# Discrete Langevin Sampler via Wasserstein Gradient Flow

arXiv: https://arxiv.org/abs/2206.14897 · HTML: https://mlsanity.com/paper/2206.14897/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 77 references that are in the map.
- First posted on arXiv 29 Jun 2022.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [MARS: Markov Molecular Sampling for Multi-objective Drug Discovery](https://mlsanity.com/paper/2103.10432/) (2021)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [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)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [A Unified Particle-Optimization Framework for Scalable Bayesian Sampling](https://mlsanity.com/paper/1805.11659/) (2018)
- [A Langevin-like Sampler for Discrete Distributions](https://mlsanity.com/paper/2206.09914/) (2022)
- [Exponential Family Estimation via Adversarial Dynamics Embedding](https://mlsanity.com/paper/1904.12083/) (2019)
- [Learning Discrete Energy-based Models via Auxiliary-variable Local Exploration](https://mlsanity.com/paper/2011.05363/) (2020)
- [Energy-based View of Retrosynthesis](https://mlsanity.com/paper/2007.13437/) (2020)
- [The Barker proposal: combining robustness and efficiency in gradient-based MCMC](https://mlsanity.com/paper/1908.11812/) (2019)
- [The Hamming Ball Sampler](https://mlsanity.com/paper/1504.08133/) (2015)
- [Stein Variational Inference for Discrete Distributions](https://mlsanity.com/paper/2003.00605/) (2020)
- [Understanding MCMC Dynamics as Flows on the Wasserstein Space](https://mlsanity.com/paper/1902.00282/) (2019)
- [Sampling in Combinatorial Spaces with SurVAE Flow Augmented MCMC](https://mlsanity.com/paper/2102.02374/) (2021)
- [LSB: Local Self-Balancing MCMC in Discrete Spaces](https://mlsanity.com/paper/2109.03867/) (2021)
- [Mixed-Variate Restricted Boltzmann Machines](https://mlsanity.com/paper/1408.1160/) (2014)

## Built on by (most cited first, 24 of 24)

- [Informed Correctors for Discrete Diffusion Models](https://mlsanity.com/paper/2407.21243/) (2024)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Plug & Play Directed Evolution of Proteins with Gradient-based Discrete MCMC](https://mlsanity.com/paper/2212.09925/) (2022)
- [Discrete Neural Flow Samplers with Locally Equivariant Transformer](https://mlsanity.com/paper/2505.17741/) (2025)
- [Gradient-based Discrete Sampling with Automatic Cyclical Scheduling](https://mlsanity.com/paper/2402.17699/) (2024)
- [Optimization by Parallel Quasi-Quantum Annealing with Gradient-Based Sampling](https://mlsanity.com/paper/2409.02135/) (2024)
- [Reheated Gradient-based Discrete Sampling for Combinatorial Optimization](https://mlsanity.com/paper/2503.04047/) (2025)
- [Improving multiple-try Metropolis with local balancing](https://mlsanity.com/paper/2211.11613/) (2022)
- [Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces](https://mlsanity.com/paper/2412.01019/) (2024)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [Divide-and-Conquer Predictive Coding: a structured Bayesian inference algorithm](https://mlsanity.com/paper/2408.05834/) (2024)
- [Entropic Matching for Expectation Propagation of Markov Jump Processes](https://mlsanity.com/paper/2309.15604/) (2023)
- [Robust Approximate Sampling via Stochastic Gradient Barker Dynamics](https://mlsanity.com/paper/2405.08999/) (2024)
- [Entropy-Guided Sampling of Flat Modes in Discrete Spaces](https://mlsanity.com/paper/2505.02296/) (2025)
- [Learning with Local Search MCMC Layers](https://mlsanity.com/paper/2505.14240/) (2025)
- [Learning Discrete Diffusion of Graphs via Free-Energy Gradient Flows](https://mlsanity.com/paper/2604.11311/) (2026)
- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Enhancing Gradient-based Discrete Sampling via Parallel Tempering](https://mlsanity.com/paper/2502.19240/) (2025)
- [Beyond Self-Repellent Kernels: History-Driven Target Towards Efficient Nonlinear MCMC on General Graphs](https://mlsanity.com/paper/2505.18300/) (2025)
- [Slithering Through Gaps: Capturing Discrete Isolated Modes via Logistic Bridging](https://mlsanity.com/paper/2604.10821/) (2026)
- [Fine-Tuning Low-Bit Models with Gradient in Quantized Code Space](https://mlsanity.com/paper/2608.30908/) (2026)
- [Training Discrete Energy-Based Models with Energy Discrepancy](https://mlsanity.com/paper/2307.07595/) (2023)
- [Discrete Wasserstein Flows for One-Step Generative Modeling](https://mlsanity.com/paper/2610.01355/) (2026)
- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

It is known that gradient-based MCMC samplers for continuous spaces, such as Langevin Monte Carlo (LMC), can be derived as particle versions of a gradient flow that minimizes KL divergence on a Wasserstein manifold. The superior efficiency of such samplers has motivated several recent attempts to generalize LMC to discrete spaces. However, a fully principled extension of Langevin dynamics to discrete spaces has yet to be achieved, due to the lack of well-defined gradients in the sample space. In this work, we show how the Wasserstein gradient flow can be generalized naturally to discrete spaces. Given the proposed formulation, we demonstrate how a discrete analogue of Langevin dynamics can subsequently be developed. With this new understanding, we reveal how recent gradient-based samplers in discrete spaces can be obtained as special cases by choosing particular discretizations. More importantly, the framework also allows for the derivation of novel algorithms, one of which, textitDiscrete Langevin Monte Carlo (DLMC), is obtained by a factorized estimate of the transition matrix. The DLMC method admits a convenient parallel implementation and time-uniform sampling that achieves larger jump distances. We demonstrate the advantages of DLMC on various binary and categorical distributions.