# Riemannian Bilevel Optimization

arXiv: https://arxiv.org/abs/2405.15816 · HTML: https://mlsanity.com/paper/2405.15816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 50 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 29 of 29)

- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [A Fully First-Order Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2301.10945/) (2023)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [iDARTS: Differentiable Architecture Search with Stochastic Implicit Gradients](https://mlsanity.com/paper/2106.10784/) (2021)
- [Stability and Generalization of Bilevel Programming in Hyperparameter Optimization](https://mlsanity.com/paper/2106.04188/) (2021)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [ZARTS: On Zero-order Optimization for Neural Architecture Search](https://mlsanity.com/paper/2110.04743/) (2021)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [Constrained Bi-Level Optimization: Proximal Lagrangian Value function Approach and Hessian-free Algorithm](https://mlsanity.com/paper/2401.16164/) (2024)
- [Efficient Gradient Approximation Method for Constrained Bilevel Optimization](https://mlsanity.com/paper/2302.01970/) (2023)
- [Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization](https://mlsanity.com/paper/2012.15467/) (2020)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level](https://mlsanity.com/paper/2202.01033/) (2022)
- [Riemannian Natural Gradient Methods](https://mlsanity.com/paper/2207.07287/) (2022)
- [Riemannian Perspective on Matrix Factorization](https://mlsanity.com/paper/2102.00937/) (2021)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Robust PCA: Optimization of the Robust Reconstruction Error over the Stiefel Manifold](https://mlsanity.com/paper/1506.00323/) (2015)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Riemannian Bilevel Optimization](https://mlsanity.com/paper/2402.02019/) (2024)

## Abstract (arXiv)

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 RF2SA, a method that leverages first-order gradient information to navigate the complex geometry of Riemannian manifolds efficiently. Notably, RF2SA 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 ε-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.