Publication Type

Conference Proceeding Article

Version

acceptedVersion

Publication Date

4-2019

Abstract

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

Discipline

Graphics and Human Computer Interfaces

Research Areas

Intelligent Systems and Optimization

Areas of Excellence

Digital transformation

Publication

Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, Naha, Okinawa, Japan, 2019 April 16-18

First Page

1

Last Page

20

Publisher

Proceedings of Machine Learning Research

City or Country

Naha, Okinawa, Japan

Additional URL

https://proceedings.mlr.press/v89/zhou19a.html

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