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 O (n^-1 t^-d+2{2} (t^d{2} 1) ) for the score function of p₀*N (0, tId), 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 O (n^-1/2 t^-d{4}) 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 -Sobolev space with 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.