# Langevin Monte Carlo without smoothness

arXiv: https://arxiv.org/abs/1905.13285 · HTML: https://mlsanity.com/paper/1905.13285/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2019.

## Built on by (most cited first, 28 of 28)

- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [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)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint](https://mlsanity.com/paper/2110.01827/) (2021)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://mlsanity.com/paper/2408.13115/) (2024)
- [Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusion](https://mlsanity.com/paper/2501.00565/) (2024)
- [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)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [A Proximal Algorithm for Sampling from Non-convex Potentials](https://mlsanity.com/paper/2205.10188/) (2022)
- [Stochastic Langevin Differential Inclusions with Applications to Machine Learning](https://mlsanity.com/paper/2206.11533/) (2022)
- [Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems](https://mlsanity.com/paper/2503.16222/) (2025)
- [Covid19 Reproduction Number: Credibility Intervals by Blockwise Proximal Monte Carlo Samplers](https://mlsanity.com/paper/2203.09142/) (2022)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Non-asymptotic analysis of Langevin-type Monte Carlo algorithms](https://mlsanity.com/paper/2303.12407/) (2023)
- [A Gentle Introduction to Gradient-Based Optimization and Variational Inequalities for Machine Learning](https://mlsanity.com/paper/2309.04877/) (2023)
- [Ito Diffusion Approximation of Universal Ito Chains for Sampling, Optimization and Boosting](https://mlsanity.com/paper/2310.06081/) (2023)
- [Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuity](https://mlsanity.com/paper/2501.04134/) (2025)
- [Estimating Multi-chirp Parameters using Curvature-guided Langevin Monte Carlo](https://mlsanity.com/paper/2501.18178/) (2025)
- [In-Context Batch Black-box Optimization via Flow Matching](https://mlsanity.com/paper/2610.20138/) (2026)

## Abstract (arXiv)

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. In this paper, we remove this limitation, providing polynomial-time convergence guarantees for a variant of LMC in the setting of nonsmooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and controlling the bias and variance that are induced by this perturbation.