  Denoising diffusion probabilistic models (DDPMs) are a class of powerful
generative models. The past few years have witnessed the great success of DDPMs
in generating high-fidelity samples. A significant limitation of the DDPMs is
the slow sampling procedure. DDPMs generally need hundreds or thousands of
sequential function evaluations (steps) of neural networks to generate a
sample. This paper aims to develop a fast sampling method for DDPMs requiring
much fewer steps while retaining high sample quality. The inference process of
DDPMs approximates solving the corresponding diffusion ordinary differential
equations (diffusion ODEs) in the continuous limit. This work analyzes how the
backward error affects the diffusion ODEs and the sample quality in DDPMs. We
propose fast sampling through the \textbf{Restricting Backward Error schedule
(RBE schedule)} based on dynamically moderating the long-time backward error.
Our method accelerates DDPMs without any further training. Our experiments show
that sampling with an RBE schedule generates high-quality samples within only 8
to 20 function evaluations on various benchmark datasets. We achieved 12.01 FID
in 8 function evaluations on the ImageNet $128\times128$, and a $20\times$
speedup compared with previous baseline samplers.
