Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riemannian Projection-free Online Learning

Zihao Hu, Guanghui Wang, Jacob Abernethy

TL;DR

The paper advances projection-free online optimization on Riemannian manifolds, specifically Hadamard spaces, by replacing costly metric projections with separation and linear-optimization oracles. It furnishes sub-linear regret guarantees for geodesically convex losses under both full-information and bandit feedback, leveraging Jacobi-field tools to manage curvature effects. Two oracle paradigms are analyzed: SO-based infeasible projections yield adaptive regrets of $O(T^{1/2})$ (full information) and $O(T^{3/4})$ (bandit), while LO-based approaches via RFW achieve $O(T^{3/4})$ for gsc-convex and $O(T^{2/3}\log T)$ for strongly gsc-convex losses, with linear optimization calls kept at $O(T)$. The results bridge Euclidean projection-free OCO and Riemannian optimization, opening pathways to efficient constrained learning on non-Euclidean manifolds. This work holds significance for tasks where feasible sets are manifold-valued and projections are expensive, enabling scalable online learning in curved spaces with firm theoretical guarantees.

Abstract

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve $O(T^{1/2}\:)$ and $O(T^{3/4}\;)$ adaptive regret guarantees in the full information setting and the bandit setting, respectively. When a linear optimization oracle is available, we obtain regret rates of $O(T^{3/4}\;)$ for geodesically convex losses and $O(T^{2/3}\; log T )$ for strongly geodesically convex losses.

Riemannian Projection-free Online Learning

TL;DR

The paper advances projection-free online optimization on Riemannian manifolds, specifically Hadamard spaces, by replacing costly metric projections with separation and linear-optimization oracles. It furnishes sub-linear regret guarantees for geodesically convex losses under both full-information and bandit feedback, leveraging Jacobi-field tools to manage curvature effects. Two oracle paradigms are analyzed: SO-based infeasible projections yield adaptive regrets of (full information) and (bandit), while LO-based approaches via RFW achieve for gsc-convex and for strongly gsc-convex losses, with linear optimization calls kept at . The results bridge Euclidean projection-free OCO and Riemannian optimization, opening pathways to efficient constrained learning on non-Euclidean manifolds. This work holds significance for tasks where feasible sets are manifold-valued and projections are expensive, enabling scalable online learning in curved spaces with firm theoretical guarantees.

Abstract

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve and adaptive regret guarantees in the full information setting and the bandit setting, respectively. When a linear optimization oracle is available, we obtain regret rates of for geodesically convex losses and for strongly geodesically convex losses.
Paper Structure (30 sections, 37 theorems, 179 equations, 10 tables)

This paper contains 30 sections, 37 theorems, 179 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Projection-free OCO in Euclidean Space
  4. Online and Projection-free Optimization on Manifolds
  5. Preliminaries and Assumptions
  6. Warm-up: the High-level Idea
  7. Riemannian OCO with a Separation Oracle
  8. Riemannian OCO with a Linear Optimization Oracle
  9. Conclusion and Perspective
  10. Omitted Proofs in Section \ref{['sec:warmup']}
  11. Proof of Lemma \ref{['lem:reg_iogd']}
  12. Proof of Lemma \ref{['lem:pull_ip']}
  13. Omitted Proofs in Section \ref{['sec:so']}
  14. Proof of Lemma \ref{['lem:ballso']}
  15. Proof of Lemma \ref{['lem:ipso']}
  16. ...and 15 more sections

Key Result

Lemma 1

(Proof in Appendix app:reg_iogd) Assume $\tilde{\mathbf{y}}_t\in\mathbb{B}_{\mathbf{p}}(R)$ and let $\nabla_t=\nabla f_t(\tilde{\mathbf{y}}_t)$ and $\mathcal{K}\subseteq \mathbb{B}_{\mathbf{p}}(R)$ be a gsc-convex subset of $\mathcal{M}$. Consider $\tilde{\mathcal{K}}\subseteq\mathcal{K}$ as a simpl

Theorems & Definitions (71)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Theorem 2
  • ...and 61 more