# Generalization Bounds for Heavy-Tailed SDEs through the Fractional Fokker-Planck Equation

arXiv: https://arxiv.org/abs/2402.07723 · HTML: https://mlsanity.com/paper/2402.07723/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2024.

## Built on by (most cited first, 5 of 5)

- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Privacy of SGD under Gaussian or Heavy-Tailed Noise: Guarantees without Gradient Clipping](https://mlsanity.com/paper/2403.02051/) (2024)
- [Generalization Bounds for Markov Algorithms through Entropy Flow Computations](https://mlsanity.com/paper/2502.07584/) (2025)
- [R'enyi Differential Privacy for Heavy-Tailed SDEs via Fractional Poincar'e Inequalities](https://mlsanity.com/paper/2511.15634/) (2025)
- [Tightening the Score Matching Gap for Diffusion Models](https://mlsanity.com/paper/2607.04442/) (2026)

## Abstract (arXiv)

Understanding the generalization properties of heavy-tailed stochastic optimization algorithms has attracted increasing attention over the past years. While illuminating interesting aspects of stochastic optimizers by using heavy-tailed stochastic differential equations as proxies, prior works either provided expected generalization bounds, or introduced non-computable information theoretic terms. Addressing these drawbacks, in this work, we prove high-probability generalization bounds for heavy-tailed SDEs which do not contain any nontrivial information theoretic terms. To achieve this goal, we develop new proof techniques based on estimating the entropy flows associated with the so-called fractional Fokker-Planck equation (a partial differential equation that governs the evolution of the distribution of the corresponding heavy-tailed SDE). In addition to obtaining high-probability bounds, we show that our bounds have a better dependence on the dimension of parameters as compared to prior art. Our results further identify a phase transition phenomenon, which suggests that heavy tails can be either beneficial or harmful depending on the problem structure. We support our theory with experiments conducted in a variety of settings.