arXiv：cs.LG· Ianis Hammani (ENS Paris Saclay, LMO), Claire Boyer (LMO, IUF)·· 3 小时前
α-Stable 分数生成模型从全局到尾部收敛的动力学分析
From Global to Tail Convergence: Dynamics of $\alpha$-Stable Score-Based Generative Models
AI 导读
研究为尺度保持的 α-stable 生成动力学建立了非渐近收敛分析，涵盖紧支撑与幂律目标。在 p-Wasserstein 距离下，量化了早停、初始化、分数近似与时间离散化的影响，并证明初始化误差呈指数衰减。与布朗动力学的对比显示，全局收敛不区分两者，但稳定动力学在任意正生成时间产生指数为 α 的幂律尾部，当 α=ν 时可正确刻画极值分位数。
正文
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.
| Comments: | 44 pages, 4 figures, 3 tables. Code: this https URL |
| Subjects: | Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.19773 [cs.LG] |
| (or arXiv:2610.19773v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.19773 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Claire Boyer [view email] [via CCSD proxy]
[v1]
Wed, 14 Oct 2026 08:23:35 UTC (312 KB)
来源：arXiv：cs.LG · arxiv.org