Optimal tuning of the Hybrid Monte-Carlo Algorithm (1001.4460v1)

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 $\Pi$ by using separable Hamiltonian dynamics with potential $-\log\Pi$. 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 $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ 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 {\em decreases} as $l$ increases, against the cost related to the average number of proposals required to obtain acceptance, which {\em increases} as $l$ increases.

• The paper analytically studies the optimal tuning of the Hybrid Monte Carlo algorithm for high-dimensional distributions, showing its superior d scaling compared to MALA and RWM.
• The study finds that for high-dimensional targets, the optimal leapfrog step-size should scale as d to maintain an O(1) acceptance probability.
• Results reveal an optimal acceptance probability of approximately 0.651 for high-dimensional HMC, providing practical guidance for algorithm implementation to maximize efficiency.

Optimal Tuning of the Hybrid Monte-Carlo Algorithm: An Analytical Study

The paper "Optimal Tuning of the Hybrid Monte-Carlo Algorithm," authored by A. Beskos, N.S. Pillai, G.O. Roberts, J. M. Sanz-Serna, and A.M. Stuart, presents a detailed investigation into the efficiency and scalability of the Hybrid Monte Carlo (HMC) algorithm when applied to high-dimensional target distributions. The primary focus of the paper is to provide theoretical insights and guidance for tuning the parameters of HMC, which has been proposed as a more efficient alternative to traditional Markov Chain Monte Carlo (MCMC) methods such as Random-Walk Metropolis (RWM) and Metropolis-adjusted Langevin Algorithm (MALA).

Overview of the Hybrid Monte Carlo Algorithm

HMC leverages Hamiltonian dynamics to generate proposals for the Markov chain, aiming to improve convergence rates and reduce the autocorrelation in sampled sequences. By incorporating additional momentum variables sampled from a Gaussian distribution, HMC transforms the sampling problem into a deterministic trajectory in an extended state space, which is then discretized using the leapfrog integrator. This step transforms the continuous-time dynamics into a numerically tractable form, and a Metropolis-Hastings acceptance criterion is applied to eliminate the bias introduced by discretization.

Key Findings and Contributions

The authors investigate the scaling properties of HMC in the context of independent and identically distributed (iid) components in high dimensions. The paper establishes that, to achieve an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ approaches infinity, the leapfrog step-size $h$ should scale as $h = l \cdot d^{-1/4}$. This indicates that HMC requires $\mathcal{O}(d^{1/4})$ steps to explore the state space, which is shown to be more efficient than the $\mathcal{O}(d^{1/3})$ and $\mathcal{O}(d)$ steps required by MALA and RWM, respectively.

Additionally, the paper reveals an optimal acceptance probability of approximately 0.651 (to three decimal places) for high-dimensional targets. This value optimally balances the computational cost of generating a proposal with the probability of acceptance, thus maximizing the efficiency of HMC. Theoretical results are supported through analytical proofs and illustrated by numerical experiments.

Implications and Future Directions

The implications of this paper are significant for the design and implementation of efficient sampling algorithms in high-dimensional spaces, particularly in areas such as Bayesian inference, statistical physics, and machine learning. The results provide a theoretical foundation for tuning the HMC parameters, notably the step-size, to achieve optimal performance across a wide range of applications.

Future research may extend this work to explore the impact of different mass matrices (M) and integration times (T) on the efficiency of HMC. Additionally, investigating the applicability of semi-implicit integrators and the use of variable step sizes for HMC could provide further avenues for enhancing the algorithm's performance, especially for target distributions with non-standard tail behaviors.

In conclusion, the paper by Beskos et al. makes substantial progress in elucidating the scaling properties and optimal tuning of the HMC algorithm, paving the way for its effective application in tackling high-dimensional sampling problems. It offers a comprehensive theoretical framework that could be valuable for researchers and practitioners implementing MCMC techniques in advanced statistical and computational settings.