For diffusions on Riemannian manifolds with boundary, the paper establishes a quantitative space-time divergence lemma and proves that randomized Hamiltonian Monte Carlo and Langevin dynamics achieve the optimal square-root reduction in relaxation time.
Convergence in total variation for the kinetic Langevin algorithm
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abstract
We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincar\'e inequality and has gradient Lipschitz potential. The main point is that the estimate improves significantly upon the corresponding bound for the non kinetic version of the algorithm, due to Dalalyan. In particular the dimension dependence drops from $O(n)$ to $O(\sqrt n)$.
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Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary
For diffusions on Riemannian manifolds with boundary, the paper establishes a quantitative space-time divergence lemma and proves that randomized Hamiltonian Monte Carlo and Langevin dynamics achieve the optimal square-root reduction in relaxation time.