Guarantees in Wasserstein Distance for the Langevin Monte Carlo Algorithm

Understand

We study the problem of sampling from a distribution $\target$ using the Langevin Monte Carlo algorithm and provide rate of convergences for this algorithm in terms of Wasserstein distance of order $2$.

• Our result holds as long as the continuous diffusion process associated with the algorithm converges exponentially fast to the target distribution along with some technical assumptions.
• While such an exponential convergence holds for example in the log-concave measure case, it also holds for the more general case of asymptoticaly log-concave measures.
• Our results thus extends the known rates of convergence in total variation and Wasserstein distances which have only been obtained in the log-concave case.