  We analyze the mixing time of Metropolized Hamiltonian Monte Carlo (HMC) with
the leapfrog integrator to sample from a distribution on $\mathbb{R}^d$ whose
log-density is smooth, has Lipschitz Hessian in Frobenius norm and satisfies
isoperimetry. We bound the gradient complexity to reach $\epsilon$ error in
total variation distance from a warm start by $\tilde
O(d^{1/4}\text{polylog}(1/\epsilon))$ and demonstrate the benefit of choosing
the number of leapfrog steps to be larger than 1. To surpass previous analysis
on Metropolis-adjusted Langevin algorithm (MALA) that has
$\tilde{O}(d^{1/2}\text{polylog}(1/\epsilon))$ dimension dependency in Wu et
al. (2022), we reveal a key feature in our proof that the joint distribution of
the location and velocity variables of the discretization of the continuous HMC
dynamics stays approximately invariant. This key feature, when shown via
induction over the number of leapfrog steps, enables us to obtain estimates on
moments of various quantities that appear in the acceptance rate control of
Metropolized HMC. Moreover, to deal with another bottleneck on the HMC proposal
distribution overlap control in the literature, we provide a new approach to
upper bound the Kullback-Leibler divergence between push-forwards of the
Gaussian distribution through HMC dynamics initialized at two different points.
Notably, our analysis does not require log-concavity or independence of the
marginals, and only relies on an isoperimetric inequality. To illustrate the
applicability of our result, several examples of natural functions that fall
into our framework are discussed.
