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

Benjamin Dupuis, Umut Simsekli
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:12087-12137, 2024.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-dupuis24a, title = {Generalization Bounds for Heavy-Tailed {SDE}s through the Fractional Fokker-Planck Equation}, author = {Dupuis, Benjamin and Simsekli, Umut}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {12087--12137}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/dupuis24a/dupuis24a.pdf}, url = {https://proceedings.mlr.press/v235/dupuis24a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Generalization Bounds for Heavy-Tailed SDEs through the Fractional Fokker-Planck Equation %A Benjamin Dupuis %A Umut Simsekli %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-dupuis24a %I PMLR %P 12087--12137 %U https://proceedings.mlr.press/v235/dupuis24a.html %V 235 %X 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.
APA
Dupuis, B. & Simsekli, U.. (2024). Generalization Bounds for Heavy-Tailed SDEs through the Fractional Fokker-Planck Equation. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:12087-12137 Available from https://proceedings.mlr.press/v235/dupuis24a.html.

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