Riemannian Online Learning

<p>Riemannian optimization is a powerful tool for decision-making in situations where the data and decision space are structured as non-flat spaces due to physical constraints and/or underlying symmetries. In emerging fields such as machine learning quantum computing biomedical imaging and robotics data and decisions often exist in curved non-Euclidean spaces due to physical constraints or underlying symmetries. Riemannian online optimization provides a new framework for handling learning tasks where data arrives sequentially in geometric spaces.</p><p></p><p>This monograph offers a comprehensive overview of online learning over Riemannian manifolds and offers a unified overview of the state-of-the-art algorithms for online optimization over Riemannian manifolds. Also presented is a detailed and systematic analysis of achievable regret for those algorithms. The study emphasizes how the curvature of manifolds influences the trade-off between exploration and exploitation and the performance of the algorithms.</p><p></p><p>After an introduction Section 2 briefly introduces Riemannian manifolds together with the preliminary knowledge of Riemannian optimization and Euclidean online optimization. In Section 3 the fundamental Riemannian online gradient descent algorithm under full information feedback is presented and the achievable regret on both Hadamard manifolds and general manifolds is analyzed. Section 4 extends the Riemannian online gradient descent algorithm to the bandit feedback setting. In Sections 5 and 6 the authors turn to two advanced Riemannian online optimization algorithms designed for dynamic regret minimization the Riemannian online extra gradient descent and the Riemannian online optimistic gradient descent.</p>