  Hamiltonian Monte Carlo (HMC) is a popular Markov Chain Monte Carlo (MCMC)
algorithm to sample from an unnormalized probability distribution. A leapfrog
integrator is commonly used to implement HMC in practice, but its performance
can be sensitive to the choice of mass matrix used therein. We develop a
gradient-based algorithm that allows for the adaptation of the mass matrix by
encouraging the leapfrog integrator to have high acceptance rates while also
exploring all dimensions jointly. In contrast to previous work that adapt the
hyperparameters of HMC using some form of expected squared jumping distance,
the adaptation strategy suggested here aims to increase sampling efficiency by
maximizing an approximation of the proposal entropy. We illustrate that using
multiple gradients in the HMC proposal can be beneficial compared to a single
gradient-step in Metropolis-adjusted Langevin proposals. Empirical evidence
suggests that the adaptation method can outperform different versions of HMC
schemes by adjusting the mass matrix to the geometry of the target distribution
and by providing some control on the integration time.
