# Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds

arXiv: https://arxiv.org/abs/2609.13646 · HTML: https://mlsanity.com/paper/2609.13646/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Distributed Online Optimization in Dynamic Environments Using Mirror Descent](https://mlsanity.com/paper/1609.02845/) (2016)
- [Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity](https://mlsanity.com/paper/1806.06373/) (2018)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Policy Optimization over Submanifolds for Linearly Constrained Feedback Synthesis](https://mlsanity.com/paper/2201.11157/) (2022)
- [Decentralized Online Riemannian Optimization with Dynamic Environments](https://mlsanity.com/paper/2410.05128/) (2024)
- [Riemannian Projection-free Online Learning](https://mlsanity.com/paper/2305.19349/) (2023)
- [Implicit Riemannian Optimism with Applications to Min-Max Problems](https://mlsanity.com/paper/2501.18381/) (2025)
- [Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms](https://mlsanity.com/paper/2505.16970/) (2025)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)
- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)

## Abstract (arXiv)

This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fr'echet-mean consensus. For h-convex and strongly h-convex local objectives, we establish O(√T) and O(log T) static regret, respectively, matching the corresponding Euclidean rates with respect to T, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.