Minimax Optimality of Score-based Diffusion Models: Beyond the Density
  Lower Bound Assumptions
Kaihong Zhang Heqi Yin Feng Liang Jingbo Liu
We study the asymptotic error of score-based diffusion model sampling in
large-sample scenarios from a non-parametric statistics perspective. We show
that a kernel-based score estimator achieves an optimal mean square error of
$\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$
for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and
$d$ represent the sample size and the dimension, $t$ is bounded above and below
by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As
a consequence, this yields an $\widetilde{O}\left(n^{-1/2}
t^{-\frac{d}{4}}\right)$ upper bound for the total variation error of the
distribution of the sample generated by the diffusion model under a mere
sub-Gaussian assumption. If in addition, $p_0$ belongs to the nonparametric
family of the $\beta$-Sobolev space with $\beta\le 2$, by adopting an early
stopping strategy, we obtain that the diffusion model is nearly (up to log
factors) minimax optimal. This removes the crucial lower bound assumption on
$p_0$ in previous proofs of the minimax optimality of the diffusion model for
nonparametric families.