# Scalable Deep Unfolding of Conic Optimizers

arXiv: https://arxiv.org/abs/2606.13825 · HTML: https://mlsanity.com/paper/2606.13825/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 46 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 29 of 29)

- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [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)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures](https://mlsanity.com/paper/1409.2574/) (2014)
- [Deep Unfolding for Communications Systems: A Survey and Some New Directions](https://mlsanity.com/paper/1906.05774/) (2019)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [DiffStack: A Differentiable and Modular Control Stack for Autonomous Vehicles](https://mlsanity.com/paper/2212.06437/) (2022)
- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [Accelerating Quadratic Optimization with Reinforcement Learning](https://mlsanity.com/paper/2107.10847/) (2021)
- [End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization](https://mlsanity.com/paper/2212.08260/) (2022)
- [Hyperparameter Tuning is All You Need for LISTA](https://mlsanity.com/paper/2110.15900/) (2021)
- [GPU Acceleration of ADMM for Large-Scale Quadratic Programming](https://mlsanity.com/paper/1912.04263/) (2019)
- [Deep Distributed Optimization for Large-Scale Quadratic Programming](https://mlsanity.com/paper/2412.12156/) (2024)
- [COSMO: A conic operator splitting method for convex conic problems](https://mlsanity.com/paper/1901.10887/) (2019)
- [Proximal Algorithm Unrolling: Flexible and Efficient Reconstruction Networks for Single-Pixel Imaging](https://mlsanity.com/paper/2505.23180/) (2025)
- [DeepKSPD: Learning Kernel-matrix-based SPD Representation for Fine-grained Image Recognition](https://mlsanity.com/paper/1711.04047/) (2017)
- [Nonlinear Uncertainty Control with Iterative Covariance Steering](https://mlsanity.com/paper/1903.10919/) (2019)
- [Differentiable Robust Model Predictive Control](https://mlsanity.com/paper/2308.08426/) (2023)
- [Deep FlexQP: Accelerated Nonlinear Programming via Deep Unfolding](https://mlsanity.com/paper/2512.01565/) (2025)
- [Solving Quadratic Programs via Deep Unrolled Douglas-Rachford Splitting](https://mlsanity.com/paper/2508.11869/) (2025)
- [Chance-Constrained Covariance Steering in a Gaussian Random Field via Successive Convex Programming](https://mlsanity.com/paper/2101.09634/) (2021)
- [Covariance Control of Discrete-Time Gaussian Linear Systems Using Affine Disturbance Feedback Control Policies](https://mlsanity.com/paper/2103.14428/) (2021)

## Built on by (most cited first, 2 of 2)

- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)
- [Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming](https://mlsanity.com/paper/2610.01546/) (2026)

## Abstract (arXiv)

Deep unfolding (DU) accelerates iterative optimizers by introducing learnable components and training them through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles that prior work on learned conic solvers has not: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size once the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection becomes numerically unstable when eigenvalues coincide. We address the first obstacle with a matrix-free implicit differentiation rule that operates entirely through matrix-vector products, reducing memory from O(n2) to O(n) and enabling backpropagation at scales where direct factorization runs out of memory. We address the second with a backward rule based on the Dalevckii--Krein representation of the Fr'echet derivative, which remains well-defined under repeated eigenvalues. Together these make it possible to learn lightweight hyperparameter policies and warm-starts for a full-update conic solver. We evaluate on nonlinear covariance steering problems solved via sequential convex programming (SCP), as well as standalone SDPs and second-order cone programs ranging from max-cut and Lov'asz vartheta SDPs to robust estimation and control problems. The learned policies outperform state-of-the-art solvers across all problems, and can provide up to a 50× speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30× speedup compared to COSMO.