# Linearly convergent bilevel optimization with single-step inner methods

arXiv: https://arxiv.org/abs/2205.04862 · HTML: https://mlsanity.com/paper/2205.04862/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2022.

## Built on by (most cited first, 7 of 7)

- [Functional Bilevel Optimization for Machine Learning](https://mlsanity.com/paper/2403.20233/) (2024)
- [Analyzing Inexact Hypergradients for Bilevel Learning](https://mlsanity.com/paper/2301.04764/) (2023)
- [An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning](https://mlsanity.com/paper/2308.10098/) (2023)
- [An Adaptively Inexact Method for Bilevel Learning Using Primal-Dual Style Differentiation](https://mlsanity.com/paper/2412.06436/) (2024)
- [Restarted contractive operators to learn at equilibrium](https://mlsanity.com/paper/2506.13239/) (2025)
- [Bilevel Learning via Inexact Stochastic Gradient Descent](https://mlsanity.com/paper/2511.06774/) (2025)
- [Automatic regularization parameter choice for tomography using a double model approach](https://mlsanity.com/paper/2602.08528/) (2026)

## Abstract (arXiv)

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.