the weight degeneracy (Del Moral et al, 2012).…”

Section: Sequential Monte Carlo and Control Variatesmentioning

confidence: 99%

Regularised Zero-Variance Control Variates for High-Dimensional Variance Reduction

South¹,

Oates²,

Mira³

et al. 2018

Preprint

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Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort is solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. We propose to exploit penalised regression to make the method more flexible and feasible, particularly in higher dimensions. Connections between this penalised ZV-CV approach and control functionals are made, providing additional motivation for our approach. Another type of regularisation based on using subsets of derivatives, or a priori regularisation as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Methods for applying ZV-CV and regularised ZV-CV to sequential Monte Carlo (SMC) are described and a new estimator for the normalising constant of the posterior is developed to aid Bayesian model choice. Several examples showing the utility and limitations of regularised ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV available at https://github.com/LeahPrice/ZVCV.

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Adaptive Tuning Of Hamiltonian Monte Carlo Within Sequential Monte Carlo

Abstract: Buchholz ( ), Nicolas Chopin ( ), Pierre E. Jacob (*) ( ) ENSAE-CREST, (

Cited by 4 publications

References 31 publications

Annealed Flow Transport Monte Carlo

Annealed Flow Transport Monte Carlo

Stochastic Gradient Annealed Importance Sampling for Efficient Online Marginal Likelihood Estimation

Regularised Zero-Variance Control Variates for High-Dimensional Variance Reduction

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