On the Robustness of Langevin Dynamics to Score Function Error
arXiv:2603.11319v1 Announce Type: new Abstract: 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 d
arXiv:2603.11319v1 Announce Type: new Abstract: 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.
Executive Summary
This article examines the robustness of Langevin dynamics to score function errors in generative modeling. The authors demonstrate that Langevin dynamics is not robust to L^2 errors in the estimate of the score function, even for simple distributions in high dimensions. This finding has significant implications for the use of Langevin dynamics in practice, particularly when learning the score function from data. The authors' results provide further justification for diffusion models over Langevin dynamics and serve as a caution against the use of Langevin dynamics with estimated scores. The article contributes to our understanding of the limitations of Langevin dynamics and highlights the importance of considering score function errors in generative modeling.
Key Points
- ▸ Langevin dynamics is not robust to L^2 errors in the estimate of the score function.
- ▸ Even small L^2 errors in the estimate of the score function can lead to significant deviations from the target distribution.
- ▸ The authors' results provide further justification for diffusion models over Langevin dynamics.
Merits
Strength
The article provides a rigorous and well-motivated analysis of the robustness of Langevin dynamics to score function errors.
Strength
The authors' results have significant implications for the use of Langevin dynamics in practice.
Demerits
Limitation
The article assumes a specific form of the score function, which may not be generalizable to all cases.
Expert Commentary
The article provides a thorough and well-motivated analysis of the robustness of Langevin dynamics to score function errors. The authors' results are significant and have important implications for the use of Langevin dynamics in practice. However, as with any research article, there are limitations to consider. The article assumes a specific form of the score function, which may not be generalizable to all cases. Nevertheless, the article makes a valuable contribution to our understanding of the limitations of Langevin dynamics and highlights the importance of considering score function errors in generative modeling.
Recommendations
- ✓ Future research should investigate the robustness of Langevin dynamics to score function errors in more general settings, including non-isotropic distributions and non-Gaussian noise.
- ✓ The development of new policies and guidelines for the use of generative models in machine learning and data science should take into account the results of this article.