# A stochastic gradient method for trilevel optimization

arXiv: https://arxiv.org/abs/2505.06805 · HTML: https://mlsanity.com/paper/2505.06805/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 11 May 2025.

## Builds on (most cited first, 19 of 19)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [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)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [When NAS Meets Robustness: In Search of Robust Architectures against Adversarial Attacks](https://mlsanity.com/paper/1911.10695/) (2019)
- [A Single-Timescale Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2102.04671/) (2021)
- [Exact and Inexact Subsampled Newton Methods for Optimization](https://mlsanity.com/paper/1609.08502/) (2016)
- [Betty: An Automatic Differentiation Library for Multilevel Optimization](https://mlsanity.com/paper/2207.02849/) (2022)
- [RC-DARTS: Resource Constrained Differentiable Architecture Search](https://mlsanity.com/paper/1912.12814/) (2019)
- [A Gradient Method for Multilevel Optimization](https://mlsanity.com/paper/2105.13954/) (2021)
- [Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems](https://mlsanity.com/paper/2110.00604/) (2021)
- [Trilevel and Multilevel Optimization using Monotone Operator Theory](https://mlsanity.com/paper/2105.09407/) (2021)
- [Provably Convergent Federated Trilevel Learning](https://mlsanity.com/paper/2312.11835/) (2023)
- [Bilevel optimization with a multi-objective lower-level problem: Risk-neutral and risk-averse formulations](https://mlsanity.com/paper/2302.05540/) (2023)
- [Unlocking TriLevel Learning with Level-Wise Zeroth Order Constraints: Distributed Algorithms and Provable Non-Asymptotic Convergence](https://mlsanity.com/paper/2412.07138/) (2024)
- [A Trilevel Model for Segmentation of the Power Transmission Grid Cyber Network](https://mlsanity.com/paper/2108.10958/) (2021)

## Built on by (most cited first, 2 of 2)

- [CHAL: Council of Hierarchical Agentic Language](https://mlsanity.com/paper/2605.12718/) (2026)
- [First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection](https://mlsanity.com/paper/2607.27632/) (2026)

## Abstract (arXiv)

With the success that the field of bilevel optimization has seen in recent years, similar methodologies have started being applied to solving more difficult applications that arise in trilevel optimization. At the helm of these applications are new machine learning formulations that have been proposed in the trilevel context and, as a result, efficient and theoretically sound stochastic methods are required. In this work, we propose the first-ever stochastic gradient descent method for solving unconstrained trilevel optimization problems and provide a convergence theory that covers all forms of inexactness of the trilevel adjoint gradient, such as the inexact solutions of the middle-level and lower-level problems, inexact computation of the trilevel adjoint formula, and noisy estimates of the gradients, Hessians, Jacobians, and tensors of third-order derivatives involved. We also demonstrate the promise of our approach by providing numerical results on both synthetic trilevel problems and trilevel formulations for hyperparameter adversarial tuning.