Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions

• URL: http://arxiv.org/abs/2607.20316v1
• Date: Wed, 22 Jul 2026 16:03:43 GMT
• Title: Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions
• Abstract summary: We develop a new type of online optimization for strongly geodesically convex (strongly g-) losses on bounded curved curvature.<n>We prove the first analysis for the two strong-point bandit feedback setting setting the smoothed versions of the bound loss functions.
• Score: 9.035866986287205
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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