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 $\sigma$):

$$H(q,p)=q^2/2\sigma +p^2/2$$

A leapfrog step with step size $ϵ$ is a linear mapping between $(q(t),p(t))$ and $(q(t+ϵ),p(t+ϵ))$ with the following transition matrix:

$$\left(\begin{array}{cc}1-{ϵ}^2/2{\sigma }^2& ϵ\\ -ϵ/{\sigma }^2+{ϵ}^3/4{\sigma }^4& 1-{ϵ}^2/2{\sigma }^2\end{array}\right)$$

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

$$\text{Missing begin{equation*} or extra end{equation*}}$$

When $ϵ/\sigma >2$ these eigenvalues are real and one will have absolute value greated than one. Trajectories computed with $ϵ<2\sigma$ 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 $K(p)=p^T{M}^{-1}p$