# Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds

arXiv: https://arxiv.org/abs/2509.07779 · HTML: https://mlsanity.com/paper/2509.07779/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 51 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 29 of 29)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization](https://mlsanity.com/paper/1404.6264/) (2014)
- [On the Convergence of Decentralized Gradient Descent](https://mlsanity.com/paper/1310.7063/) (2013)
- [Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs](https://mlsanity.com/paper/1607.03218/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Decentralized Riemannian Gradient Descent on the Stiefel Manifold](https://mlsanity.com/paper/2102.07091/) (2021)
- [Distributed Online Optimization in Dynamic Environments Using Mirror Descent](https://mlsanity.com/paper/1609.02845/) (2016)
- [A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimization](https://mlsanity.com/paper/1910.10782/) (2019)
- [From Nesterov's Estimate Sequence to Riemannian Acceleration](https://mlsanity.com/paper/2001.08876/) (2020)
- [Multi-Agent Distributed Optimization via Inexact Consensus ADMM](https://mlsanity.com/paper/1402.6065/) (2014)
- [Online Distributed Optimization on Dynamic Networks](https://mlsanity.com/paper/1412.7215/) (2014)
- [An accelerated first-order method for non-convex optimization on manifolds](https://mlsanity.com/paper/2008.02252/) (2020)
- [Distributed Linearized Alternating Direction Method of Multipliers for Composite Convex Consensus Optimization](https://mlsanity.com/paper/1512.08122/) (2015)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds](https://mlsanity.com/paper/2304.08241/) (2023)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Retraction-Free Decentralized Non-convex Optimization with Orthogonal Constraints](https://mlsanity.com/paper/2405.11590/) (2024)
- [Riemannian Optimistic Algorithms](https://mlsanity.com/paper/2308.16004/) (2023)
- [Decentralized Gradient-Free Methods for Stochastic Non-Smooth Non-Convex Optimization](https://mlsanity.com/paper/2310.11973/) (2023)
- [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)
- [Online Optimization Perspective on First-Order and Zero-Order Decentralized Nonsmooth Nonconvex Stochastic Optimization](https://mlsanity.com/paper/2406.01484/) (2024)
- [Convergence of Riemannian Stochastic Gradient Descent on Hadamard Manifold](https://mlsanity.com/paper/2312.07990/) (2023)
- [Consensus on Lie groups for the Riemannian Center of Mass](https://mlsanity.com/paper/2308.08054/) (2023)
- [Achieving Consensus over Compact Submanifolds](https://mlsanity.com/paper/2306.04769/) (2023)

## Built on by (most cited first, 4 of 4)

- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)
- [Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds](https://mlsanity.com/paper/2609.13646/) (2026)

## Abstract (arXiv)

We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a O(√T) regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same O(√T) regret bound through the subconvexity analysis of smoothed objectives.