Step size for stable Hamiltonian trajectories

tags

Hamiltonian Monte Carlo

Let us consider the following Hamiltonian (corresponds to sampling from a univariate Gaussian with standard deviation \(\sigma\)):

\begin{equation*} H(q, p) = q^2\,/\,2\sigma + p^2\,/\,2 \end{equation*}

A leapfrog step with step size \(\epsilon\) is a linear mapping between \(\left(q(t), p(t)\right)\) and \(\left(q(t+\epsilon), p(t+\epsilon)\right)\) with the following transition matrix:

\begin{pmatrix} 1-\epsilon^2\,/\, 2\sigma^2 & \epsilon\\ -\epsilon/\sigma^2+\epsilon^3/4\sigma^4 & 1-\epsilon^2/2\sigma^2 \end{pmatrix}

Whether this leads to a stable trajectory or diverges depends on the magnitude of thge eigenvalues:

\begin{equation*} \left(1 - \epsilon^2/2\sigma^2) \pm (\epsilon/\sigma) \sqrt{\epsilon^2/4\sigma^2-1} \end{equation*}

When \(\epsilon/\sigma > 2\) these eigenvalues are real and one will have absolute value greated than one. Trajectories computed with \(\epsilon < 2\sigma\) are thus stable. For multi-dimensional problems, the stability will be determined by the width of the distribution in the most constrained region. Stability for general quadratic hamiltonian \(K(p) = p^T M^{-1} p\)