  Diffusion models, which convert noise into new data instances by learning to
reverse a Markov diffusion process, have become a cornerstone in contemporary
generative modeling. While their practical power has now been widely
recognized, the theoretical underpinnings remain far from mature. In this work,
we develop a suite of non-asymptotic theory towards understanding the data
generation process of diffusion models in discrete time, assuming access to
$\ell_2$-accurate estimates of the (Stein) score functions. For a popular
deterministic sampler (based on the probability flow ODE), we establish a
convergence rate proportional to $1/T$ (with $T$ the total number of steps),
improving upon past results; for another mainstream stochastic sampler (i.e., a
type of the denoising diffusion probabilistic model), we derive a convergence
rate proportional to $1/\sqrt{T}$, matching the state-of-the-art theory.
Imposing only minimal assumptions on the target data distribution (e.g., no
smoothness assumption is imposed), our results characterize how $\ell_2$ score
estimation errors affect the quality of the data generation processes. In
contrast to prior works, our theory is developed based on an elementary yet
versatile non-asymptotic approach without resorting to toolboxes for SDEs and
ODEs. Further, we design two accelerated variants, improving the convergence to
$1/T^2$ for the ODE-based sampler and $1/T$ for the DDPM-type sampler, which
might be of independent theoretical and empirical interest.
