Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds

Generative modeling aims to produce new random examples from an unknown target distribution, given access to a finite collection of examples. Among the leading approaches, denoising diffusion probabilistic models (DDPMs) construct such examples by mapping a Brownian motion via a diffusion process driven by an estimated score function. In this work, we first provide empirical evidence that DDPMs are robust to constant-variance noise in the score evaluations. We then establish finite-sample guarantees in Wasserstein-2 distance that exhibit two key features: (i) they characterize and quantify the robustness of DDPMs to noisy score estimates, and (ii) they achieve faster convergence rates than previously known results. Furthermore, we observe that the obtained rates match those known in the Gaussian case, implying their optimality.

Guaranteed Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution

Generative modeling is a widely-used machine learning method with various applications in scientific and industrial fields. Its primary objective is to simulate new examples drawn from an unknown distribution given training data while ensuring diversity and avoiding replication of examples from the training data. This paper presents theoretical insights into training a generative model with two properties: (i) the error of replacing the true data-generating distribution with the trained data-generating distribution should optimally converge to zero as the sample size approaches infinity, and (ii) the trained data-generating distribution should be far enough from any distribution replicating examples in the training data. We provide non-asymptotic results in the form of finite sample risk bounds that quantify these properties and depend on relevant parameters such as sample size, the dimension of the ambient space, and the dimension of the latent space. Our results are applicable to general integral probability metrics used to quantify errors in probability distribution spaces, with the Wasserstein-$1$ distance being the central example. We also include numerical examples to illustrate our theoretical findings.