# Analyzing and Enhancing the Backward-Pass Convergence of Unrolled Optimization

arXiv: https://arxiv.org/abs/2312.17394 · HTML: https://mlsanity.com/paper/2312.17394/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 21 of 21)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [Gradient Estimation with Stochastic Softmax Tricks](https://mlsanity.com/paper/2006.08063/) (2020)
- [Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization](https://mlsanity.com/paper/2205.02640/) (2022)
- [Smooth Loss Functions for Deep Top-k Classification](https://mlsanity.com/paper/1802.07595/) (2018)
- [The Limited Multi-Label Projection Layer](https://mlsanity.com/paper/1906.08707/) (2019)
- [Solution Refinement at Regular Points of Conic Problems](https://mlsanity.com/paper/1811.02157/) (2018)
- [End-to-end Learning for Fair Ranking Systems](https://mlsanity.com/paper/2111.10723/) (2021)
- [Backpropagation of Unrolled Solvers with Folded Optimization](https://mlsanity.com/paper/2301.12047/) (2023)
- [NESTA, The NICTA Energy System Test Case Archive](https://mlsanity.com/paper/1411.0359/) (2014)

## Built on by (most cited first, 1 of 1)

- [Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling](https://mlsanity.com/paper/2605.12754/) (2026)

## Abstract (arXiv)

The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. A central challenge in this setting is backpropagation through the solution of an optimization problem, which often lacks a closed form. One typical strategy is algorithm unrolling, which relies on automatic differentiation through the entire chain of operations executed by an iterative optimization solver. This paper provides theoretical insights into the backward pass of unrolled optimization, showing that it is asymptotically equivalent to the solution of a linear system by a particular iterative method. Several practical pitfalls of unrolling are demonstrated in light of these insights, and a system called Folded Optimization is proposed to construct more efficient backpropagation rules from unrolled solver implementations. Experiments over various end-to-end optimization and learning tasks demonstrate the advantages of this system both computationally, and in terms of flexibility over various optimization problem forms.