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Optimal tuning of the hybrid monte-carlo algorithm
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Beskos, Alexandros, Pillai, Natesh S., Roberts, Gareth O., Sanz-Serna, Jesus and Stuart, A. M. (2013) Optimal tuning of the hybrid monte-carlo algorithm. Bernoulli, Volume 19 (Number 5a). pp. 1501-1534. doi:10.3150/12-BEJ414
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Official URL: http://dx.doi.org/10.3150/12-BEJ414
Abstract
We investigate the properties of the Hybrid Monte Carlo algorithm (HMC) in high dimensions.
HMC develops a Markov chain reversible w.r.t. a given target distribution . by using separable Hamiltonian dynamics with potential -log .. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-
Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an O(1) acceptance probability as the dimension d of the state space tends to ., the leapfrog step-size h should be scaled as h=l ×d−1/
4 . Therefore, in high dimensions, HMC requires O(d1/
4 ) steps to traverse the state space. We also identify
analytically the asymptotically optimal acceptance probability, which turns out to be 0.651
(to three decimal places). This is the choice which optimally balances the cost of generating a
proposal, which decreases as l increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as l increases
| Item Type: | Journal Article | ||||
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| Alternative Title: | |||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Statistics |
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| Journal or Publication Title: | Bernoulli | ||||
| Publisher: | Int Statistical Institute | ||||
| ISSN: | 1350-7265 | ||||
| Official Date: | 2013 | ||||
| Dates: |
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| Volume: | Volume 19 | ||||
| Number: | Number 5a | ||||
| Page Range: | pp. 1501-1534 | ||||
| DOI: | 10.3150/12-BEJ414 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published |
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