  Diffusion models are a powerful method for generating approximate samples
from high-dimensional data distributions. Several recent results have provided
polynomial bounds on the convergence rate of such models, assuming
$L^2$-accurate score estimators. However, up until now the best known such
bounds were either superlinear in the data dimension or required strong
smoothness assumptions. We provide the first convergence bounds which are
linear in the data dimension (up to logarithmic factors) assuming only finite
second moments of the data distribution. We show that diffusion models require
at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps to
approximate an arbitrary data distribution on $\mathbb{R}^d$ corrupted with
Gaussian noise of variance $\delta$ to within $\varepsilon^2$ in
Kullback--Leibler divergence. Our proof builds on the Girsanov-based methods of
previous works. We introduce a refined treatment of the error arising from the
discretization of the reverse SDE, which is based on tools from stochastic
localization.

Machine Learning;Machine Learning