  We provide theoretical convergence guarantees for score-based generative
models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which
constitute the backbone of large-scale real-world generative models such as
DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates,
such SGMs can efficiently sample from essentially any realistic data
distribution. In contrast to prior works, our results (1) hold for an
$L^2$-accurate score estimate (rather than $L^\infty$-accurate); (2) do not
require restrictive functional inequality conditions that preclude substantial
non-log-concavity; (3) scale polynomially in all relevant problem parameters;
and (4) match state-of-the-art complexity guarantees for discretization of the
Langevin diffusion, provided that the score error is sufficiently small. We
view this as strong theoretical justification for the empirical success of
SGMs. We also examine SGMs based on the critically damped Langevin diffusion
(CLD). Contrary to conventional wisdom, we provide evidence that the use of the
CLD does not reduce the complexity of SGMs.
