How to Set $\beta_1, \beta_2$ in Adam: An Online Learning Perspective

Quan M. Nguyen
Proceedings of The 37th International Conference on Algorithmic Learning Theory, PMLR 313:1-16, 2026.

Abstract

While Adam is one of the most effective optimizer for training large-scale machine learning models, a theoretical understanding of how to optimally set its momentum factors, $\beta_1$ and $\beta_2$, remains largely incomplete. Prior works have shown that Adam can be seen as an instance of Follow-the-Regularized-Leader (FTRL), one of the most important class of algorithms in online learning. The prior analyses in these works required setting $\beta_1 = \sqrt{\beta_2}$, which does not cover the more practical cases with $\beta_1 \neq \sqrt{\beta_2}$. We derive novel, more general analyses that hold for both $\beta_1 \geq \sqrt{\beta_2}$ and $\beta_1 \leq \sqrt{\beta_2}$. In both cases, our results strictly generalize the existing bounds. Furthermore, we show that our bounds are tight in the worst case. We also prove that setting $\beta_1 = \sqrt{\beta_2}$ is optimal for an oblivious adversary, but sub-optimal for an non-oblivious adversary.

Cite this Paper

BibTeX
@InProceedings{pmlr-v313-nguyen26b, title = {How to Set $\beta_1, \beta_2$ in Adam: An Online Learning Perspective}, author = {Nguyen, Quan M.}, booktitle = {Proceedings of The 37th International Conference on Algorithmic Learning Theory}, pages = {1--16}, year = {2026}, editor = {Telgarsky, Matus and Ullman, Jonathan}, volume = {313}, series = {Proceedings of Machine Learning Research}, month = {23--26 Feb}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v313/main/assets/nguyen26b/nguyen26b.pdf}, url = {https://proceedings.mlr.press/v313/nguyen26b.html}, abstract = {While Adam is one of the most effective optimizer for training large-scale machine learning models, a theoretical understanding of how to optimally set its momentum factors, $\beta_1$ and $\beta_2$, remains largely incomplete. Prior works have shown that Adam can be seen as an instance of Follow-the-Regularized-Leader (FTRL), one of the most important class of algorithms in online learning. The prior analyses in these works required setting $\beta_1 = \sqrt{\beta_2}$, which does not cover the more practical cases with $\beta_1 \neq \sqrt{\beta_2}$. We derive novel, more general analyses that hold for both $\beta_1 \geq \sqrt{\beta_2}$ and $\beta_1 \leq \sqrt{\beta_2}$. In both cases, our results strictly generalize the existing bounds. Furthermore, we show that our bounds are tight in the worst case. We also prove that setting $\beta_1 = \sqrt{\beta_2}$ is optimal for an oblivious adversary, but sub-optimal for an non-oblivious adversary.} }
Endnote
%0 Conference Paper %T How to Set $\beta_1, \beta_2$ in Adam: An Online Learning Perspective %A Quan M. Nguyen %B Proceedings of The 37th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Matus Telgarsky %E Jonathan Ullman %F pmlr-v313-nguyen26b %I PMLR %P 1--16 %U https://proceedings.mlr.press/v313/nguyen26b.html %V 313 %X While Adam is one of the most effective optimizer for training large-scale machine learning models, a theoretical understanding of how to optimally set its momentum factors, $\beta_1$ and $\beta_2$, remains largely incomplete. Prior works have shown that Adam can be seen as an instance of Follow-the-Regularized-Leader (FTRL), one of the most important class of algorithms in online learning. The prior analyses in these works required setting $\beta_1 = \sqrt{\beta_2}$, which does not cover the more practical cases with $\beta_1 \neq \sqrt{\beta_2}$. We derive novel, more general analyses that hold for both $\beta_1 \geq \sqrt{\beta_2}$ and $\beta_1 \leq \sqrt{\beta_2}$. In both cases, our results strictly generalize the existing bounds. Furthermore, we show that our bounds are tight in the worst case. We also prove that setting $\beta_1 = \sqrt{\beta_2}$ is optimal for an oblivious adversary, but sub-optimal for an non-oblivious adversary.
APA
Nguyen, Q.M.. (2026). How to Set $\beta_1, \beta_2$ in Adam: An Online Learning Perspective. Proceedings of The 37th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 313:1-16 Available from https://proceedings.mlr.press/v313/nguyen26b.html.

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