Step-size and acceptance-rate theory for HMC with parallel leapfrog integration

Determine appropriate optimal step sizes and acceptance rates for Hamiltonian Monte Carlo when the leapfrog integrator is evaluated via parallel Newton iterations that yield sublinear time complexity in the number of leapfrog steps and a step-size-dependent number of Newton iterations to convergence; develop a formal analysis that provides tuning guidance under these parallel-integration conditions.

Background

Standard HMC tuning analyses assume linear cost in the number of leapfrog steps, guiding step-size and acceptance-rate targets. Parallelizing leapfrog integration with Newton-based methods (DEER/quasi-DEER) breaks this assumption by enabling sublinear cost in the number of steps, and the number of Newton iterations to convergence can vary with step size. This changes the tuning landscape and motivates new theory to set step sizes and acceptance rates under parallel integration.

References

New analysis investigating the appropriate optimal step sizes and acceptance rates need to be devised under these conditions. We hypothesize that in some cases, these conditions will favor the use of smaller step sizes and more leapfrog steps, thereby increasing the optimal acceptance rate. We currently leave a formal analysis of this to future work.

— Parallelizing MCMC Across the Sequence Length (2508.18413 - Zoltowski et al., 25 Aug 2025) in Appendix, Subsection “Step-size for HMC with parallel leapfrog”