Sample-Efficient Training for Diffusion
Shivam Gupta Aditya Parulekar Eric Price Zhiyang Xun
Score-based diffusion models have become the most popular approach to deep
generative modeling of images, largely due to their empirical performance and
reliability. Recently, a number of theoretical works \citep{chen2022,
Chen2022ImprovedAO, Chenetal23flowode, benton2023linear} have shown that
diffusion models can efficiently sample, assuming $L^2$-accurate score
estimates. The score-matching objective naturally approximates the true score
in $L^2$, but the sample complexity of existing bounds depends
\emph{polynomially} on the data radius and desired Wasserstein accuracy. By
contrast, the time complexity of sampling is only logarithmic in these
parameters. We show that estimating the score in $L^2$ \emph{requires} this
polynomial dependence, but that a number of samples that scales
polylogarithmically in the Wasserstein accuracy actually do suffice for
sampling. We show that with a polylogarithmic number of samples, the ERM of the
score-matching objective is $L^2$ accurate on all but a probability $\delta$
fraction of the true distribution, and that this weaker guarantee is sufficient
for efficient sampling.