Riemannian Bilevel Optimization

TL;DR

This paper introduces RF^2SA, a first-order, single-loop algorithm for Riemannian bilevel optimization that avoids second-order derivatives, with proven convergence rates and applicability to constrained manifold problems.

Contribution

It presents RF^2SA, a novel first-order, single-loop method for Riemannian bilevel optimization, simplifying implementation and extending to constrained manifolds without second-order information.

Findings

RF^2SA achieves explicit convergence rates for stationary points.

The method is applicable to stochastic and constrained Riemannian problems.

It avoids second-order derivatives, simplifying computations.

Abstract

We develop new algorithms for Riemannian bilevel optimization. We focus in particular on batch and stochastic gradient-based methods, with the explicit goal of avoiding second-order information such as Riemannian hyper-gradients. We propose and analyze $\mathrm{RF^2SA}$, a method that leverages first-order gradient information to navigate the complex geometry of Riemannian manifolds efficiently. Notably, $\mathrm{RF^2SA}$ is a single-loop algorithm, and thus easier to implement and use. Under various setups, including stochastic optimization, we provide explicit convergence rates for reaching $\epsilon$-stationary points. We also address the challenge of optimizing over Riemannian manifolds with constraints by adjusting the multiplier in the Lagrangian, ensuring convergence to the desired solution without requiring access to second-order derivatives.