Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance

TL;DR

This paper establishes new convergence guarantees in Wasserstein distance for both stochastic and deterministic diffusion model samplers, providing a unified analysis framework and novel bounds for popular algorithms.

Contribution

It introduces a simple analysis framework for diffusion sampler convergence, deriving the first Wasserstein bound for the Heun sampler and improving bounds for the Euler sampler.

Findings

First Wasserstein convergence bound for Heun sampler

Improved convergence bounds for Euler sampler

Highlights importance of score function regularity

Abstract

We provide new convergence guarantees in Wasserstein distance for diffusion-based generative models, covering both stochastic (DDPM-like) and deterministic (DDIM-like) sampling methods. We introduce a simple framework to analyze discretization, initialization, and score estimation errors. Notably, we derive the first Wasserstein convergence bound for the Heun sampler and improve existing results for the Euler sampler of the probability flow ODE. Our analysis emphasizes the importance of spatial regularity of the learned score function and argues for controlling the score error with respect to the true reverse process, in line with denoising score matching. We also incorporate recent results on smoothed Wasserstein distances to sharpen initialization error bounds.