Riemannian Online Optimistic Algorithms With Dynamic Regret

In this article, we consider Riemannian online convex optimization with dynamic regret, which involves minimizing the cumulative loss difference between a learner's decisions and a sequence of adaptive decisions across dynamic environments over time on Riemannian manifolds. First, we propose Riemannian online optimistic gradient descent (R-OOGD), which extends the classical optimistic algorithms to Riemannian manifolds, requiring one gradient inquiry per round. We then propose Riemannian adaptive online optimistic gradient descent (R-AOOGD) by incorporating the meta-expert framework into R-OOGD. We analyze the dynamic regret of R-OOGD and R-AOOGD in terms of the regularity of the sequence of cost functions and comparators. The dynamic regret bound matches the results in Euclidean space and with a scale adjustment of $\mathcal {O}\left(\frac{1+\kappa D^{2}}{\sqrt{1-KD^{2}}}\right)$ related to the sectional curvature lower bound $\kappa$, sectional curvature upper bound $K$ and feasible set diameter $D$. Finally, we provide numerical experiments to illustrate the performance of the proposed algorithms.