Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fréchet Inception Distance (FID) on the latter.

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

Sampling from an unnormalized Boltzmann density requires proposals that move probability mass globally while retaining enough path-probability information for statistical correction. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), a train-then-correct learned Hamiltonian sampler. Starting from a tractable base distribution, NHMC learns stochastic Hamiltonian-style paths toward the target. Once training is complete, the learned proposal parameters are fixed; the proposal then generates complete paths and endpoint configurations, which are statistically corrected using the recorded non-equilibrium work. This dimensionless generalized work is determined by the probability ratio between the forward proposal path and a reverse reference path. During training, minimizing its mean reduces a path-space KL divergence and controls an upper bound on endpoint mismatch. During evaluation, the same quantity defines weights for self-normalized importance sampling on paths (path-SNIS), estimates normalizing constants or free-energy differences, and gives the acceptance ratio for path-space independent Metropolis-Hastings (path-IMH). The same forward-reverse laws also define a shared-bridge round-trip Metropolis kernel that acts directly on configurations and preserves the Boltzmann target. On double-well and finite-volume lattice $φ^4$ targets, the NHMC construction gives corrected estimates when path overlap is sufficient; when overlap is poor, weight degeneracy, low acceptance, and long autocorrelation expose proposal failure. We additionally report a molecular internal-coordinate feasibility study using an MD prior and learned-force path proposal.