Publication Type

Journal Article

Version

acceptedVersion

Publication Date

8-2019

Abstract

First-order non-convex Riemannian optimization algorithms have gained recent popularity in structured machine learning problems including principal component analysis and low-rank matrix completion. The current paper presents an efficient Riemannian Stochastic Path Integrated Differential EstimatoR (R-SPIDER) algorithm to solve the finite-sum and online Riemannian non-convex minimization problems. At the core of R-SPIDER is a recursive semi-stochastic gradient estimator that can accurately estimate Riemannian gradient under not only exponential mapping and parallel transport, but also general retraction and vector transport operations. Compared with prior Riemannian algorithms, such a recursive gradient estimation mechanism endows R-SPIDER with higher computational efficiency in first-order oracle complexity. Specifically, for finite-sum problems with n components, R-SPIDER is proved to converge to an -accuracy stationary point within O min n + √n 2 , 1 3 stochastic gradient evaluations, beating the best-known complexity O n + 1 4 ; 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. For the special case of gradient dominated functions, we further develop a variant of R-SPIDER with improved linear rate of convergence. Extensive experimental results demonstrate the advantage of the proposed algorithms over the state-of-the-art Riemannian non-convex optimization methods.

Keywords

Riemannian Optimization, Stochastic Variance-Reduced Algorithm, Non-convex Optimization, Online Lear

Discipline

Theory and Algorithms

Research Areas

Intelligent Systems and Optimization

Areas of Excellence

Digital transformation

Publication

IEEE Transactions on Pattern Analysis and Machine Intelligence

Volume

43

Issue

2

First Page

459

Last Page

472

ISSN

0162-8828

Identifier

10.1109/TPAMI.2019.2933841

Publisher

Institute of Electrical and Electronics Engineers

Additional URL

https://doi.org/10.1109/TPAMI.2019.2933841

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