High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion
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The efficiency of a markov sampler based on the underdamped Langevin diffusion is studied for high dimensionial targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proved for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adjusted and unadjusted chains. In terms of the dimension d and the desired accuracy ε, the Wasserstein efficiency bounds are of order √(d) / ε in the general case, √(d/ε) if the Hessian of the potential is Lipschitz, and d^1/4/√(ε) in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.
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