On the Robustness of Langevin Dynamics to Score Function Error

We consider the robustness of score-based generative modeling to errors in the estimate of the score function. In particular, we show that Langevin dynamics is not robust to the L 2 errors (more generally L p errors) in the estimate of the score function. It is well-established that with small L 2 errors in the estimate of the score function, diffusion models can sample faithfully from the target distribution under fairly mild regularity assumptions in a polynomial time horizon. In contrast, our work shows that even for simple distributions in high dimensions, Langevin dynamics run for any polynomial time horizon will produce a distribution far from the target distribution in Total Variation (TV) distance, even when the L 2 error (more generally L p ) of the estimate of the score function is arbitrarily small. Considering such an error in the estimate of the score function is unavoidable in practice when learning the score function from data, our results provide further justification for diffusion models over Langevin dynamics and serve to caution against the use of Langevin dynamics with estimated scores. Introduction Many sampling algorithms ubiquitous in statistics and Machine Learning (ML) -used from Bayesian inference to modern generative modeling -are score-based sampling algorithms. These sampling algorithms are widely used in a range of scientific and engineering applications, such as image generation (see e.g. Croitoru et al. (2023)), inverse problems in applied mathematics (e.g. Sanz-Alonso et al. (2023)), physical sciences (e.g. Zheng et al. (2025)), protein design and computational biology (e.g. Guo et al. (2024)), and medical image reconstruction (e.g. Chung et al. (2022)). Many such score-based or related sampling algorithms exist. An incomplete list includes Langevin dynamics (see e.g. Welling and Teh (2011); Durmus et al. (2018); Chewi (2025)), diffusion models (e.g. Ho et al. (2020); Song and Ermon (2019); Song et al. (2021a); Song and Ermon (2020); Song et al. (2021b) for some foundational early works), flow matching (e.g. Lipman et al. (2023)), and stochastic interpolants (e.g. Albergo et al. ( 2025 )). Among these sampling algorithms, we investigate Langevin dynamics and diffusion models. Langevin dynamics is a classical technique for sampling from target distributions in statistics and ML, while diffusion models are a modern, popular and effective method for generative AI. At a high level, these algorithms approximately sample from a target distribution π tar in R d by running a stochastic process driven by the score function ∇ log π tar in the case of Langevin dynamics, or a suitable sequence of score functions ∇ log π 0 , ∇ log π 1 , . . . , ∇ log π k (the annealed score functions) in the case of diffusion models, where π 0 ≈ π tar and π 0 , . . . , π k forms a gradually noised version of the target distribution π tar . The continuoustime idealization of both these processes converges to π tar under mild assumptions on π tar , see Section 2.