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Novel convergence analyses are presented of Riemannian stochastic gradientdescent (RSGD) on a Hadamard manifold. RSGD is the most basic Riemannianstochastic optimization algorithm and is used in many applications in the fieldof machine learning. The analyses incorporate the concept of mini-batchlearning used in deep learning and overcome several problems in previousanalyses. Four types of convergence analysis are described for both constantand decreasing step sizes. The number of steps needed for RSGD convergence isshown to be a convex monotone decreasing function of the batch size.Application of RSGD with several batch sizes to a Riemannian stochasticoptimization problem on a symmetric positive definite manifold theoreticallyshows that increasing the batch size improves RSGD performance. Numericalevaluation of the relationship between batch size and RSGD performance providesevidence supporting the theoretical results.

Convergence of Riemannian Stochastic Gradient Descent on Hadamard Manifold