From Proximal Point Method to Accelerated Methods on Riemannian Manifolds

Recently, there has been significant effort to generalize successful ideas in Euclidean optimization to Riemannian optimization. However, one landmark result of Euclidean optimization has eluded the Riemannian setting: namely, a Riemannian analog of Nesterov's accelerated gradient method (AGM). In this thesis, we establish the first globally accelerated gradient method for Riemannian manifolds. Toward establishing our result, the first part of the thesis revisits Nesterov's AGM and develops a conceptually simple understanding of it based on the proximal point method (PPM). The main observation is that AGM is in fact an approximation of PPM, which results in simple derivations and analyses of different versions of AGM. The second part of the thesis then extends our simple approach to the Riemannian case. In our extension, we handle a technical hurdle inherent to the Riemannian case by introducing an appropriate notion of ``metric distortion.'' We control this distortion via a novel geometric inequality, which enables us to formulate and analyze global Riemannian acceleration.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2021

Author dc:creator

• Ahn, Kwangjun

Advisor dc:contributor.advisor

• Sra, Suvrit

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright MIT

Licence dc:rights.uri

• http://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/139219

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/139219