Faster Hamiltonian Monte Carlo by Learning Leapfrog Scale: a self-calibrated randomized solution

We introduce a Hamiltonian Monte Carlo (HMC) methodology based on a randomized selection of integration times, referred to as eHMC, where "e" stands for empirical. The approach relies on an offline calibration phase that leverages importance sampling to construct an empirical distribution on discretization parameters, thereby eliminating the need for manual burn-in diagnostics and online adaptation. The proposal distribution used in the calibration stage is obtained via a Population Monte Carlo scheme combined with tempering and flexible parametric variational families such as normalizing flows. The resulting algorithm defines a mixture of HMC kernels with a fixed mixing distribution, preserving the target distribution. Numerical experiments on benchmarks demonstrate that eHMC achieves competitive or improved efficiency compared to the No-U-Turn Sampler (NUTS) when accounting for computational cost. These results suggest that offline calibration combined with randomized integration schemes provides a viable alternative to adaptive HMC methods.