# A Fully First-Order Layer for Differentiable Optimization

arXiv: https://arxiv.org/abs/2512.02494 · HTML: https://mlsanity.com/paper/2512.02494/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 81 references that are in the map.
- First posted on arXiv 2 Dec 2025.

## Builds on (most cited first, 30 of 59)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Recent Advances in Adversarial Training for Adversarial Robustness](https://mlsanity.com/paper/2102.01356/) (2021)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [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)
- [Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities](https://mlsanity.com/paper/2307.13565/) (2023)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [A Fully First-Order Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2301.10945/) (2023)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [Theseus: A Library for Differentiable Nonlinear Optimization](https://mlsanity.com/paper/2207.09442/) (2022)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization](https://mlsanity.com/paper/2106.07991/) (2021)
- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis](https://mlsanity.com/paper/2301.00712/) (2023)
- [Non-convex Min-Max Optimization: Applications, Challenges, and Recent Theoretical Advances](https://mlsanity.com/paper/2006.08141/) (2020)

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

- [Learning to Optimize by Differentiable Programming](https://mlsanity.com/paper/2601.16510/) (2026)
- [lapanda: A Matrix-Free Differentiable Solver for Nonconvex Constrained Optimization Layers](https://mlsanity.com/paper/2609.33630/) (2026)
- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

Differentiable optimization layers enable learning systems to make decisions by solving embedded optimization problems. However, computing gradients via implicit differentiation requires solving a linear system with Hessian terms, which is both compute- and memory-intensive. To address this challenge, we propose a novel algorithm that computes the gradient using only first-order information. The key insight is to rewrite the differentiable optimization as a bilevel optimization problem and leverage recent advances in bilevel methods. Specifically, we introduce an active-set Lagrangian hypergradient oracle that avoids Hessian evaluations and provides finite-time, non-asymptotic approximation guarantees. We show that an approximate hypergradient can be computed using only first-order information in tildeO(1) time, leading to an overall complexity of tildeO(δ-1ε-3) for constrained bilevel optimization, which matches the best known rate for non-smooth non-convex optimization. Furthermore, we release an open-source Python library that can be easily adapted from existing solvers. The source code is available at https://github.com/guaguakai/FFOLayer.