Convergence of Riemannian Stochastic Gradient Descent on Hadamard Manifold

Novel convergence analyses are presented of Riemannian stochastic gradient descent (RSGD) on a Hadamard manifold. RSGD is the most basic Riemannian stochastic optimization algorithm and is used in many applications in the field of machine learning. The analyses incorporate the concept of mini-batch learning used in deep learning and overcome several problems in previous analyses. Four types of convergence analysis are described for both constant and decreasing step sizes. The number of steps needed for RSGD convergence is shown to be a convex monotone decreasing function of the batch size. Application of RSGD with several batch sizes to a Riemannian stochastic optimization problem on a symmetric positive definite manifold theoretically shows that increasing the batch size improves RSGD performance. Numerical evaluation of the relationship between batch size and RSGD performance provides evidence supporting the theoretical results.