Faster first-order methods for stochastic non-convex optimization on Riemannian manifolds

• Main Authors: ZHOU, Pan, YUAN, Xiao-Tong, FENG, Jiashi
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2019
• Subjects: Graphics and Human Computer Interfaces
• Online Access: https://ink.library.smu.edu.sg/sis_research/9004; https://ink.library.smu.edu.sg/context/sis_research/article/10007/viewcontent/2019_AIS_Riemannian.pdf
• Institution: Singapore Management University

SPIDER (Stochastic Path Integrated Differential EstimatoR) is an efficient gradient estimation technique developed for non-convex stochastic optimization. Although having been shown to attain nearly optimal computational complexity bounds, the SPIDERtype methods are limited to linear metric spaces. In this paper, we introduce the Riemannian SPIDER (R-SPIDER) method as a novel nonlinear-metric extension of SPIDER for efficient non-convex optimization on Riemannian manifolds. We prove that for finitesum problems with n components, R-SPIDER converges to an -accuracy stationary point within O min n + √ n 2 , 1 3 stochastic gradient evaluations, which is sharper in magnitude than the prior Riemannian first-order methods. For online optimization, R-SPIDER is shown to converge with O 1 3 complexity which is, to the best of our knowledge, the first non-asymptotic result for online Riemannian optimization. Especially, for gradient dominated functions, we further develop a variant of R-SPIDER and prove its linear convergence rate. Numerical results demonstrate the computational efficiency of the proposed method