# The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions

arXiv: https://arxiv.org/abs/2502.03458 · HTML: https://mlsanity.com/paper/2502.03458/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 55 references that are in the map.
- First posted on arXiv 5 Feb 2025.

## Builds on (most cited first, 30 of 32)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization](https://mlsanity.com/paper/1910.02008/) (2019)
- [Non-asymptotic bounds for sampling algorithms without log-concavity](https://mlsanity.com/paper/1808.07105/) (2018)
- [The Tamed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1710.05559/) (2017)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Langevin Monte Carlo without smoothness](https://mlsanity.com/paper/1905.13285/) (2019)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [The Langevin Monte Carlo algorithm in the non-smooth log-concave case](https://mlsanity.com/paper/2101.10695/) (2021)
- [Taming neural networks with TUSLA: Non-convex learning via adaptive stochastic gradient Langevin algorithms](https://mlsanity.com/paper/2006.14514/) (2020)
- [On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case](https://mlsanity.com/paper/1812.02709/) (2018)
- [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)
- [Proximal nested sampling for high-dimensional Bayesian model selection](https://mlsanity.com/paper/2106.03646/) (2021)
- [Subgradient Langevin Methods for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2308.01417/) (2023)
- [Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting](https://mlsanity.com/paper/2207.02600/) (2022)
- [Fractal Landscapes in Policy Optimization](https://mlsanity.com/paper/2310.15418/) (2023)
- [Taming under isoperimetry](https://mlsanity.com/paper/2311.09003/) (2023)
- [Tamed Langevin sampling under weaker conditions](https://mlsanity.com/paper/2405.17693/) (2024)

## Built on by (most cited first, 4 of 4)

- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Gradient Flow Sampler-based Distributionally Robust Optimization](https://mlsanity.com/paper/2510.25956/) (2025)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials](https://mlsanity.com/paper/2610.02158/) (2026)

## Abstract (arXiv)

In this article, we study the problem of sampling from distributions whose densities are not necessarily smooth nor logconcave. We propose a simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such as the Moreau-Yosida envelope or Gaussian smoothing, and show consequently that the performance of samplers like ULA does not necessarily degenerate arbitrarily with low regularity. In particular, we show that the Lipschitz or H"older continuity assumption can be replaced by a geometric one-sided Lipschitz condition that allows even for discontinuous log-gradients. We derive non-asymptotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non-asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.