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The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models

Saravanan Kandasamy, Dheeraj Nagaraj

TL;DR

This work tackles the inefficiency of Langevin-based sampling, notably diffusion-model schedulers, by introducing the Poisson Midpoint Method (PLMC) that replaces K steps of size α/K with a single step of size α. The authors prove a KL-bound between PLMC and the fine-grained discretization under very mild conditions and, under λ-LSI, establish a quadratic speedup for both OLMC and ULMC. They further show that PLMC can maintain DDPM-like sample quality with far fewer neural-network evaluations, delivering practical compute gains and competitive performance against ODE-based schedulers across standard image datasets. The theoretical guarantees combined with empirical diffusion-model results suggest PLMC as a robust, general-purpose scheduler for diffusion and Langevin-based sampling tasks. Overall, the paper bridges rigorous stochastic-approximation analysis with impactful empirical gains in modern generative modeling.

Abstract

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.

The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models

TL;DR

This work tackles the inefficiency of Langevin-based sampling, notably diffusion-model schedulers, by introducing the Poisson Midpoint Method (PLMC) that replaces K steps of size α/K with a single step of size α. The authors prove a KL-bound between PLMC and the fine-grained discretization under very mild conditions and, under λ-LSI, establish a quadratic speedup for both OLMC and ULMC. They further show that PLMC can maintain DDPM-like sample quality with far fewer neural-network evaluations, delivering practical compute gains and competitive performance against ODE-based schedulers across standard image datasets. The theoretical guarantees combined with empirical diffusion-model results suggest PLMC as a robust, general-purpose scheduler for diffusion and Langevin-based sampling tasks. Overall, the paper bridges rigorous stochastic-approximation analysis with impactful empirical gains in modern generative modeling.

Abstract

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.
Paper Structure (60 sections, 35 theorems, 152 equations, 1 figure, 7 tables)

This paper contains 60 sections, 35 theorems, 152 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Our Contributions
  4. Notation:
  5. Problem Setup
  6. Overdamped Langevin Monte Carlo:
  7. Underdamped Langevin Monte Carlo
  8. Denoising Diffusion Models:
  9. Technical Notes
  10. Scaling Relations:
  11. Randomized Midpoint Method and Stochastic Approximation
  12. The Poisson Midpoint Method
  13. The Algorithm and Computational Complexity
  14. Main Results
  15. Quadratic Speedup
  16. ...and 45 more sections

Key Result

Proposition 1

When the scaling relations hold (Section subsec:scaling_relations), the trajectory $(X_t^{\mathsf{P}})_{t\geq 0}$ in Equation eq:unfurl has the same joint distribution as the trajectory $(\tilde{X}_{tK})_{t\geq 0}$ given in Equation eq:stoc_approx. In expectation, one step of $\mathsf{PS}(A,G,\Gamma

Figures (1)

  • Figure 1: Comparison of different variants of DDPM

Theorems & Definitions (56)

  • Remark 1
  • Proposition 1
  • Theorem 1
  • Theorem 2: OLMC
  • Theorem 3: ULMC
  • Remark 2
  • Lemma 1
  • Lemma 2: Chain Rule for KL divergence
  • Lemma 3: Data Processing Inequality
  • Lemma 4
  • ...and 46 more