Abstract
The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional 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 proven 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 particular, for the unadjusted chain, 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 d1∕4∕ε in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.
Highlights
The Langevin diffusion is the Markov process on Rd × Rd that solves the SDE dXt = Vtdt √dVt = −∇U (Xt)dt − γVtdt + 2γdBt (1)where B is a standard d-dimensional Brownian motion, U ∈ C2(Rd) is called the potential and γ > 0 is a friction parameter
In this work we focus on the use of the Langevin diffusion in Markov Chain Monte Carlo (MCMC) algorithms in order to estimate expectations with respect to π
This process has been used for decades in molecular dynamics (MD), in particular due to physical motivations related to the motion of particles in classical physics
Summary
In this work we focus on the use of the Langevin diffusion in Markov Chain Monte Carlo (MCMC) algorithms in order to estimate expectations with respect to π This process has been used for decades in molecular dynamics (MD), in particular due to physical motivations related to the motion of particles in classical physics (see [25, 38, 55, 30, 32, 31, 51, 8, 7, 6] and references within). We will focus on the case where U is m-convex and L-smooth (see Assumptions (∇Lip) and (Conv) below) These are usual conditions in the statistics community to compare MCMC methods, the objective being to obtain nonasymptotic explicit estimates in term of the dimension d, see e.g.
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