Wasserstein bounds for denoising diffusion probabilistic models via the F\"ollmer process

TL;DR

This paper derives sharp, dimension- and step-optimal Wasserstein distance bounds for denoising diffusion models, linking score function conditions to transportation inequalities and analyzing the Föllmer process as a sampler.

Contribution

It establishes new optimal error bounds under broad conditions, connects score function properties to transportation inequalities, and analyzes the Föllmer process for sampling.

Findings

Sharp upper bounds for sampling error in Wasserstein distance.

Lipschitz conditions imply transportation inequalities for DDPMs.

Optimal Wasserstein bounds are achievable for log-concave targets without transportation inequalities.

Abstract

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 F\"ollmer process rather than the conventional reverse Ornstein-Uhlenbeck process.