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# Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

*Proceedings of the 2017 Conference on Learning Theory*, PMLR 65:319-342, 2017.

#### Abstract

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $K$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $K$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.