  Score-based generative modeling (SGM) has grown to be a hugely successful
method for learning to generate samples from complex data distributions such as
that of images and audio. It is based on evolving an SDE that transforms white
noise into a sample from the learned distribution, using estimates of the score
function, or gradient log-pdf. Previous convergence analyses for these methods
have suffered either from strong assumptions on the data distribution or
exponential dependencies, and hence fail to give efficient guarantees for the
multimodal and non-smooth distributions that arise in practice and for which
good empirical performance is observed. We consider a popular kind of SGM --
denoising diffusion models -- and give polynomial convergence guarantees for
general data distributions, with no assumptions related to functional
inequalities or smoothness. Assuming $L^2$-accurate score estimates, we obtain
Wasserstein distance guarantees for any distribution of bounded support or
sufficiently decaying tails, as well as TV guarantees for distributions with
further smoothness assumptions.
