# Learning Variational Models with Unrolling and Bilevel Optimization

arXiv: https://arxiv.org/abs/2209.12651 · HTML: https://mlsanity.com/paper/2209.12651/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 70 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification](https://mlsanity.com/paper/1507.03003/) (2015)
- [A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications](https://mlsanity.com/paper/1705.06270/) (2017)
- [Deep Learning on Image Denoising: An overview](https://mlsanity.com/paper/1912.13171/) (2019)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Learning step sizes for unfolded sparse coding](https://mlsanity.com/paper/1905.11071/) (2019)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Deep Unfolding of a Proximal Interior Point Method for Image Restoration](https://mlsanity.com/paper/1812.04276/) (2018)
- [Unrolling of Deep Graph Total Variation for Image Denoising](https://mlsanity.com/paper/2010.11290/) (2020)
- [Learning to solve inverse problems using Wasserstein loss](https://mlsanity.com/paper/1710.10898/) (2017)
- [Understanding approximate and unrolled dictionary learning for pattern recovery](https://mlsanity.com/paper/2106.06338/) (2021)
- [Data splitting improves statistical performance in overparametrized regimes](https://mlsanity.com/paper/2110.10956/) (2021)
- [Theoretical Interpretation of Learned Step Size in Deep-Unfolded Gradient Descent](https://mlsanity.com/paper/2001.05142/) (2020)
- [Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming](https://mlsanity.com/paper/1402.6185/) (2014)
- [A higher-order MRF based variational model for multiplicative noise reduction](https://mlsanity.com/paper/1404.5344/) (2014)

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

- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)
- [Why the noise model matters: A performance gap in learned regularization](https://mlsanity.com/paper/2510.12521/) (2025)
- [Bridging Constraints and Stochasticity: A Fully First-Order Method for Stochastic Bilevel Optimization with Linear Constraints](https://mlsanity.com/paper/2511.09845/) (2025)

## Abstract (arXiv)

In this paper we consider the problem of learning variational models in the context of supervised learning via risk minimization. Our goal is to provide a deeper understanding of the two approaches of learning of variational models via bilevel optimization and via algorithm unrolling. The former considers the variational model as a lower level optimization problem below the risk minimization problem, while the latter replaces the lower level optimization problem by an algorithm that solves said problem approximately. Both approaches are used in practice, but unrolling is much simpler from a computational point of view. To analyze and compare the two approaches, we consider a simple toy model, and compute all risks and the respective estimators explicitly. We show that unrolling can be better than the bilevel optimization approach, but also that the performance of unrolling can depend significantly on further parameters, sometimes in unexpected ways: While the stepsize of the unrolled algorithm matters a lot (and learning the stepsize gives a significant improvement), the number of unrolled iterations plays a minor role.