From optimal score matching to optimal sampling

The recent, impressive advances in algorithmic generation of high-fidelity image, audio, and video are largely due to great successes in score-based diffusion models. A key implementing step is score matching, that is, the estimation of the score function of the forward diffusion process from training data. As shown in earlier literature, the total variation distance between the law of a sample generated from the trained diffusion model and the ground truth distribution can be controlled by the score matching risk. Despite the widespread use of score-based diffusion models, basic theoretical questions concerning exact optimal statistical rates for score estimation and its application to density estimation remain open. We establish the sharp minimax rate of score estimation for smooth, compactly supported densities. Formally, given $n$ i.i.d. samples from an unknown $\alpha$-H\"{o}lder density $f$ supported on $[-1, 1]$, we prove the minimax rate of estimating the score function of the diffused distribution $f * \mathcal{N}(0, t)$ with respect to the score matching loss is $\frac{1}{nt^2} \wedge \frac{1}{nt^{3/2}} \wedge (t^{\alpha-1} + n^{-2(\alpha-1)/(2\alpha+1)})$ for all $\alpha > 0$ and $t \ge 0$. As a consequence, it is shown the law $\hat{f}$ of a sample generated from the diffusion model achieves the sharp minimax rate $\bE(\dTV(\hat{f}, f)^2) \lesssim n^{-2\alpha/(2\alpha+1)}$ for all $\alpha > 0$ without any extraneous logarithmic terms which are prevalent in the literature, and without the need for early stopping which has been required for all existing procedures to the best of our knowledge.