# Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions

arXiv: https://arxiv.org/abs/2607.20316 · HTML: https://mlsanity.com/paper/2607.20316/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 42 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 24 of 24)

- [Online Optimization : Competing with Dynamic Comparators](https://mlsanity.com/paper/1501.06225/) (2015)
- [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)
- [Online Distributed Optimization on Dynamic Networks](https://mlsanity.com/paper/1412.7215/) (2014)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Global Riemannian Acceleration in Hyperbolic and Spherical Spaces](https://mlsanity.com/paper/2012.03618/) (2020)
- [Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis](https://mlsanity.com/paper/2202.02036/) (2022)
- [Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds](https://mlsanity.com/paper/2304.08241/) (2023)
- [A Linearly Convergent Algorithm for Distributed Principal Component Analysis](https://mlsanity.com/paper/2101.01300/) (2021)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature](https://mlsanity.com/paper/1609.02901/) (2016)
- [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)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [Distributed Riemannian Optimization with Lazy Communication for Collaborative Geometric Estimation](https://mlsanity.com/paper/2203.00851/) (2022)
- [Riemannian Optimization for Active Mapping with Robot Teams](https://mlsanity.com/paper/2404.18321/) (2024)
- [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 Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking](https://mlsanity.com/paper/2506.07351/) (2025)
- [Distributed Riemannian Optimization in Geodesically Non-convex Environments](https://mlsanity.com/paper/2512.04915/) (2025)
- [Intrinsic Decentralized Stochastic Riemannian Optimization on Manifolds with Bounded Sectional Curvature](https://mlsanity.com/paper/2603.17096/) (2026)

## Built on by (most cited first, 1 of 1)

- [Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds](https://mlsanity.com/paper/2609.13646/) (2026)

## Abstract (arXiv)

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from O(√T) to O(log T), where T is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first O(log T) static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same O(log T) regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.