Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions

2026-07-22

AI Summary

This work addresses the absence of optimal regret bounds for strongly geodesically convex loss functions over positively curved Riemannian manifolds in decentralized online optimization. The authors propose a decentralized online Riemannian gradient descent algorithm with time-varying step sizes, which carefully balances network error and convergence behavior. By leveraging strong geodesic convexity and smoothness properties, they establish—for the first time—an $O(\log T)$ static regret bound under both full-information and two-point bandit feedback settings. This result not only overcomes the limitation of existing methods that apply solely to merely geodesically convex functions but also matches the minimax-optimal rate known for strongly convex online optimization in Euclidean spaces.

Abstract

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 $O(\sqrt{T})$ 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.

Problem

• decentralized online optimization
• Riemannian manifolds
• strongly geodesically convex
• regret bound
• bandit feedback

Innovation

• decentralized online optimization
• Riemannian manifolds
• strong geodesic convexity
• time-varying step size
• bandit feedback