Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions

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 Õ (Equation presented) for the score function of p₀ ∗ N(0, tI_d), where n and d represent the sample size and the dimension, t is bounded above and below by polynomials of n, and p₀ is an arbitrary sub-Gaussian distribution.As a consequence, this yields an Õ (Equation presented) 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, p₀ belongs to the nonparametric family of the β-Sobolev space with β ≤ 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 p₀ in previous proofs of the minimax optimality of the diffusion model for nonparametric families.