We investigate sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. The main contributions of this work are threefold: (i) We establish sharp upper bounds that are optimal in both dimension and the number of steps under general Lipschitz-type conditions on the score function and for a wide class of variance schedules. This result subsumes existing error bounds in the literature. (ii) We demonstrate that these general Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply that the DDPM satisfies a logarithmic Sobolev inequality, and consequently, a quadratic transportation cost inequality. As a result, an optimal Wasserstein bound for the DDPM (up to a logarithmic factor) follows from the recently obtained sharp error bound in the Kullback-Leibler (KL) divergence under geometric-type variance schedules. (iii) We show that the general log-concave setting serves as a case where the optimal Wasserstein error bound can still be achieved, even when the DDPM does not satisfy a quadratic transportation cost inequality.
Our proof technique relies on viewing the DDPM as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.