Implicit Langevin Algorithms for Sampling From Log-concave Densities

For sampling from a log-concave density, we study implicit integrators resulting from $\theta$-method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the resulting sampling methods for $\theta \in [0,1]$ and a range of step sizes are established. Our results generalize and extend prior works in several directions. In particular, for $\theta\ge1/2$, we prove geometric ergodicity and stability of the resulting methods for all step sizes. We show that obtaining subsequent samples amounts to solving a strongly-convex optimization problem, which is readily achievable using one of numerous existing methods. Numerical examples supporting our theoretical analysis are also presented

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

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 $\mathsf{K}$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $\mathsf{K}$. 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