# Decentralized Online Riemannian Optimization with Dynamic Environments

arXiv: https://arxiv.org/abs/2410.05128 · HTML: https://mlsanity.com/paper/2410.05128/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2024.

## Built on by (most cited first, 5 of 5)

- [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)
- [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)

This paper develops the first decentralized online Riemannian optimization algorithm on Hadamard manifolds. Our algorithm, the decentralized projected Riemannian gradient descent, iteratively performs local updates using projected Riemannian gradient descent and a consensus step via weighted Frechet mean. Theoretically, we establish linear variance reduction for the consensus step. Building on this, we prove a dynamic regret bound of order cal O(√T(1+PT)/√(1-σ2(W))), where T is the time horizon, PT represents the path variation measuring nonstationarity, and σ2(W) measures the network connectivity. The weighted Frechet mean in our algorithm incurs a minimization problem, which can be computationally expensive. To further alleviate this cost, we propose a simplified consensus step with a closed-form, replacing the weighted Frechet mean. We then establish linear variance reduction for this alternative and prove that the decentralized algorithm, even with this simple consensus step, achieves the same dynamic regret bound. Finally, we validate our approach with experiments on nonstationary decentralized Frechet mean computation over hyperbolic spaces and the space of symmetric positive definite matrices, demonstrating the effectiveness of our methods.