# On the Convergence of Adam-Type Algorithm for Bilevel Optimization under Unbounded Smoothness

arXiv: https://arxiv.org/abs/2503.03908 · HTML: https://mlsanity.com/paper/2503.03908/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 65 references that are in the map.
- First posted on arXiv 5 Mar 2025.

## Builds on (most cited first, 30 of 45)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [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)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [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)
- [Coresets via Bilevel Optimization for Continual Learning and Streaming](https://mlsanity.com/paper/2006.03875/) (2020)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [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)
- [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)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Large-scale Robust Deep AUC Maximization: A New Surrogate Loss and Empirical Studies on Medical Image Classification](https://mlsanity.com/paper/2012.03173/) (2020)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [Stochastic AUC Maximization with Deep Neural Networks](https://mlsanity.com/paper/1908.10831/) (2019)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Convex and Non-convex Optimization Under Generalized Smoothness](https://mlsanity.com/paper/2306.01264/) (2023)

## Built on by (most cited first, 1 of 1)

- [Adaptive Algorithms with Sharp Convergence Rates for Stochastic Hierarchical Optimization](https://mlsanity.com/paper/2509.15399/) (2025)

## Abstract (arXiv)

Adam has become one of the most popular optimizers for training modern deep neural networks, such as transformers. However, its applicability is largely restricted to single-level optimization problems. In this paper, we aim to extend vanilla Adam to tackle bilevel optimization problems, which have important applications in machine learning, such as meta-learning. In particular, we study stochastic bilevel optimization problems where the lower-level function is strongly convex and the upper-level objective is nonconvex with potentially unbounded smoothness. This unbounded smooth objective function covers a broad class of neural networks, including transformers, which may exhibit non-Lipschitz gradients. In this work, we introduce AdamBO, a single-loop Adam-type method that achieves widetildeO(ε-4) oracle complexity to find ε-stationary points, where the oracle calls involve stochastic gradient or Hessian/Jacobian-vector product evaluations. The key to our analysis is a novel randomness decoupling lemma that provides refined control over the lower-level variable. We conduct extensive experiments on various machine learning tasks involving bilevel formulations with recurrent neural networks (RNNs) and transformers, demonstrating the effectiveness of our proposed Adam-type algorithm.