# The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models arXiv: https://arxiv.org/abs/2405.17068 ยท HTML: https://mlsanity.com/paper/2405.17068/ ## Key facts - Cited by 6 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 34 of its 52 references that are in the map. - First posted on arXiv 27 May 2024. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 34) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021) - [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015) - [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022) - [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022) - [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022) - [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022) - [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019) - [The Role of ImageNet Classes in Fr'echet Inception Distance](https://mlsanity.com/paper/2203.06026/) (2022) - [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) - [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015) - [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023) - [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) - [Parallel Sampling of Diffusion Models](https://mlsanity.com/paper/2305.16317/) (2023) - [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019) - [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) - [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020) - [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022) - [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019) - [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019) - [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017) - [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023) ## Built on by (most cited first, 6 of 6) - [Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms](https://mlsanity.com/paper/2502.00234/) (2025) - [Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel](https://mlsanity.com/paper/2406.00924/) (2024) - [Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration](https://mlsanity.com/paper/2502.04849/) (2025) - [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025) - [Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Bounds](https://mlsanity.com/paper/2506.07614/) (2025) - [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026) ## Abstract (arXiv) Langevin Dynamics is a Stochastic Differential Equation (SDE) central to sampling and generative modeling and is implemented via time discretization. Langevin Monte Carlo (LMC), based on the Euler-Maruyama discretization, is the simplest and most studied algorithm. LMC can suffer from slow convergence - requiring a large number of steps of small step-size to obtain good quality samples. This becomes stark in the case of diffusion models where a large number of steps gives the best samples, but the quality degrades rapidly with smaller number of steps. Randomized Midpoint Method has been recently proposed as a better discretization of Langevin dynamics for sampling from strongly log-concave distributions. However, important applications such as diffusion models involve non-log concave densities and contain time varying drift. We propose its variant, the Poisson Midpoint Method, which approximates a small step-size LMC with large step-sizes. We prove that this can obtain a quadratic speed up of LMC under very weak assumptions. We apply our method to diffusion models for image generation and show that it maintains the quality of DDPM with 1000 neural network calls with just 50-80 neural network calls and outperforms ODE based methods with similar compute.