From Global to Tail Convergence: Dynamics of $\alpha$-Stable Score-Based Generative Models

arXiv:2610.19773v1 Announce Type: new Abstract: Score-based generative models driven by $\alpha$-stable L{\'e}vy 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 $\alpha$-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 $\alpha$, whereas Brownian dynamics remains light-tailed. For a target with tail index $\nu$, asymptotically correct extreme-quantile scaling is possible precisely when $\alpha$=$\nu$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.

Abstract:Score-based generative models driven by $\alpha$-stable L{é}vy 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 $\alpha$-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 $\alpha$, whereas Brownian dynamics remains light-tailed. For a target with tail index $\nu$, asymptotically correct extreme-quantile scaling is possible precisely when $\alpha$=$\nu$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.

44 pages, 4 figures, 3 tables. Code: this https URL

Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)

arXiv:2610.19773 [cs.LG]

(or arXiv:2610.19773v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.19773 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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From Global to Tail Convergence: Dynamics of $\alpha$-Stable Score-Based Generative Models

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