# Implicit Langevin Algorithms for Sampling From Log-concave Densities

arXiv: https://arxiv.org/abs/1903.12322 · HTML: https://mlsanity.com/paper/1903.12322/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2019.

## Built on by (most cited first, 6 of 6)

- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Parallel Simulation for Log-concave Sampling and Score-based Diffusion Models](https://mlsanity.com/paper/2412.07435/) (2024)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)
- [Beyond Scores: Proximal Diffusion Models](https://mlsanity.com/paper/2507.08956/) (2025)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [Score-based Metropolis-Hastings for Fractional Langevin Algorithms](https://mlsanity.com/paper/2602.00835/) (2026)

## Abstract (arXiv)

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.