# Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms

arXiv: https://arxiv.org/abs/2305.15988 · HTML: https://mlsanity.com/paper/2305.15988/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 107 of its 204 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 30 of 107)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [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)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Oops I Took A Gradient: Scalable Sampling for Discrete Distributions](https://mlsanity.com/paper/2102.04509/) (2021)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)

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

- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Beyond Scores: Proximal Diffusion Models](https://mlsanity.com/paper/2507.08956/) (2025)
- [Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data](https://mlsanity.com/paper/2503.10428/) (2025)
- [Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental study](https://mlsanity.com/paper/2606.14800/) (2026)

## Abstract (arXiv)

We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte Carlo (MCMC) methods derived from discretizations of the overdamped Langevin diffusions, which are commonly known as Langevin Monte Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases for which a large class of proximal MCMC methods have been developed: (i) a nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling tasks arise from a wide range of applications to Bayesian inference and imaging inverse problems such as image deconvolution. We perform numerical simulations to compare the performance of most commonly used Langevin Monte Carlo algorithms.