# A Generalized Alternating Method for Bilevel Learning under the Polyak-Lojasiewicz Condition

arXiv: https://arxiv.org/abs/2306.02422 · HTML: https://mlsanity.com/paper/2306.02422/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 73 references that are in the map.
- First posted on arXiv 4 Jun 2023.

## Builds on (most cited first, 30 of 49)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [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)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Coresets via Bilevel Optimization for Continual Learning and Streaming](https://mlsanity.com/paper/2006.03875/) (2020)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Lower Bounds for Finding Stationary Points I](https://mlsanity.com/paper/1710.11606/) (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)
- [A framework for bilevel optimization that enables stochastic and global variance reduction algorithms](https://mlsanity.com/paper/2201.13409/) (2022)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [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)
- [Revisiting and Advancing Fast Adversarial Training Through The Lens of Bi-Level Optimization](https://mlsanity.com/paper/2112.12376/) (2021)
- [A Single-Timescale Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2102.04671/) (2021)
- [A First Order Method for Solving Convex Bi-Level Optimization Problems](https://mlsanity.com/paper/1702.03999/) (2017)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [Amortized Implicit Differentiation for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2111.14580/) (2021)

## Built on by (most cited first, 5 of 5)

- [On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization](https://mlsanity.com/paper/2507.23155/) (2025)
- [Bridging MARL to SARL: An Order-Independent Multi-Agent Transformer via Latent Consensus](https://mlsanity.com/paper/2604.13472/) (2026)
- [Safe Gradient Flow for Bilevel Optimization](https://mlsanity.com/paper/2501.16520/) (2025)
- [Alchemist: Unlocking Efficiency in Text-to-Image Model Training via Meta-Gradient Data Selection](https://mlsanity.com/paper/2512.16905/) (2025)
- [To Solve Bilevel Optimization with Nonconvex Lower Levels, We Need Second-Order Stationarity](https://mlsanity.com/paper/2609.30501/) (2026)

## Abstract (arXiv)

Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can match the convergence rate of single-level gradient descent (GD) when addressing bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric, for a nonconvex lower-level objective that satisfies the Polyak-Lojasiewicz (PL) condition. We then propose a Generalized ALternating mEthod for bilevel opTimization (GALET) tailored to BLO with convex PL LL problem and establish that GALET achieves an ε-stationary point for the considered problem within tildecal O(ε-1) iterations, which matches the iteration complexity of GD for single-level smooth nonconvex problems.