  Hamiltonian Monte Carlo (HMC) is a very popular and generic collection of
Markov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity
of HMC algorithms is their excellent performance as the dimension $d$ of the
target becomes large: under conditions that are satisfied for many common
statistical models, optimally-tuned HMC algorithms have a running time that
scales like $d^{0.25}$. In stark contrast, the running time of the usual
Random-Walk Metropolis (RWM) algorithm, optimally tuned, scales like $d$. This
superior scaling of the HMC algorithm with dimension is attributed to the fact
that it, unlike RWM, incorporates the gradient information in the proposal
distribution. In this paper, we investigate a different scaling question: does
HMC beat RWM for highly $\textit{multimodal}$ targets? We find that the answer
is often $\textit{no}$. We compute the spectral gaps for both the algorithms
for a specific class of multimodal target densities, and show that they are
identical. The key reason is that, within one mode, the gradient is effectively
ignorant about other modes, thus negating the advantage the HMC algorithm
enjoys in unimodal targets. We also give heuristic arguments suggesting that
the above observation may hold quite generally. Our main tool for answering
this question is a novel simple formula for the conductance of HMC using
Liouville's theorem. This result allows us to compute the spectral gap of HMC
algorithms, for both the classical HMC with isotropic momentum and the recent
Riemannian HMC, for multimodal targets.
