Implicit Langevin Algorithms for Sampling From Log-concave Densities
This provides a stable sampling method for probabilistic modeling and Bayesian inference, but it is incremental as it generalizes prior works.
The paper tackles the problem of sampling from log-concave densities by studying implicit integrators from θ-method discretization of the overdamped Langevin diffusion, establishing geometric ergodicity and stability for θ≥1/2 across all step sizes.
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\in [0,1] $ and a range of step sizes are established. Our results generalize and extend prior works in several directions. In particular, for $θ\ge1/2$, 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.