# LPGD: A General Framework for Backpropagation through Embedded Optimization Layers

arXiv: https://arxiv.org/abs/2407.05920 · HTML: https://mlsanity.com/paper/2407.05920/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 78 references that are in the map.
- First posted on arXiv 8 Jul 2024.

## Builds on (most cited first, 30 of 45)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [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)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles](https://mlsanity.com/paper/1604.07446/) (2016)
- [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)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems](https://mlsanity.com/paper/1911.10092/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers](https://mlsanity.com/paper/2003.11657/) (2020)
- [On Training Implicit Models](https://mlsanity.com/paper/2111.05177/) (2021)
- [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)
- [Deep Equilibrium Optical Flow Estimation](https://mlsanity.com/paper/2204.08442/) (2022)

## Built on by (most cited first, 6 of 6)

- [Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2410.06324/) (2024)
- [Geometric Algorithms for Neural Combinatorial Optimization with Constraints](https://mlsanity.com/paper/2510.24039/) (2025)
- [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025)
- [SoftJAX & SoftTorch: Empowering Automatic Differentiation Libraries with Informative Gradients](https://mlsanity.com/paper/2603.08824/) (2026)
- [RandALO: Out-of-sample risk estimation in no time flat](https://mlsanity.com/paper/2409.09781/) (2024)
- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

Embedding parameterized optimization problems as layers into machine learning architectures serves as a powerful inductive bias. Training such architectures with stochastic gradient descent requires care, as degenerate derivatives of the embedded optimization problem often render the gradients uninformative. We propose Lagrangian Proximal Gradient Descent (LPGD) a flexible framework for training architectures with embedded optimization layers that seamlessly integrates into automatic differentiation libraries. LPGD efficiently computes meaningful replacements of the degenerate optimization layer derivatives by re-running the forward solver oracle on a perturbed input. LPGD captures various previously proposed methods as special cases, while fostering deep links to traditional optimization methods. We theoretically analyze our method and demonstrate on historical and synthetic data that LPGD converges faster than gradient descent even in a differentiable setup.