Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds

2025-09-09

AI Summary

This paper addresses decentralized online optimization on Riemannian manifolds of positive curvature—extending beyond the conventional restriction to Hadamard (nonpositively curved) manifolds. The core challenge lies in the absence of global geodesic convexity of distance functions under positive curvature, which impedes consensus convergence. To overcome this, we propose a curvature-aware decentralized consensus protocol, achieving linear convergence of consensus steps on non-Hadamard manifolds for the first time. Integrated with Riemannian gradient descent, two-point bandit gradient estimation, and smoothing techniques, our framework establishes a sub-convexity analysis and attains an $O(sqrt{T})$ dynamic regret bound. Experiments validate both theoretical guarantees and computational efficiency on positively curved manifolds. This work provides the first provably convergent decentralized Riemannian learning framework applicable to general positive-curvature settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 $O(sqrt{T})$ 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 $O(sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives.

Problem

• Decentralized online optimization on positively curved manifolds
• Overcoming non-convexity from geodesic distances in Riemannian spaces
• Achieving sublinear regret with bandit feedback and efficient estimators

Innovation

• Curvature-aware Riemannian consensus for non-Hadamard manifolds
• Decentralized online Riemannian gradient descent with O(√T) regret
• Bandit feedback with efficient gradient estimators via smoothing