# Riemannian Optimistic Algorithms

arXiv: https://arxiv.org/abs/2308.16004 · HTML: https://mlsanity.com/paper/2308.16004/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Aug 2023.

## Built on by (most cited first, 5 of 5)

- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Distribution-informed Online Conformal Prediction](https://mlsanity.com/paper/2512.07770/) (2025)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold](https://mlsanity.com/paper/2405.13392/) (2024)

## Abstract (arXiv)

In this paper, we consider Riemannian online convex optimization with dynamic regret. First, we propose two novel algorithms, namely the Riemannian Online Optimistic Gradient Descent (R-OOGD) and the Riemannian Adaptive Online Optimistic Gradient Descent (R-AOOGD), which combine the advantages of classical optimistic algorithms with the rich geometric properties of Riemannian manifolds. We analyze the dynamic regrets of the R-OOGD and R-AOOGD in terms of regularity of the sequence of cost functions and comparators. Next, we apply the R-OOGD to Riemannian zero-sum games, leading to the Riemannian Optimistic Gradient Descent Ascent algorithm (R-OGDA). We analyze the average iterate and best-iterate of the R-OGDA in seeking Nash equilibrium for a two-player, zero-sum, g-convex-concave games. We also prove the last-iterate convergence of the R-OGDA for g-strongly convex-strongly concave problems. Our theoretical analysis shows that all proposed algorithms achieve results in regret and convergence that match their counterparts in Euclidean spaces. Finally, we conduct several experiments to verify our theoretical findings.