Sampling from a log-concave distribution with Projected Langevin Monte Carlo

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported via a projection step, akin to projected Stochastic Gradient Descent(SGD). We show that (projected) LMC allows to sample in polynomial time from a-concave distribution with smooth potential. This gives a new Markov chain sample from a log-concave distribution. Our main result shows in particular when the target distribution is uniform, LMC mixes in ̃(n⁷) (where n is the dimension). We also provide preliminary experimental that LMC performs at least as well as hit-and-run, for which a better time of ̃(n⁴) was proved by Lov{\\'a}sz and Vempala.