# Intrinsic Decentralized Stochastic Riemannian Optimization on Manifolds with Bounded Sectional Curvature

arXiv: https://arxiv.org/abs/2603.17096 · HTML: https://mlsanity.com/paper/2603.17096/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2026.

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

- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)

## Abstract (arXiv)

Decentralized optimization on Riemannian manifolds is foundational for many modern machine learning and signal processing applications in which data are non-Euclidean and generated and processed in a distributed manner. Although intrinsic Riemannian methods exploit manifold geometry without relying on Euclidean embeddings, existing decentralized Riemannian optimization algorithms typically use constant step sizes and therefore converge only to a neighborhood of steady-state error. In this paper, we study the decentralized stochastic Riemannian gradient method in the diminishing step-size regime on manifolds with (possibly positive) bounded sectional curvature. We prove an O(1/T) bound for the network consensus error and an O(log T/√T) ergodic bound for the global optimality gap. To the best of our knowledge, this is the first exact, non-asymptotic optimality-gap guarantee for an intrinsic decentralized stochastic Riemannian method in the geodesically convex setting. Furthermore, the diminishing step-size schedule allows substantially larger initial gradient steps than fixed-step baselines, leading to better performance in practice. We illustrate this on the problem of distributed PCA over a Grassmann manifold.