# On the Convergence Theory for Hessian-Free Bilevel Algorithms

arXiv: https://arxiv.org/abs/2110.07004 · HTML: https://mlsanity.com/paper/2110.07004/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 63 references that are in the map.
- First posted on arXiv 13 Oct 2021.

## Builds on (most cited first, 30 of 42)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [How to train your MAML](https://mlsanity.com/paper/1810.09502/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Adversarial Attacks on Graph Neural Networks via Meta Learning](https://mlsanity.com/paper/1902.08412/) (2019)
- [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)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [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)
- [On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms](https://mlsanity.com/paper/1908.10400/) (2019)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [Zeroth-Order Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1805.10367/) (2018)
- [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)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton](https://mlsanity.com/paper/2006.04045/) (2020)

## Built on by (most cited first, 30 of 35)

- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [ScaleBiO: Scalable Bilevel Optimization for LLM Data Reweighting](https://mlsanity.com/paper/2406.19976/) (2024)
- [Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions](https://mlsanity.com/paper/2306.12067/) (2023)
- [Decentralized Stochastic Bilevel Optimization with Improved per-Iteration Complexity](https://mlsanity.com/paper/2210.12839/) (2022)
- [Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles](https://mlsanity.com/paper/2306.14853/) (2023)
- [A Lower Bound and a Near-Optimal Algorithm for Bilevel Empirical Risk Minimization](https://mlsanity.com/paper/2302.08766/) (2023)
- [Moreau Envelope for Nonconvex Bi-Level Optimization: A Single-loop and Hessian-free Solution Strategy](https://mlsanity.com/paper/2405.09927/) (2024)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [Optimal Hessian/Jacobian-Free Nonconvex-PL Bilevel Optimization](https://mlsanity.com/paper/2407.17823/) (2024)
- [On The Sample Complexity Bounds In Bilevel Reinforcement Learning](https://mlsanity.com/paper/2503.17644/) (2025)
- [A Primal-Dual-Assisted Penalty Approach to Bilevel Optimization with Coupled Constraints](https://mlsanity.com/paper/2406.10148/) (2024)
- [Fully Zeroth-Order Bilevel Programming via Gaussian Smoothing](https://mlsanity.com/paper/2404.00158/) (2024)
- [BiSLS/SPS: Auto-tune Step Sizes for Stable Bi-level Optimization](https://mlsanity.com/paper/2305.18666/) (2023)
- [Stochastic Bilevel Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2509.14952/) (2025)
- [Effective Bilevel Optimization via Minimax Reformulation](https://mlsanity.com/paper/2305.13153/) (2023)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Lazy Queries Can Reduce Variance in Zeroth-order Optimization](https://mlsanity.com/paper/2206.07126/) (2022)
- [Achieving O(ε-1.5) Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization](https://mlsanity.com/paper/2312.03807/) (2023)
- [Online Nonconvex Bilevel Optimization with Bregman Divergences](https://mlsanity.com/paper/2409.10470/) (2024)
- [qNBO: quasi-Newton Meets Bilevel Optimization](https://mlsanity.com/paper/2502.01076/) (2025)
- [Leveraging Machine Unlearning for Cost-Efficient Preference Alignment](https://mlsanity.com/paper/2504.06659/) (2025)
- [Computational Budget Should Be Considered in Data Selection](https://mlsanity.com/paper/2510.16806/) (2025)
- [Multi-Objective Bilevel Learning](https://mlsanity.com/paper/2511.07824/) (2025)
- [Soft-Label Integration for Robust Toxicity Classification](https://mlsanity.com/paper/2410.14894/) (2024)
- [Fully First-Order Methods for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2410.19319/) (2024)
- [Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization](https://mlsanity.com/paper/2602.06457/) (2026)
- [A Provably Convergent Plug-and-Play Framework for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2505.01258/) (2025)
- [Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning](https://mlsanity.com/paper/2602.00282/) (2026)
- [Fully First-Order Algorithms for Online Bilevel Optimization](https://mlsanity.com/paper/2602.11665/) (2026)
- [Binomial Gradient-Based Meta-Learning for Enhanced Meta-Gradient Estimation](https://mlsanity.com/paper/2604.13263/) (2026)

## Abstract (arXiv)

Bilevel optimization has arisen as a powerful tool in modern machine learning. However, due to the nested structure of bilevel optimization, even gradient-based methods require second-order derivative approximations via Jacobian- or/and Hessian-vector computations, which can be costly and unscalable in practice. Recently, Hessian-free bilevel schemes have been proposed to resolve this issue, where the general idea is to use zeroth- or first-order methods to approximate the full hypergradient of the bilevel problem. However, we empirically observe that such approximation can lead to large variance and unstable training, but estimating only the response Jacobian matrix as a partial component of the hypergradient turns out to be extremely effective. To this end, we propose a new Hessian-free method, which adopts the zeroth-order-like method to approximate the response Jacobian matrix via taking difference between two optimization paths. Theoretically, we provide the convergence rate analysis for the proposed algorithms, where our key challenge is to characterize the approximation and smoothness properties of the trajectory-dependent estimator, which can be of independent interest. This is the first known convergence rate result for this type of Hessian-free bilevel algorithms. Experimentally, we demonstrate that the proposed algorithms outperform baseline bilevel optimizers on various bilevel problems. Particularly, in our experiment on few-shot meta-learning with ResNet-12 network over the miniImageNet dataset, we show that our algorithm outperforms baseline meta-learning algorithms, while other baseline bilevel optimizers do not solve such meta-learning problems within a comparable time frame.