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Riemannian gradient methods for stochastic composition problems.

Feihu Huang1, Shangqian Gao2

  • 1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China; Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA.

Neural Networks : the Official Journal of the International Neural Network Society
|June 26, 2022
PubMed
Summary
This summary is machine-generated.

This paper introduces new algorithms for stochastic composition optimization on Riemannian manifolds, crucial for machine learning. The proposed methods achieve improved sample complexity for finding optimal solutions.

Keywords:
Composition optimizationDeep neural networksGrassmann manifoldPrincipal component analysisRiemannian manifoldStiefel manifold

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Area of Science:

  • Optimization Theory
  • Machine Learning
  • Differential Geometry

Background:

  • Stochastic composition optimization problems are increasingly relevant in machine learning, particularly in distributionally robust learning on Riemannian manifolds.
  • Existing methods lack efficient solutions for these complex optimization tasks in non-Euclidean settings.

Purpose of the Study:

  • To introduce novel algorithms for solving stochastic composition optimization problems over Riemannian manifolds.
  • To analyze and improve the sample complexity of these optimization problems.

Main Methods:

  • Development of a Riemannian compositional gradient (RCG) algorithm.
  • Introduction of an accelerated momentum-based Riemannian compositional gradient (M-RCG) algorithm.
  • Theoretical analysis of sample complexity for both algorithms.

Main Results:

  • The RCG algorithm achieves a sample complexity of O(ϵ-4) for finding ϵ-stationary points.
  • The M-RCG algorithm achieves a near-optimal sample complexity of Õ(ϵ-3), outperforming existing methods.
  • Demonstrated effectiveness of the algorithms on deep neural networks (DNNs) over Stiefel manifold and principal component analysis (PCA) over Grassmann manifold.

Conclusions:

  • The proposed RCG and M-RCG algorithms are effective for stochastic composition optimization on Riemannian manifolds.
  • M-RCG offers the best known sample complexity for such problems, matching Euclidean counterparts.
  • This work pioneers the study of composition optimization on Riemannian manifolds.