Wasserstein bounds for denoising diffusion probabilistic models via the Föllmer process
Provides rigorous theoretical guarantees for DDPM sampling error, benefiting practitioners and theorists by clarifying optimal convergence rates under realistic assumptions.
This paper establishes sharp upper bounds on the 2-Wasserstein error for denoising diffusion probabilistic models under Lipschitz score conditions, achieving optimal dimension and step dependence. It also proves that these conditions imply a logarithmic Sobolev inequality, yielding optimal Wasserstein bounds from KL divergence results, and shows optimal bounds hold for general log-concave targets without transportation cost inequalities.
This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.