No-regret Online Learning over Riemannian ManifoldsDownload PDF

21 May 2021, 20:46 (edited 09 Nov 2021)NeurIPS 2021 PosterReaders: Everyone
  • Keywords: Online optimization, Riemannian manifolds, Riemannian optimization
  • TL;DR: We study online convex optimization over Riemannian manifolds, and establish tight regret bounds that match their Euclidean counterparts.
  • Abstract: We consider online optimization over Riemannian manifolds, where a learner attempts to minimize a sequence of time-varying loss functions defined on Riemannian manifolds. Though many Euclidean online convex optimization algorithms have been proven useful in a wide range of areas, less attention has been paid to their Riemannian counterparts. In this paper, we study Riemannian online gradient descent (R-OGD) on Hadamard manifolds for both geodesically convex and strongly geodesically convex loss functions, and Riemannian bandit algorithm (R-BAN) on Hadamard homogeneous manifolds for geodesically convex functions. We establish upper bounds on the regrets of the problem with respect to time horizon, manifold curvature, and manifold dimension. We also find a universal lower bound for the achievable regret by constructing an online convex optimization problem on Hadamard manifolds. All the obtained regret bounds match the corresponding results are provided in Euclidean spaces. Finally, some numerical experiments validate our theoretical results.
  • Supplementary Material: pdf
  • Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
  • Code: zip
12 Replies

Loading