First-Order Stationarity of Reverse Diffusions
Abstract
Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex---a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds---the sampling analog of averaged gradient-norm guarantees in nonconvex optimization---for samplers of both overdamped and underdamped diffusion models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.
Related
- Terminal Localization of Empirical Reverse Diffusions under Singular Score Errors
- Second-Order Stationarity with Common Random Losses: Matching Tolerance Bounds
- A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers
- Finding Simple Proofs for First-Order Optimization
- CRANDALL-PAZY RIGIDITY FOR C 1 CONVEX FUNCTIONS: A FIRST-ORDER MEDIAN-FLOW ARGUMENT
BibTeX
@misc{chen2026firstorder,
title = {{First-Order Stationarity of Reverse Diffusions}},
author = {Zhifeng Chen and Chenyang Jiang and Yazhen Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31612},
eprint = {2609.31612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31612},
}
RIS
TY - UNPB
TI - First-Order Stationarity of Reverse Diffusions
AU - Zhifeng Chen
AU - Chenyang Jiang
AU - Yazhen Wang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31612
UR - https://arxiv.org/abs/2609.31612
AB - Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex---a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds---the sampling analog of averaged gradient-norm guarantees in nonconvex optimization---for samplers of both overdamped and underdamped diffusion models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.
ER -