# Revisiting Implicit Differentiation for Learning Problems in Optimal Control

arXiv: https://arxiv.org/abs/2310.14468 · HTML: https://mlsanity.com/paper/2310.14468/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [SuperGlue: Learning Feature Matching with Graph Neural Networks](https://mlsanity.com/paper/1911.11763/) (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)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Safe Reinforcement Learning Using Robust MPC](https://mlsanity.com/paper/1906.04005/) (2019)
- [Deep Declarative Networks: A New Hope](https://mlsanity.com/paper/1909.04866/) (2019)
- [Differentiable Dynamic Programming for Structured Prediction and Attention](https://mlsanity.com/paper/1802.03676/) (2018)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [Solving the Blind Perspective-n-Point Problem End-To-End With Robust Differentiable Geometric Optimization](https://mlsanity.com/paper/2007.14628/) (2020)
- [iPlanner: Imperative Path Planning](https://mlsanity.com/paper/2302.11434/) (2023)
- [Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control](https://mlsanity.com/paper/1909.04947/) (2019)
- [Differentiable Gaussian Process Motion Planning](https://mlsanity.com/paper/1907.09591/) (2019)
- [Safe Pontryagin Differentiable Programming](https://mlsanity.com/paper/2105.14937/) (2021)
- [Differentiable Factor Graph Optimization for Learning Smoothers](https://mlsanity.com/paper/2105.08257/) (2021)
- [Generalized Rank Pooling for Activity Recognition](https://mlsanity.com/paper/1704.02112/) (2017)
- [LEO: Learning Energy-based Models in Factor Graph Optimization](https://mlsanity.com/paper/2108.02274/) (2021)
- [Deep Declarative Dynamic Time Warping for End-to-End Learning of Alignment Paths](https://mlsanity.com/paper/2303.10778/) (2023)
- [Inverse Optimal Control from Incomplete Trajectory Observations](https://mlsanity.com/paper/1803.07696/) (2018)
- [A Differentiable Augmented Lagrangian Method for Bilevel Nonlinear Optimization](https://mlsanity.com/paper/1902.03319/) (2019)

## Built on by (most cited first, 8 of 8)

- [DiffTORI: Differentiable Trajectory Optimization for Deep Reinforcement and Imitation Learning](https://mlsanity.com/paper/2402.05421/) (2024)
- [End-to-End Training of High-Dimensional Optimal Control with Implicit Hamiltonians via Jacobian-Free Backpropagation](https://mlsanity.com/paper/2510.00359/) (2025)
- [Shadow Program Inversion with Differentiable Planning: A Framework for Unified Robot Program Parameter and Trajectory Optimization](https://mlsanity.com/paper/2409.08678/) (2024)
- [Learning Set Functions with Implicit Differentiation](https://mlsanity.com/paper/2412.11239/) (2024)
- [Accelerating Optimization via Differentiable Stopping Time](https://mlsanity.com/paper/2505.22509/) (2025)
- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)
- [Neural-Surrogate Co-Design of Aerodynamic Shape and Control](https://mlsanity.com/paper/2603.06760/) (2026)
- [Explaining Process Control Optimisation Recommendations via GradientSHAP and Implicit Differentiation](https://mlsanity.com/paper/2607.14970/) (2026)

## Abstract (arXiv)

This paper proposes a new method for differentiating through optimal trajectories arising from non-convex, constrained discrete-time optimal control (COC) problems using the implicit function theorem (IFT). Previous works solve a differential Karush-Kuhn-Tucker (KKT) system for the trajectory derivative, and achieve this efficiently by solving an auxiliary Linear Quadratic Regulator (LQR) problem. In contrast, we directly evaluate the matrix equations which arise from applying variable elimination on the Lagrange multiplier terms in the (differential) KKT system. By appropriately accounting for the structure of the terms within the resulting equations, we show that the trajectory derivatives scale linearly with the number of timesteps. Furthermore, our approach allows for easy parallelization, significantly improved scalability with model size, direct computation of vector-Jacobian products and improved numerical stability compared to prior works. As an additional contribution, we unify prior works, addressing claims that computing trajectory derivatives using IFT scales quadratically with the number of timesteps. We evaluate our method on a both synthetic benchmark and four challenging, learning from demonstration benchmarks including a 6-DoF maneuvering quadrotor and 6-DoF rocket powered landing.