Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training

TL;DR

This paper investigates the impact of heavy-tailed noise in diffusion models, revealing that it complicates estimation and may not enhance generative diversity as previously assumed.

Contribution

The study provides a theoretical and empirical analysis showing heavy-tailed noise worsens estimation errors in diffusion models, challenging recent design trends.

Findings

Heavy-tailed noise increases sampling-error bounds in diffusion models.

Experiments confirm that heavy-tailed noise leads to higher estimation errors.

Results question the effectiveness of heavy-tailed noise for rare-region exploration.

Abstract

Recent works have proposed incorporating heavy-tailed (HT) noise into diffusion- and flow-based generative models, with the goals of better recovering the tails of target distributions and improving generative diversity. This motivation is intuitive: if the data are heavy-tailed, HT noise may appear better matched than light-tailed (LT) Gaussian noise. However, replacing Gaussian noise by HT noise also changes the underlying estimation problem. In this paper, we revisit this paradigm through a combined theoretical and empirical study, establishing sampling-error bounds for two representative diffusion models driven by HT and LT noise. We show that HT noise makes the statistical estimation problem harder, leading to less favorable sampling-error bounds. We support these findings with experiments on synthetic and real-world datasets, empirically recovering the predicted error trade-off. Our results call into question a growing design trend in generative modeling and challenge the use of HT noise to improve rare-region exploration.