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Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling

Ruoyu Wang, Beier Zhu, Junzhi Li, Liangyu Yuan, Chi Zhang

TL;DR

This work analyzes the error dynamics of ODE- and SDE-based diffusion samplers, showing ODEs accumulate gradient error while SDEs require many steps to suppress discretization error. It introduces AdaSDE, a single-step SDE solver with an adaptive per-step stochastic coefficient $oldsymbol{\\gamma_i}$ and a process-supervision training framework, enabling efficient few-step diffusion sampling. Theoretical bounds demonstrate gradient-error contraction under AdaSDE and synthetic results validate reduced total error; empirically, AdaSDE achieves state-of-the-art FID scores at low NFEs across CIFAR-10, FFHQ, LSUN Bedroom, and MSCOCO with Stable Diffusion, while remaining a lightweight plug-in for existing solvers. These findings offer a practical path to fast, high-quality diffusion sampling with minimal additional training or parameter overhead.

Abstract

Diffusion-based generative processes, formulated as differential equation solving, frequently balance computational speed with sample quality. Our theoretical investigation of ODE- and SDE-based solvers reveals complementary weaknesses: ODE solvers accumulate irreducible gradient error along deterministic trajectories, while SDE methods suffer from amplified discretization errors when the step budget is limited. Building upon this insight, we introduce AdaSDE, a novel single-step SDE solver that aims to unify the efficiency of ODEs with the error resilience of SDEs. Specifically, we introduce a single per-step learnable coefficient, estimated via lightweight distillation, which dynamically regulates the error correction strength to accelerate diffusion sampling. Notably, our framework can be integrated with existing solvers to enhance their capabilities. Extensive experiments demonstrate state-of-the-art performance: at 5 NFE, AdaSDE achieves FID scores of 4.18 on CIFAR-10, 8.05 on FFHQ and 6.96 on LSUN Bedroom. Codes are available in https://github.com/WLU-wry02/AdaSDE.

Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling

TL;DR

This work analyzes the error dynamics of ODE- and SDE-based diffusion samplers, showing ODEs accumulate gradient error while SDEs require many steps to suppress discretization error. It introduces AdaSDE, a single-step SDE solver with an adaptive per-step stochastic coefficient and a process-supervision training framework, enabling efficient few-step diffusion sampling. Theoretical bounds demonstrate gradient-error contraction under AdaSDE and synthetic results validate reduced total error; empirically, AdaSDE achieves state-of-the-art FID scores at low NFEs across CIFAR-10, FFHQ, LSUN Bedroom, and MSCOCO with Stable Diffusion, while remaining a lightweight plug-in for existing solvers. These findings offer a practical path to fast, high-quality diffusion sampling with minimal additional training or parameter overhead.

Abstract

Diffusion-based generative processes, formulated as differential equation solving, frequently balance computational speed with sample quality. Our theoretical investigation of ODE- and SDE-based solvers reveals complementary weaknesses: ODE solvers accumulate irreducible gradient error along deterministic trajectories, while SDE methods suffer from amplified discretization errors when the step budget is limited. Building upon this insight, we introduce AdaSDE, a novel single-step SDE solver that aims to unify the efficiency of ODEs with the error resilience of SDEs. Specifically, we introduce a single per-step learnable coefficient, estimated via lightweight distillation, which dynamically regulates the error correction strength to accelerate diffusion sampling. Notably, our framework can be integrated with existing solvers to enhance their capabilities. Extensive experiments demonstrate state-of-the-art performance: at 5 NFE, AdaSDE achieves FID scores of 4.18 on CIFAR-10, 8.05 on FFHQ and 6.96 on LSUN Bedroom. Codes are available in https://github.com/WLU-wry02/AdaSDE.
Paper Structure (30 sections, 12 theorems, 75 equations, 8 figures, 8 tables, 2 algorithms)

This paper contains 30 sections, 12 theorems, 75 equations, 8 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Diffusion Models with Differential Equations
  5. Analysis of ODE and SDE
  6. Trade-offs Between ODE and SDE Solvers
  7. Error Propagation in Deterministic and Stochastic Sampling
  8. Synthetic Validation
  9. Methodology
  10. Sampling Trajectory Geometry
  11. Fast SDE-based Sampling
  12. Experiments
  13. Experiment Setup
  14. Ablation Studies
  15. Conclusion and Limitation
  16. ...and 15 more sections

Key Result

Theorem 1

(ODE Error Boundxu2023restartsamplingimprovinggenerative) Let $\Delta t>0$ denote the discretization step size. Over the interval $[t,t+\Delta t]$, the trajectory $\mathbf{x}_t=\mathsf{ODE}_\theta\left(\mathbf{x}_{t+\Delta t}, t+\Delta t \rightarrow t\right)$ is generated by the learned drift $s_\th where $\mathsf{TV}(\cdot, \cdot)$ denotes the total variation distance.

Figures (8)

  • Figure 1: Gradient error, Discretization error and Total error on synthetic dataset across various steps (measured in 1-Wasserstein Distance). $\gamma=0$ indicates adding no stochasticity (ODE), $\gamma>0$ indicates SDE variants, figures are plotted in Pareto Frontier.
  • Figure 2: Illustration of the 2D double-circles: two concentric rings with radii $0.8$ (outer, blue) and $0.6$ (inner, green). We uniformly sample $20{,}000$ points and add isotropic Gaussian noise ($\sigma=0.1$).
  • Figure 3: The proposed $\texttt{AdaSDE}$ framework. $\texttt{AdaSDE}$ diverges from traditional heuristic noise injection methods used in DDPM ho2020denoisingdiffusionprobabilisticmodels and EDM-SDE karras2022elucidating. Instead, we use intermediary supervision from a teacher sampling path to learn and optimize the noise injection process.
  • Figure 4: Comparison of image synthesis quality under identical NFE constraints using AdaSDE (ours) and DPM-Solver++ (2M). Both methods generate images with Stable Diffusion v1.5 rombach2022high and classifier-free guidance (scale = 7.5) for the prompt "A photo of some flowers in a ceramic vase".
  • Figure 5: Qualitative result on CIFAR10 32$\times$32 ($\text{5}$ and $\text{9}$ NFEs)
  • ...and 3 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof sketch
  • Remark 1
  • Lemma 1: Upper Bound on ODE Discretization Error
  • proof
  • Lemma 2: TV Distance Between Gaussian Perturbations
  • proof
  • Lemma 3
  • ...and 13 more