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Modified Armijo line search in optimization on Riemannian submanifolds with reduced computational cost

Hiroyuki Sato, Yuya Yamakawa, Kensuke Aihara

TL;DR

This paper addresses the high cost of Riemannian Armijo line searches on embedded submanifolds by introducing a modified Armijo strategy that tests trial steps in the ambient Euclidean space via $\mathbf{x}_k + \alpha\mathbf{p}_k$ and only computes the retraction when necessary, ensuring a true Riemannian Armijo condition holds afterward. The authors develop a class of optimization methods, including steepest descent and Newton variants, equipped with this Euclidean Armijo-based test and prove their global convergence under standard assumptions. Numerical experiments on the sphere, Stiefel, and SPD manifolds demonstrate substantial reductions in retraction evaluations and overall runtime, especially for large-scale problems, while maintaining convergence guarantees. The approach reduces computational overhead without sacrificing robustness, making Riemannian optimization more scalable for manifold-constrained problems.

Abstract

For optimization problems on Riemannian manifolds, many types of globally convergent algorithms have been proposed, and they are often equipped with the Riemannian version of the Armijo line search for global convergence. Such existing methods need to compute the value of a retraction mapping regarding the search direction several times at each iteration; this may result in high computational costs, particularly if computing the value of the retraction is expensive. To address this issue, this study focuses on Riemannian submanifolds of the Euclidean spaces and proposes a novel Riemannian line search that achieves lower computational cost by incorporating a new strategy that computes the retraction only when inevitable. A class of Riemannian optimization algorithms, including the steepest descent and Newton methods, with the new line search strategy is proposed and proved to be globally convergent. Furthermore, numerical experiments on solving optimization problems on several types of Riemannian submanifolds illustrate that the proposed methods are superior to the standard Riemannian Armijo line search-based methods.

Modified Armijo line search in optimization on Riemannian submanifolds with reduced computational cost

TL;DR

This paper addresses the high cost of Riemannian Armijo line searches on embedded submanifolds by introducing a modified Armijo strategy that tests trial steps in the ambient Euclidean space via and only computes the retraction when necessary, ensuring a true Riemannian Armijo condition holds afterward. The authors develop a class of optimization methods, including steepest descent and Newton variants, equipped with this Euclidean Armijo-based test and prove their global convergence under standard assumptions. Numerical experiments on the sphere, Stiefel, and SPD manifolds demonstrate substantial reductions in retraction evaluations and overall runtime, especially for large-scale problems, while maintaining convergence guarantees. The approach reduces computational overhead without sacrificing robustness, making Riemannian optimization more scalable for manifold-constrained problems.

Abstract

For optimization problems on Riemannian manifolds, many types of globally convergent algorithms have been proposed, and they are often equipped with the Riemannian version of the Armijo line search for global convergence. Such existing methods need to compute the value of a retraction mapping regarding the search direction several times at each iteration; this may result in high computational costs, particularly if computing the value of the retraction is expensive. To address this issue, this study focuses on Riemannian submanifolds of the Euclidean spaces and proposes a novel Riemannian line search that achieves lower computational cost by incorporating a new strategy that computes the retraction only when inevitable. A class of Riemannian optimization algorithms, including the steepest descent and Newton methods, with the new line search strategy is proposed and proved to be globally convergent. Furthermore, numerical experiments on solving optimization problems on several types of Riemannian submanifolds illustrate that the proposed methods are superior to the standard Riemannian Armijo line search-based methods.
Paper Structure (14 sections, 5 theorems, 45 equations, 2 figures, 4 tables, 3 algorithms)

This paper contains 14 sections, 5 theorems, 45 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminology
  4. Existing concepts and techniques in Riemannian optimization
  5. Armijo line search in Riemannian optimization and its computational cost improvement
  6. Overview of the Armijo line search
  7. Modification of the Armijo line search in Riemannian optimization
  8. Global convergence of Algorithm \ref{['alg:Newton_method']}
  9. Well-definedness of Algorithm \ref{['alg:Newton_method']}
  10. Global convergence of Algorithm \ref{['alg:Newton_method']}
  11. Numerical experiments
  12. Optimization problems on several manifolds and retractions on them
  13. Numerical results
  14. Concluding remarks

Key Result

Lemma 1

Let $\cal M$ be a Riemannian submanifold of $\mathbb{R}^n$ and let $x \in {\cal M}$ and $p \in T_{x} {\cal M}$. Then, $\langle \mathop{\rm grad} f(x), p \rangle_{x} = \langle \nabla f(x), p \rangle$ and $\mathop{\rm grad} f(x) = {\cal P}_{x}(\nabla f(x))$ hold.

Figures (2)

  • Figure 1: Concept of the proposed line search strategy. We approximate $R_{x_k}(\alpha p_k)$ by $x_k + \alpha p_k$.
  • Figure 2: Time comparison of the steepest descent methods with the existing and proposed line search strategies for Problem \ref{['prob:sphere']} on $S^{n-1}$.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 5 more