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This research establishes fundamental theoretical limits on parallelizing diffusion sampling, proving that certain sequential dependencies are unavoidable even with approximate score functions. By deriving polynomial lower bounds for specific distribution families, the study clarifies the inherent trade-offs between query complexity and generation rounds in high-dimensional spaces. These findings provide critical constraints for algorithm designers aiming to accelerate generative models without sacrificing accuracy.
Lower Bounds for Parallel Diffusion Sampling
arXiv:2610.09166v1 Announce Type: cross Abstract: Standard diffusion samplers generate samples through repeated evaluations of a learned score function. Parallel sampling methods seek to accelerate generation by trading additional evaluations for fewer sequential rounds. This raises the question of how much sequential dependence is unavoidable, even when many score queries can be made simultaneously. We establish the first polynomial parallel-round lower bounds for diffusion sampling with approximate scores. Specifically, we prove (1) a $\widetilde{\Omega}(d^{1/3})$-round lower bound for sampling smooth, near-isotropic Gaussian mixtures in $R^d$, and (2) an $\Omega(d)$-round lower bound for uniform sampling from anisotropic axis-aligned boxes contained in the unit ball. Both bounds hold for arbitrary randomized algorithms making polynomially many queries per round at arbitrary locations and noise levels, with inverse-polynomial score error and constant total variation accuracy. The linear bound is tight for our box family. Our constructions use fixed approximate score oracles that enforce sequential access to hidden information while satisfying the accuracy guarantee at every noise level.
Abstract:Standard diffusion samplers generate samples through repeated evaluations of a learned score function. Parallel sampling methods seek to accelerate generation by trading additional evaluations for fewer sequential rounds. This raises the question of how much sequential dependence is unavoidable, even when many score queries can be made simultaneously. We establish the first polynomial parallel-round lower bounds for diffusion sampling with approximate scores. Specifically, we prove (1) a $\widetilde{\Omega}(d^{1/3})$-round lower bound for sampling smooth, near-isotropic Gaussian mixtures in $R^d$, and (2) an $\Omega(d)$-round lower bound for uniform sampling from anisotropic axis-aligned boxes contained in the unit ball. Both bounds hold for arbitrary randomized algorithms making polynomially many queries per round at arbitrary locations and noise levels, with inverse-polynomial score error and constant total variation accuracy. The linear bound is tight for our box family. Our constructions use fixed approximate score oracles that enforce sequential access to hidden information while satisfying the accuracy guarantee at every noise level.
Data Structures and Algorithms (cs.DS); Artificial Intelligence (cs.AI); Computational Complexity (cs.CC); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.09166 [cs.DS]
(or arXiv:2610.09166v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2610.09166 arXiv-issued DOI via DataCite (pending registration)
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