Nearly $d$-Linear Convergence Bounds for Diffusion Models via Stochastic Localization

TL;DR

This paper establishes nearly linear convergence bounds for diffusion models in high-dimensional data generation, requiring fewer steps than previous bounds and relying only on finite second moments of the data.

Contribution

It provides the first nearly linear in dimension convergence bounds for diffusion models under minimal assumptions, extending existing methods with a new error analysis approach.

Findings

Diffusion models need at most  O(d log^2(1/
))/^2) steps for high-dimensional data.

The bounds are nearly linear in the data dimension, improving over previous superlinear bounds.

The analysis introduces a refined discretization error treatment inspired by stochastic localization.

Abstract

Denoising diffusions are a powerful method to generate approximate samples from high-dimensional data distributions. Recent results provide polynomial bounds on their convergence rate, assuming $L^2$-accurate scores. Until now, the tightest bounds were either superlinear in the data dimension or required strong smoothness assumptions. We provide the first convergence bounds which are linear in the data dimension (up to logarithmic factors) assuming only finite second moments of the data distribution. We show that diffusion models require at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps to approximate an arbitrary distribution on $\mathbb{R}^d$ corrupted with Gaussian noise of variance $\delta$ to within $\varepsilon^2$ in KL divergence. Our proof extends the Girsanov-based methods of previous works. We introduce a refined treatment of the error from discretizing the reverse SDE inspired by stochastic localization.