Score-based generative models are provably robust: an uncertainty quantification perspective

Score-based generative models (SGMs) are powerful tools for generating high-quality data samples, but their robustness to errors in practical implementation has been underexplored. Existing analyses often rely on strong assumptions, such as the manifold hypothesis and absolute continuity assumptions about the target distribution, which limits their applicability in real-world settings. Furthermore, existing generalization bounds are usually obtained indirectly which makes them hard to compute and interpret in practical settings.

This paper addresses these limitations by using an uncertainty quantification (UQ) perspective. It introduces a novel model-form UQ bound, the Wasserstein uncertainty propagation (WUP) theorem, which directly shows how L2 errors in learning the score function map to the Wasserstein-1 distance (a measure of discrepancy) between the true data distribution and that generated by the SGM. The WUP theorem establishes robust generalization bounds for SGMs under minimal assumptions. Importantly, these bounds capture the impact of different error sources (finite sample effects, early stopping, etc.) on the quality of the generated samples. The approach uses regularity theory of nonlinear PDEs to derive computable bounds which is agnostic to manifold hypothesis and does not rely on strong divergences.