Projection-free nonconvex stochastic optimization on Riemannian manifolds
Name
1910.04194.pdf
Description
Submitted version
Size
1.33 MB
Format
Adobe PDF
Checksum (MD5)
87bdb44b97ed532c08de97511833c943
Author(s)
Weber, Melanie
Sra, Suvrit
Date Issued
2021
Journal
IMA Journal of Numerical Analysis
Publisher
Oxford University Press (OUP)
Citation
Weber, Melanie and Sra, Suvrit. 2021. "Projection-free nonconvex stochastic optimization on Riemannian manifolds." IMA Journal of Numerical Analysis.
Version
Original manuscript
Abstract
Abstract
We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank–Wolfe (Fw) methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: the computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.
We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank–Wolfe (Fw) methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: the computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1093/IMANUM/DRAB066
