Implicit Riemannian Optimism with Applications to Min-Max Problems

Christophe Roux, David Martı́nez-Rubio, Sebastian Pokutta
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:52139-52172, 2025.

Abstract

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-roux25a, title = {Implicit {R}iemannian Optimism with Applications to Min-Max Problems}, author = {Roux, Christophe and Mart\'{\i}nez-Rubio, David and Pokutta, Sebastian}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {52139--52172}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/roux25a/roux25a.pdf}, url = {https://proceedings.mlr.press/v267/roux25a.html}, abstract = {We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.} }
Endnote
%0 Conference Paper %T Implicit Riemannian Optimism with Applications to Min-Max Problems %A Christophe Roux %A David Martı́nez-Rubio %A Sebastian Pokutta %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-roux25a %I PMLR %P 52139--52172 %U https://proceedings.mlr.press/v267/roux25a.html %V 267 %X We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.
APA
Roux, C., Martı́nez-Rubio, D. & Pokutta, S.. (2025). Implicit Riemannian Optimism with Applications to Min-Max Problems. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:52139-52172 Available from https://proceedings.mlr.press/v267/roux25a.html.

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