Iterating Consistency Models: Stability, Error Bounds and Noise Schedules
Abstract
Consistency models (CMs) have become a leading approach for generating high-quality samples in few steps. However, adding steps can improve or degrade sample quality in ways that are highly sensitive to the schedule and that existing theory does not fully explain. To provide accuracy guarantees and guide CM sampler design, we analyze multistep CM sampling as a composition of noising and approximate denoising operators. Under explicit, verifiable stability assumptions, we derive a non-asymptotic error bound that separates contraction of the initialization error from accumulation of approximation error. The bound assigns distinct roles to the schedule: large early noise levels drive contraction, while small late noise levels control the residual bias. As a corollary, we obtain explicit constants for strongly log-concave and semi-log-concave targets. We further establish a complementary guarantee whose assumptions, one-step accuracy and stability, can be estimated for a given trained model. Experiments show that the contraction and approximation profiles entering our bounds can be reliably measured and closely match the predicted functional forms. Together, these results provide a meaningful convergence theory for multi-step CMs and a practical route to sampler design.
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BibTeX
@misc{spagnoletti2026iterating,
title = {{Iterating Consistency Models: Stability, Error Bounds and Noise Schedules}},
author = {Alessio Spagnoletti and Abdul-Lateef Haji-Ali and Andrés Almansa and Alain Oliviero Durmus and Eric Moulines and Marcelo Pereyra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03414},
eprint = {2610.03414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03414},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Iterating Consistency Models: Stability, Error Bounds and Noise Schedules
AU - Alessio Spagnoletti
AU - Abdul-Lateef Haji-Ali
AU - Andrés Almansa
AU - Alain Oliviero Durmus
AU - Eric Moulines
AU - Marcelo Pereyra
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03414
UR - https://arxiv.org/abs/2610.03414
AB - Consistency models (CMs) have become a leading approach for generating high-quality samples in few steps. However, adding steps can improve or degrade sample quality in ways that are highly sensitive to the schedule and that existing theory does not fully explain. To provide accuracy guarantees and guide CM sampler design, we analyze multistep CM sampling as a composition of noising and approximate denoising operators. Under explicit, verifiable stability assumptions, we derive a non-asymptotic error bound that separates contraction of the initialization error from accumulation of approximation error. The bound assigns distinct roles to the schedule: large early noise levels drive contraction, while small late noise levels control the residual bias. As a corollary, we obtain explicit constants for strongly log-concave and semi-log-concave targets. We further establish a complementary guarantee whose assumptions, one-step accuracy and stability, can be estimated for a given trained model. Experiments show that the contraction and approximation profiles entering our bounds can be reliably measured and closely match the predicted functional forms. Together, these results provide a meaningful convergence theory for multi-step CMs and a practical route to sampler design.
ER -