# The Tamed Unadjusted Langevin Algorithm

arXiv: https://arxiv.org/abs/1710.05559 · HTML: https://mlsanity.com/paper/1710.05559/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2017.

## Built on by (most cited first, 30 of 31)

- [Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors](https://mlsanity.com/paper/2403.11407/) (2024)
- [Diffusion posterior sampling for simulation-based inference in tall data settings](https://mlsanity.com/paper/2404.07593/) (2024)
- [Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation](https://mlsanity.com/paper/2303.13429/) (2023)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks](https://mlsanity.com/paper/2105.13937/) (2021)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Lagrangian Manifold Monte Carlo on Monge Patches](https://mlsanity.com/paper/2202.00755/) (2022)
- [Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function](https://mlsanity.com/paper/2107.08649/) (2021)
- [Taming under isoperimetry](https://mlsanity.com/paper/2311.09003/) (2023)
- [Tamed Langevin sampling under weaker conditions](https://mlsanity.com/paper/2405.17693/) (2024)
- [Kinetic Interacting Particle Langevin Monte Carlo](https://mlsanity.com/paper/2407.05790/) (2024)
- [Langevin dynamics based algorithm e-THvarepsilonO POULA for stochastic optimization problems with discontinuous stochastic gradient](https://mlsanity.com/paper/2210.13193/) (2022)
- [From Global to Local: A Scalable Benchmark for Local Posterior Sampling](https://mlsanity.com/paper/2507.21449/) (2025)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials](https://mlsanity.com/paper/2112.09311/) (2021)
- [Taming the Interacting Particle Langevin Algorithm: The Superlinear case](https://mlsanity.com/paper/2403.19587/) (2024)
- [kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients](https://mlsanity.com/paper/2506.04878/) (2025)
- [Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization](https://mlsanity.com/paper/2605.24937/) (2026)
- [Deterministic Envelopes for Tamed SGLD: Decoupling Stochastic Gradient Noise and Localizing Taming](https://mlsanity.com/paper/2606.05242/) (2026)
- [Contractive kinetic Langevin samplers beyond global Lipschitz continuity](https://mlsanity.com/paper/2509.12031/) (2025)
- [Theoretical guidelines for annealed Langevin dynamics in compositional simulation-based inference](https://mlsanity.com/paper/2605.21253/) (2026)
- [Deterministic Denominator Design for Localized Tamed Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2606.10559/) (2026)
- [Non-asymptotic analysis of Langevin-type Monte Carlo algorithms](https://mlsanity.com/paper/2303.12407/) (2023)
- [STANLEY: Stochastic Gradient Anisotropic Langevin Dynamics for Learning Energy-Based Models](https://mlsanity.com/paper/2310.12667/) (2023)
- [Briding Diffusion Posterior Sampling and Monte Carlo methods: a survey](https://mlsanity.com/paper/2510.14114/) (2025)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)
- [Event Generation with Parallel Langevin Sampling and Learned Stein Diagnostics](https://mlsanity.com/paper/2606.14854/) (2026)
- [RELTA-SGLD: Relative-Growth Localized Taming for Nonconvex Stochastic-Gradient Langevin Learning](https://mlsanity.com/paper/2607.19544/) (2026)
- [Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity](https://mlsanity.com/paper/2608.11486/) (2026)

## Abstract (arXiv)

In this article, we consider the problem of sampling from a probability measure π having a density on ℝd known up to a normalizing constant, xmapsto e-U(x) / intℝd e-U(y) d y. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable in a precise sense, when the potential U is superlinear, i.e. liminfVert x Vert→+∞ Vert nabla U(x) Vert / Vert x Vert = +∞. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.