# A Bilevel Approach for Parameter Learning in Inverse Problems

arXiv: https://arxiv.org/abs/1802.01365 · HTML: https://mlsanity.com/paper/1802.01365/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2018.

## Built on by (most cited first, 9 of 9)

- [Functional Bilevel Optimization for Machine Learning](https://mlsanity.com/paper/2403.20233/) (2024)
- [Efficient Prior Calibration From Indirect Data](https://mlsanity.com/paper/2405.17955/) (2024)
- [On Optimal Regularization Parameters via Bilevel Learning](https://mlsanity.com/paper/2305.18394/) (2023)
- [Bilevel learning of l1-regularizers with closed-form gradients(BLORC)](https://mlsanity.com/paper/2111.10858/) (2021)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Bilevel Learning with Inexact Stochastic Gradients](https://mlsanity.com/paper/2412.12049/) (2024)
- [Learning Theory for Kernel Bilevel Optimization](https://mlsanity.com/paper/2502.08457/) (2025)
- [Non-Stationary Functional Bilevel Optimization](https://mlsanity.com/paper/2601.15363/) (2026)
- [Meta-Prior: Meta learning for Adaptive Inverse Problem Solvers](https://mlsanity.com/paper/2311.18710/) (2023)

## Abstract (arXiv)

A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem of interest are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.