From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models
Abstract
Score-based generative models driven by α-stable Lévy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving α-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index α, whereas Brownian dynamics remains light-tailed. For a target with tail index ν, asymptotically correct extreme-quantile scaling is possible precisely when α=ν. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.
Related
- Non-asymptotic Convergence of Stochastic Gradient Descent in Score-based Generative Models
- Enhanced convergence guarantees of score-based generative models in 𝒲2-distance beyond log-concavity
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- Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing
BibTeX
@misc{hammani2026from,
title = {{From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models}},
author = {Ianis Hammani and Claire Boyer},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19773},
eprint = {2610.19773},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05790341v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models
AU - Ianis Hammani
AU - Claire Boyer
PY - 2026
DA - 2026/10/13
PB - HAL
DO - 10.48550/arxiv.2610.19773
UR - https://hal.science/hal-05790341v1
AB - Score-based generative models driven by α-stable Lévy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving α-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index α, whereas Brownian dynamics remains light-tailed. For a target with tail index ν, asymptotically correct extreme-quantile scaling is possible precisely when α=ν. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.
ER -