Analyzing the Error of Generative Diffusion Models: From Euler-Maruyama to Higher-Order Schemes

TL;DR

This paper provides a comprehensive theoretical analysis of discretization errors in generative diffusion models, demonstrating that higher-order schemes can outperform Euler-Maruyama methods in sampling accuracy.

Contribution

It establishes the first error bounds for higher-order SDE discretization methods in GDMs, extending convergence analysis beyond Euler-Maruyama schemes.

Findings

Higher-order schemes can outperform Euler-Maruyama in practice.

All-at-once error bounds are derived for EM method under various score-matching errors.

Numerical experiments confirm the theoretical advantage of higher-order discretization methods.

Abstract

Although generative diffusion models (GDMs) are widely used in practice, their theoretical foundations remain limited, especially concerning the impact of different discretization schemes applied to the underlying stochastic differential equation (SDE). Existing convergence analysis largely focuses on Euler-Maruyama (EM)-like methods and does not extend to higher-order schemes, which are naturally expected to provide improved discretization accuracy. In this paper, we establish asymptotic 2-Wasserstein convergence results for SDE-based discretization methods employed in sampling for GDMs. We provide an all-at-once error bound analysis of the EM method that accounts for all error sources and establish convergence under all prevalent score-matching error assumptions in the literature, assuming a strongly log-concave data distribution. Moreover, we present the first error bound result for arbitrary higher-order SDE-discretization methods with known strong L_2 convergence in dependence on the discretization grid and the score-matching error. Finally, we complement our theoretical findings with an extensive numerical study, providing comprehensive experimental evidence and showing that, contrary to popular believe, higher order discretization methods can in fact retain their theoretical advantage over EM for sampling GDMs.