# Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets

arXiv: https://arxiv.org/abs/1906.08530 · HTML: https://mlsanity.com/paper/1906.08530/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2019.

## Built on by (most cited first, 16 of 16)

- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compression](https://mlsanity.com/paper/2303.04622/) (2023)
- [Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions](https://mlsanity.com/paper/2301.03077/) (2023)
- [Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped](https://mlsanity.com/paper/2510.03949/) (2025)
- [Local and Global Contraction Principles for MCMC Mixing](https://mlsanity.com/paper/2606.03033/) (2026)

## Abstract (arXiv)

In this paper, we provide non-asymptotic upper bounds on the error of sampling from a target density using three schemes of discretized Langevin diffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, the Euler discretization of the Langevin diffusion. The second and the third schemes are, respectively, the kinetic Langevin Monte Carlo (KLMC) for differentiable potentials and the kinetic Langevin Monte Carlo for twice-differentiable potentials (KLMC2). The main focus is on the target densities that are smooth and log-concave on mathbb Rp, but not necessarily strongly log-concave. Bounds on the computational complexity are obtained under two types of smoothness assumption: the potential has a Lipschitz-continuous gradient and the potential has a Lipschitz-continuous Hessian matrix. The error of sampling is measured by Wasserstein-q distances. We advocate for the use of a new dimension-adapted scaling in the definition of the computational complexity, when Wasserstein-q distances are considered. The obtained results show that the number of iterations to achieve a scaled-error smaller than a prescribed value depends only polynomially in the dimension.