Riemannian Stochastic Recursive Gradient Algorithm with Retraction and Vector Transport and Its Convergence Analysis


Hiroyuki Kasai, Hiroyuki Sato, Bamdev Mishra ;
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2521-2529, 2018.


Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions on a Riemannian manifold. The present paper proposes a Riemannian stochastic recursive gradient algorithm (R-SRG), which does not require the inverse of retraction between two distant iterates on the manifold. Convergence analyses of R-SRG are performed on both retraction-convex and non-convex functions under computationally efficient retraction and vector transport operations. The key challenge is analysis of the influence of vector transport along the retraction curve. Numerical evaluations reveal that R-SRG competes well with state-of-the-art Riemannian batch and stochastic gradient algorithms.

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