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 different values of theta and a range of step sizes are established. Our results generalize and extend prior works in several directions. In particular, for theta greater than or equal to 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.