# A Gradient Method for Multilevel Optimization

arXiv: https://arxiv.org/abs/2105.13954 · HTML: https://mlsanity.com/paper/2105.13954/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 28 May 2021.

## Builds on (most cited first, 9 of 9)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Learning Discrete Structures for Graph Neural Networks](https://mlsanity.com/paper/1903.11960/) (2019)
- [Manipulating Machine Learning: Poisoning Attacks and Countermeasures for Regression Learning](https://mlsanity.com/paper/1804.00308/) (2018)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications](https://mlsanity.com/paper/1705.06270/) (2017)
- [Trilevel and Multilevel Optimization using Monotone Operator Theory](https://mlsanity.com/paper/2105.09407/) (2021)

## Built on by (most cited first, 12 of 12)

- [PR-Attack: Coordinated Prompt-RAG Attacks on Retrieval-Augmented Generation in Large Language Models via Bilevel Optimization](https://mlsanity.com/paper/2504.07717/) (2025)
- [Betty: An Automatic Differentiation Library for Multilevel Optimization](https://mlsanity.com/paper/2207.02849/) (2022)
- [A Single-Timescale Analysis For Stochastic Approximation With Multiple Coupled Sequences](https://mlsanity.com/paper/2206.10414/) (2022)
- [Generative AI Enables Medical Image Segmentation in Ultra Low-Data Regimes](https://mlsanity.com/paper/2408.17421/) (2024)
- [Provably Convergent Federated Trilevel Learning](https://mlsanity.com/paper/2312.11835/) (2023)
- [Single-Timescale Multi-Sequence Stochastic Approximation Without Fixed Point Smoothness: Theories and Applications](https://mlsanity.com/paper/2410.13743/) (2024)
- [A stochastic gradient method for trilevel optimization](https://mlsanity.com/paper/2505.06805/) (2025)
- [Fast and robust single particle reconstruction in 3D fluorescence microscopy](https://mlsanity.com/paper/2301.09452/) (2023)
- [Federated Multi-Sequence Stochastic Approximation with Local Hypergradient Estimation](https://mlsanity.com/paper/2306.01648/) (2023)
- [Towards Differentiable Multilevel Optimization: A Gradient-Based Approach](https://mlsanity.com/paper/2410.11312/) (2024)
- [Riemannian Manifold Learning for Stackelberg Games with Neural Flow Representations](https://mlsanity.com/paper/2502.05498/) (2025)
- [LLM-Driven Multi-Turn Task-Oriented Dialogue Synthesis for Realistic Reasoning](https://mlsanity.com/paper/2602.23610/) (2026)

## Abstract (arXiv)

Although application examples of multilevel optimization have already been discussed since the 1990s, the development of solution methods was almost limited to bilevel cases due to the difficulty of the problem. In recent years, in machine learning, Franceschi et al. have proposed a method for solving bilevel optimization problems by replacing their lower-level problems with the T steepest descent update equations with some prechosen iteration number T. In this paper, we have developed a gradient-based algorithm for multilevel optimization with n levels based on their idea and proved that our reformulation asymptotically converges to the original multilevel problem. As far as we know, this is one of the first algorithms with some theoretical guarantee for multilevel optimization. Numerical experiments show that a trilevel hyperparameter learning model considering data poisoning produces more stable prediction results than an existing bilevel hyperparameter learning model in noisy data settings.