# Efficient Curvature-Aware Hypergradient Approximation for Bilevel Optimization

arXiv: https://arxiv.org/abs/2505.02101 · HTML: https://mlsanity.com/paper/2505.02101/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 61 references that are in the map.
- First posted on arXiv 4 May 2025.

## Builds on (most cited first, 30 of 46)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [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)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [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)
- [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)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [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)
- [DARTS+: Improved Differentiable Architecture Search with Early Stopping](https://mlsanity.com/paper/1909.06035/) (2019)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [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)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [Revisiting and Advancing Fast Adversarial Training Through The Lens of Bi-Level Optimization](https://mlsanity.com/paper/2112.12376/) (2021)
- [Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters](https://mlsanity.com/paper/2006.09486/) (2020)
- [Amortized Implicit Differentiation for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2111.14580/) (2021)
- [A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse](https://mlsanity.com/paper/2112.04660/) (2021)
- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning](https://mlsanity.com/paper/2308.00788/) (2023)
- [On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis](https://mlsanity.com/paper/2301.00712/) (2023)
- [Exact and Inexact Subsampled Newton Methods for Optimization](https://mlsanity.com/paper/1609.08502/) (2016)
- [Will Bilevel Optimizers Benefit from Loops](https://mlsanity.com/paper/2205.14224/) (2022)
- [Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions](https://mlsanity.com/paper/2306.12067/) (2023)

## Built on by (most cited first, 3 of 3)

- [On the Difficulty of Learning a Meta-network for Training Data Selection](https://mlsanity.com/paper/2606.00571/) (2026)
- [Efficient Bilevel Optimization with KFAC-Based Hypergradients](https://mlsanity.com/paper/2603.29108/) (2026)
- [Sharper Analysis of Single-Loop Methods for Bilevel Optimization](https://mlsanity.com/paper/2607.10263/) (2026)

## Abstract (arXiv)

Bilevel optimization is a powerful tool for many machine learning problems, such as hyperparameter optimization and meta-learning. Estimating hypergradients (also known as implicit gradients) is crucial for developing gradient-based methods for bilevel optimization. In this work, we propose a computationally efficient technique for incorporating curvature information into the approximation of hypergradients and present a novel algorithmic framework based on the resulting enhanced hypergradient computation. We provide convergence rate guarantees for the proposed framework in both deterministic and stochastic scenarios, particularly showing improved computational complexity over popular gradient-based methods in the deterministic setting. This improvement in complexity arises from a careful exploitation of the hypergradient structure and the inexact Newton method. In addition to the theoretical speedup, numerical experiments demonstrate the significant practical performance benefits of incorporating curvature information.