A novel randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) in place of the usual Verlet integrator; namely, a stratified Monte Carlo (sMC) integrator which involves a minor modification to Verlet, and hence, is easy to implement. For target distributions of the form where is both -strongly convex and -gradient Lipschitz, and initial distributions with finite second moment, coupling proofs reveal that an -accurate approximation of the target distribution in -Wasserstein distance can be achieved by the uHMC algorithm with sMC time integration using gradient evaluations; whereas without additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general . Duration randomization, which has a similar effect as partial momentum refreshment, is also treated. In this case, without additional assumptions on the target distribution, the complexity of duration-randomized uHMC with sMC time integration improves to up to logarithmic factors. The improvement due to duration randomization turns out to be analogous to that of time integrator randomization.
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