# On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level

arXiv: https://arxiv.org/abs/2303.03944 · HTML: https://mlsanity.com/paper/2303.03944/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 50 references that are in the map.
- First posted on arXiv 7 Mar 2023.

## Builds on (most cited first, 30 of 32)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [Stability and Generalization of Learning Algorithms that Converge to Global Optima](https://mlsanity.com/paper/1710.08402/) (2017)
- [A Fully First-Order Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2301.10945/) (2023)
- [A framework for bilevel optimization that enables stochastic and global variance reduction algorithms](https://mlsanity.com/paper/2201.13409/) (2022)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton](https://mlsanity.com/paper/2006.04045/) (2020)
- [A Single-Timescale Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2102.04671/) (2021)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization](https://mlsanity.com/paper/2106.07991/) (2021)
- [A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse](https://mlsanity.com/paper/2112.04660/) (2021)
- [First-order penalty methods for bilevel optimization](https://mlsanity.com/paper/2301.01716/) (2023)
- [Will Bilevel Optimizers Benefit from Loops](https://mlsanity.com/paper/2205.14224/) (2022)
- [Penalty Method for Inversion-Free Deep Bilevel Optimization](https://mlsanity.com/paper/1911.03432/) (2019)
- [BiAdam: Fast Adaptive Bilevel Optimization Methods](https://mlsanity.com/paper/2106.11396/) (2021)
- [Enhanced Bilevel Optimization via Bregman Distance](https://mlsanity.com/paper/2107.12301/) (2021)
- [A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1907.03793/) (2019)
- [Subquadratic Overparameterization for Shallow Neural Networks](https://mlsanity.com/paper/2111.01875/) (2021)
- [Averaged Method of Multipliers for Bi-Level Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2302.03407/) (2023)
- [Non-Convex Bilevel Games with Critical Point Selection Maps](https://mlsanity.com/paper/2207.04888/) (2022)
- [Value-Function-based Sequential Minimization for Bi-level Optimization](https://mlsanity.com/paper/2110.04974/) (2021)
- [Proxy Convexity: A Unified Framework for the Analysis of Neural Networks Trained by Gradient Descent](https://mlsanity.com/paper/2106.13792/) (2021)

## Built on by (most cited first, 14 of 14)

- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [PARL: A Unified Framework for Policy Alignment in Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2308.02585/) (2023)
- [Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions](https://mlsanity.com/paper/2306.12067/) (2023)
- [Constrained Bi-Level Optimization: Proximal Lagrangian Value function Approach and Hessian-free Algorithm](https://mlsanity.com/paper/2401.16164/) (2024)
- [Moreau Envelope for Nonconvex Bi-Level Optimization: A Single-loop and Hessian-free Solution Strategy](https://mlsanity.com/paper/2405.09927/) (2024)
- [Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem](https://mlsanity.com/paper/2308.07536/) (2023)
- [Optimal Hessian/Jacobian-Free Nonconvex-PL Bilevel Optimization](https://mlsanity.com/paper/2407.17823/) (2024)
- [Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity](https://mlsanity.com/paper/2405.19697/) (2024)
- [A Generalized Alternating Method for Bilevel Learning under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/2306.02422/) (2023)
- [Nested Stochastic Algorithm for Generalized Sinkhorn distance-Regularized Distributionally Robust Optimization](https://mlsanity.com/paper/2503.22923/) (2025)
- [Adaptive Mirror Descent Bilevel Optimization](https://mlsanity.com/paper/2311.04520/) (2023)
- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)
- [A Provably Convergent Plug-and-Play Framework for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2505.01258/) (2025)
- [Fully First-Order Algorithms for Online Bilevel Optimization](https://mlsanity.com/paper/2602.11665/) (2026)

## Abstract (arXiv)

Bilevel optimization is a popular two-level hierarchical optimization, which has been widely applied to many machine learning tasks such as hyperparameter learning, meta learning and continual learning. Although many bilevel optimization methods recently have been developed, the bilevel methods are not well studied when the lower-level problem is nonconvex. To fill this gap, in the paper, we study a class of nonconvex bilevel optimization problems, where both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-Lojasiewicz (PL) condition. We propose an efficient momentum-based gradient bilevel method (MGBiO) to solve these deterministic problems. Meanwhile, we propose a class of efficient momentum-based stochastic gradient bilevel methods (MSGBiO and VR-MSGBiO) to solve these stochastic problems. Moreover, we provide a useful convergence analysis framework for our methods. Specifically, under some mild conditions, we prove that our MGBiO method has a sample (or gradient) complexity of O(ε-2) for finding an ε-stationary solution of the deterministic bilevel problems (i.e., |nabla F(x)|≤ ε), which improves the existing best results by a factor of O(ε-1). Meanwhile, we prove that our MSGBiO and VR-MSGBiO methods have sample complexities of tildeO(ε-4) and tildeO(ε-3), respectively, in finding an ε-stationary solution of the stochastic bilevel problems (i.e., mathbbE|nabla F(x)|≤ ε), which improves the existing best results by a factor of tildeO(ε-3). Extensive experimental results on bilevel PL game and hyper-representation learning demonstrate the efficiency of our algorithms. This paper commemorates the mathematician Boris Polyak (1935 -2023).