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Riemannian Dueling Optimization

Yuxuan Ren, Abhishek Roy, Shiqian Ma

TL;DR

This work proposes a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establishes its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex and proposes a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited.

Abstract

Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.

Riemannian Dueling Optimization

TL;DR

This work proposes a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establishes its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex and proposes a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited.

Abstract

Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.
Paper Structure (34 sections, 13 theorems, 100 equations, 14 figures, 1 table, 3 algorithms)

This paper contains 34 sections, 13 theorems, 100 equations, 14 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Review of Euclidean Dueling Optimization
  3. Motivating Examples
  4. Contributions
  5. Preliminaries
  6. Riemannian Dueling Normalized Gradient Descent Method
  7. Riemannian Dueling Frank-Wolfe Method
  8. Numerical Experiments
  9. Synthetic Problems
  10. Rayleigh Quotient Maximization
  11. Karcher Mean Problem
  12. Karcher Mean with Constraints
  13. Real Applications
  14. Attack on Deep Neural Network
  15. Horizon leveling
  16. ...and 19 more sections

Key Result

Lemma 3.1

Assume $f$ is geodesically $L$-smooth. Let $u \sim \mathrm{Unif(\mathcal{S}_{\mathrm{T}_x\mathcal{M}}(1))}$, and $\nu \in (0,1)$. Then with probability at least $1 - \gamma_x$, we have where $h_{\nu}(x)$ is defined in eq:graddirestimator and

Figures (14)

  • Figure 1: Numerical results for Rayleigh quotient maximization.
  • Figure 2: Numerical result for Karcher mean problem.
  • Figure 3: Numerical result for constrained Karcher mean problem.
  • Figure 4: Attack results on CIFAR-10. (a) Generated adversarial images. (b) and (c) show the convergence curves. Our estimator yields accurate gradient estimation with only $10$ samples (vs. $500$ for ZO-RGD), demonstrating superior query efficiency.
  • Figure 5: Horizon leveling results. The loss stabilizes at $10^{-5}$ due to a large constant step size, chosen to balance convergence speed and the optimality gap.
  • ...and 9 more figures

Theorems & Definitions (45)

  • Remark 1.1
  • Definition 2.1: Geodesic $L$-smoothness
  • Definition 2.2: Geodesically uniquely convex set
  • Definition 2.3: Geodesic (strong) convexity
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Remark 3.4
  • Definition 3.5: Cosine Annealing step size
  • ...and 35 more