Implicit langevin algorithms for sampling from log-concave densities

For sampling from a log-concave density, we study implicit integrators resulting from θ- method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the resulting sampling methods for θ ∈ [0; 1] and a range of step sizes are established. Our results generalize and extend prior works in several directions. In particular, for θ ≥ 1/2, 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.