mlsanityData to 14 Oct 2026

Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives

arXiv 2509.11236 ↗First posted 14 Sep 2025cs.LG math.OC stat.MLcited by 3

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Abstract (arXiv)

We study online Riemannian optimization on Hadamard manifolds under the framework of horospherical convexity (h-convexity). Prior work mostly relies on the geodesic convexity (g-convexity), leading to regret bounds scaling poorly with the manifold curvature. To address this limitation, we analyze Riemannian online gradient descent for h-convex and strongly h-convex functions and establish O(√T) and O(log(T)) regret guarantees, respectively. These bounds are curvature-independent and match the results in the Euclidean setting. We validate our approach with experiments on the manifold of symmetric positive definite (SPD) matrices equipped with the affine-invariant metric. In particular, we investigate online Tyler's M-estimation and online Fr'echet mean computation, showing the application of h-convexity in practice.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 20 of its 35 references that are in the map. First posted on arXiv 14 Sep 2025.

Builds on

20 of 35 references.

References · 20

  1. 2019Introduction to Online Convex Optimizationcited by 803
  2. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 684
  3. 2019Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Groupcited by 134
  4. 2016First-order Methods for Geodesically Convex Optimizationcited by 93
  5. 2016Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifoldscited by 64
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  7. 2019A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimizationcited by 25
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  11. 2024Retraction-Free Decentralized Non-convex Optimization with Orthogonal Constraintscited by 6
  12. 2022Policy Optimization over Submanifolds for Linearly Constrained Feedback Synthesiscited by 5
8 more
  1. 2024Decentralized Online Riemannian Optimization with Dynamic Environmentscited by 5
  2. 2023Riemannian Projection-free Online Learningcited by 4
  3. 2025Implicit Riemannian Optimism with Applications to Min-Max Problemscited by 4
  4. 2025Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithmscited by 4
  5. 2025Decentralized Online Riemannian Optimization Beyond Hadamard Manifoldscited by 4
  6. 2022Convex analysis on Hadamard spaces and scaling problemscited by 2
  7. 2022Riemannian optimization on unit sphere with p-norm and its applicationscited by 1
  8. 2024Horoballs and the subgradient methodcited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifoldscited by 2
  2. 2026Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functionscited by 1
  3. 2026Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifoldscited by 0