Nonconvex-nonconcave min-max optimization on Riemannian manifolds

Andi Han, Bamdev Mishra, Pratik Jawanpuria, Junbin Gao

Abstract: This work studies nonconvex-nonconcave min-max problems on Riemannian manifolds. We first characterize the local optimality of nonconvex-nonconcave problems on manifolds with a generalized notion of local minimax points. We then define the stability and convergence criteria of dynamical systems on manifolds and provide necessary and sufficient conditions of strictly stable equilibrium points for both continuous and discrete dynamics. Additionally, we propose several novel second-order methods on manifolds that provably converge to local minimax points asymptotically. We validate the empirical benefits of the proposed methods with extensive experiments.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

As per suggestions from Reviewer Aujm, we have added a schematic figure illustrating the process of Riemannian optimization, added textual guidance for Lemmas in Appendix, included a separate section in Appendix outlining the organization of the Appendix, expanded the conclusion section on future challenges.

We have modified Theorem 1 regarding Reviewer z28z's concerns. We have added references suggested by Reviewer z28z. We have modified the manuscript to improve clarity as commented by Reviewer z28z and yZgL. We have corrected typos.

We have revised for the camera ready version.

Assigned Action Editor: ~Zhihui_Zhu1

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 1039